/*
    Copyright (C) 2020 Alessandro Languasco
*/
 
/****************   	A. LANGUASCO  		******************** 
************* COMPUTATION OF THE min_chi |L'/L| MOD q(PRIME) *******/ 

/*************************  Building the  S approach ****************/

\\ Global variables:

 
global(Deninger_matrix_S_DIF);  
global(log_gamma_matrix);
global(ak_matrix); 
global(tab);   \\global variable used to initialise sumnum
global(phir);
global(phirhalf);
global(pir); 
global(defaultprecision);

 /************* COMPUTATION OF THE min_chi |L'/L| MOD q(PRIME) ******* 
*************                   FOR r1<=q<=r2                        *************
*************          Using the S-function (decimated in frequency)         ********************/

  
{minS_DIF(r1,r2,defaultprecision)=local(minutes, millisec, seconds, rows,
columns, elaptimefinalcomp, elaptimeprecomp, M, derlog, evenresult,oddresult,
qminusone, qminustwo, correctionderlog, logtwopi, qminusonehalf, qminusonehalfminusone,
twoPi, twoPiI, u, conjv, chi, denom, numer, i
);

 
\\ minutes,millisec,seconds: used just to compute the elapsed computation time; local variables

\\ defaultprecision: used to fix the precision used in the computations; global variable


print("************ A. LANGUASCO *************");
print("********* COMPUTATION OF THE MIN |L'/L|**********");
 print("******* FOR q IN ONE INTERVAL ********");
 print("********* with the S-FUNCTION (decimated in frequency)  **********");
 
\\ precision setting

default(realprecision,defaultprecision);       


r1=nextprime(r1);
r2=precprime(r2);
if (r1>r2, print("error: no odd primes in this interval. END PROGRAM");return);
if (r1 < 3, r1=3);
print("Interval=[",r1,",",r2,"]");
phir=r2-1;
phirhalf=phir/2;
pir=primepi(r2)-primepi(r1)+1; 

tab = sumnuminit(); \\ needed for the sumnum functions

gettime(); 
print("Performing precomputations of S(a/q) (decimated in frequency)");
print("and log(Gamma(a/q)) for this interval");

precompS_DIF(r1,r2);

elaptimeprecomp=gettime();

twoPi=2*Pi;
logtwopi=log(twoPi);
twoPiI=twoPi*I;
 

i=0;

gettime();
print("Starting computation of min_q");
forprime(q=r1,r2,
	i+=1; 
	qminusone=q-1; 
	qminustwo=qminusone-1;
	qminusonehalf=qminusone/2;   
	qminusonehalfminusone=qminusonehalf-1;
	correctionderlog= Euler + logtwopi; 
 	
	u=twoPiI/qminusone;  
	
	M=10^(10000);derlog=0;
	\\ even characters
	for (m=1, qminusonehalfminusone, 
					numer=0; 
					denom=0;  
			        \\ v=exp(u*2*m);  \\ j=2m+1; j-1 even (j odd) means an even character
			        conjv=  exp(-u*2*m); \\ conjugate of v
			        chi=1; \\ k=0 ; chi = v^0 
			        for (k=0,qminusonehalfminusone,   \\chi=v^(-k) built via subsequent products
 						numer += chi * Deninger_matrix_S_DIF[i,k+1]; \\ DeningerS(g^k%q/q)	\\ the output is -(S(a/q)+S(1-a/q))						 
 						denom += chi * log_gamma_matrix[i,k+1]; \\log(gamma(g^k%q/q))
 						chi= chi * conjv;
						);
					for (k=qminusonehalf,qminustwo, 		
						denom += chi * log_gamma_matrix[i,k+1]; \\log(gamma(g^k%q/q))
						chi= chi * conjv;
						);
					derlog=abs(correctionderlog+numer/(2*denom));
					if (derlog<M, M=derlog);
			);
	evenresult= M;
	if(evenresult==10^(10000), print("no even characters mod ",q),
	print("min_{chi even}|L'/L(1,chi)|(",q")  = ",   evenresult)); 
	
	M=10^(10000);derlog=0;
	\\ odd characters
	for(m=1, qminusonehalf, 
					numer=0; 
					denom=0;  
					\\ v=exp(u*(2*m-1));  \\ j=2m; j-1 odd (j even) means an odd character  
					conjv=  exp(-u*(2*m-1));\\ conjugate of v
					chi = 1; 	\\ k=0 ; chi = v^0	
					 for (k=0,qminustwo,   \\chi=v^(-k) built via subsequent products
						denom += chi * ak_matrix[i,k+1]; \\ a= g^k%q
						numer += chi * log_gamma_matrix[i,k+1]; \\log(gamma(g^k%q/q)) 
						chi= chi * conjv;
						\\print(numer); print(denom);
	 					); 
					derlog=abs(correctionderlog+q*numer/denom);
					if (derlog<M, M=derlog);
			);
	oddresult=M;
	if(oddresult==10^(10000), print("no odd characters mod ",q),
	print("min_{chi odd}|L'/L(1,chi)|(",q")  = ", oddresult)); 
	print("min_{chi}|L'/L(1,chi)|(",q")  = ", min(oddresult,evenresult)); 
 
);


elaptimefinalcomp=gettime();

seconds=floor(elaptimeprecomp/1000)%60;
minutes=floor(elaptimeprecomp/60000);
millisec=elaptimeprecomp- minutes*60000 - seconds*1000;
\\print(elaptimeprecomp); 
print("Precomputation time: ", minutes, " min, ", seconds, " sec, ", millisec, " millisec");

\\print(elaptimefinalcomp);
seconds=floor(elaptimefinalcomp/1000)%60;
minutes=floor(elaptimefinalcomp/60000);
millisec=elaptimefinalcomp- minutes*60000 - seconds*1000;
 
print("Final step min computation time (output time included): ", minutes, " min, ", seconds, " sec, ", millisec, " millisec");

seconds=floor((elaptimefinalcomp+elaptimeprecomp)/1000)%60;
minutes=floor((elaptimefinalcomp+elaptimeprecomp)/60000);
millisec=(elaptimefinalcomp+elaptimeprecomp)- minutes*60000 - seconds*1000;

print("Total elapsed time: ", minutes, " min, ", seconds, " sec, ", millisec, " millisec");

print("******  END PROGRAM   ********"); 
}
 
 

/*****************   the S-FUNCTION      ****************/

{DeningerS_DIF(x)= local(S,n,ord,eps);  
\\ DECIMATION IN FREQUENCY (with Dilcher formula about log(Gamma_1) )
 
eps=1/100.;
ord=min(x, 1-x); \\rate of decay for the integral formula of S-DIF
if (ord < eps, 		
      \\ if ord is small, uses the series def of S-DIF; otherwise uses the integral formula of S-DIF 
	  \\ using log1p in the second log gives large differences with the previous def; bug ?
	  \\ sqr(x) is a bit faster than x^2
	    \\ then
	  S = sumnum(n=1, -sqr(log1p(x/n)) - sqr(log(1-x/n)) - 2*log(n)*log1p(-sqr(x/n)) , tab) - sqr(log(x)),
	    \\ else
	  S = (-2)*intnum(t=0,[+oo,ord],(-3 + exp(-t) + exp(x*t) + exp((1-x)*t))*(Euler+log(t))/(t*(exp(t)-1)));
	);

\\ the output is -(S(a_k/q) + S(a_{k+m}/q))
return(S)
} 


{DeningerS_DIF_tab(x,tabstandalone)= local(S,n,ord,eps);
\\ DECIMATION IN FREQUENCY (with Dilcher formula about log(Gamma_1) )
 
eps=1/100.;
ord=min(x, 1-x); \\rate of decay for the integral formula of S-DIF
if (ord < eps, 		 
	  \\ if ord is small, uses the series def of S-DIF; otherwise uses the integral formula of S-DIF 
	  \\ using log1p in the second log gives large differences with the previous def; bug ?
	  \\ sqr(x) is a bit faster than x^2
	    \\ then
	  S = sumnum(n=1, -sqr(log1p(x/n)) - sqr(log(1-x/n)) - 2*log(n)*log1p(-sqr(x/n)) , tabstandalone) - sqr(log(x)),
	    \\ else
	  S = (-2)*intnum(t=0,[+oo,ord],(-3 + exp(-t) + exp(x*t) + exp((1-x)*t))*(Euler+log(t))/(t*(exp(t)-1)));
	);

\\ the output is -(S(a_k/q) + S(1 - a_k/q))
return(S)
} 

/*****************   for precomputations    ****************/

{precompS_DIF(r1,r2)= local(rows, columns, i, aoverq, g, g1, a, nhalf,logpi); 

Deninger_matrix_S_DIF=matrix(pir,phirhalf+1,rows,columns,0);

ak_matrix=matrix(pir,phir+1,rows,columns,0); 
log_gamma_matrix=matrix(pir,phir+1,rows,columns,0);

 logpi = log(Pi);
i=0;
forprime(q=r1,r2,
			print("Precomputation for q = ",q);
			g1=znprimroot(q);  \\ needed here to sort the sequences S(a,q) log(Gamma(a/q)) using a=a=g^k%q
			g=lift(g1);  
			i+=1; 
			Deninger_matrix_S_DIF[i,phirhalf+1]=q;		
			ak_matrix[i,phir+1]=q;
            log_gamma_matrix[i,phir+1]=q;
            
			a=1;  
			nhalf= (q-1)/2;
			for(k=0, nhalf-1, aoverq = a/q; \\ already sorted for the final summation 
				log_gamma_matrix[i,k+1] = log(gamma(aoverq));  \\ using lngamma is slower !!
				log_gamma_matrix[i,k+nhalf+1] = -log_gamma_matrix[i,k+1] + logpi - log(sin(Pi* aoverq));		
				Deninger_matrix_S_DIF[i,k+1] = DeningerS_DIF(aoverq);
				ak_matrix[i,k+1]= a;	
				ak_matrix[i,k+nhalf+1]= q-a;	\\ g^(k+m)= g^k(q-1) \equiv - g^k \equiv q-g^k 
				a=(a*g)%q; \\a=g^(k+1)%q
				);

		);

\\print(Deninger_vector_S);
\\return(Deninger_vector_S)

}

 
 
/**** RESULTS ****

 gp2.11.2 and gp2c0.0.11 compiled by myself, see below 

*******************
 gp 2.11.2  on the Dell Optiplex
 
languasc@languasco1:~/Desktop/EK/minderlog$ gp2c-run -pmy_ -g -W  Min-S-final.gp

GP/PARI CALCULATOR Version 2.11.2 (released)
amd64 running linux (x86-64/GMP-6.1.2 kernel) 64-bit version
compiled: May 21 2019, gcc version 7.4.0 (Ubuntu 7.4.0-1ubuntu1~18.04)
threading engine: single
(readline v7.0 enabled, extended help enabled)

Copyright (C) 2000-2018 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and comes 
WITHOUT ANY WARRANTY WHATSOEVER.

Type ? for help, \q to quit.
Type ?17 for how to get moral (and possibly technical) support.

parisize = 8000000, primelimit = 500000

***************  with the S-function  ************
? allocatemem
  ***   Warning: new stack size = 256000000 (244.141 Mbytes).
? init_Min_S_final();minS_DIF(3,1000,30)
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L'/L|**********
******* FOR q IN ONE INTERVAL ********
********* with the S-FUNCTION (decimated in frequency)  **********
Interval=[3,997]
Performing precomputations of S(a/q) (decimated in frequency)
and log(Gamma(a/q)) for this interval 
Starting computation of min_q
no even characters mod 3
min_{chi odd}|L'/L(1,chi)|(3)  = 0.368281615970147842633237904076
min_{chi}|L'/L(1,chi)|(3)  = 0.368281615970147842633237904076
min_{chi even}|L'/L(1,chi)|(5)  = 0.827679476715504887991046989677
min_{chi odd}|L'/L(1,chi)|(5)  = 0.180899098585657908884214228728
min_{chi}|L'/L(1,chi)|(5)  = 0.180899098585657908884214228728
min_{chi even}|L'/L(1,chi)|(7)  = 0.693743252999179022242316373933
min_{chi odd}|L'/L(1,chi)|(7)  = 0.0156356899937203789566227513506
min_{chi}|L'/L(1,chi)|(7)  = 0.0156356899937203789566227513506
min_{chi even}|L'/L(1,chi)|(11)  = 0.519437385739669755636364225908
min_{chi odd}|L'/L(1,chi)|(11)  = 0.0842183042970409250936873839957
min_{chi}|L'/L(1,chi)|(11)  = 0.0842183042970409250936873839957
min_{chi even}|L'/L(1,chi)|(13)  = 0.448347394284393253770799907743
min_{chi odd}|L'/L(1,chi)|(13)  = 0.300105391273262471564455827946
min_{chi}|L'/L(1,chi)|(13)  = 0.300105391273262471564455827946
min_{chi even}|L'/L(1,chi)|(17)  = 0.331091434355506117257156504931
min_{chi odd}|L'/L(1,chi)|(17)  = 0.215168738351581113325995469061
min_{chi}|L'/L(1,chi)|(17)  = 0.215168738351581113325995469061
min_{chi even}|L'/L(1,chi)|(19)  = 0.328509621985788199179915073994
min_{chi odd}|L'/L(1,chi)|(19)  = 0.0849136817877115885069793938261
min_{chi}|L'/L(1,chi)|(19)  = 0.0849136817877115885069793938261
min_{chi even}|L'/L(1,chi)|(23)  = 0.222264564054341426307285821914
min_{chi odd}|L'/L(1,chi)|(23)  = 0.333941126688613216643374977650
min_{chi}|L'/L(1,chi)|(23)  = 0.222264564054341426307285821914
min_{chi even}|L'/L(1,chi)|(29)  = 0.186466418002260383262831736558
min_{chi odd}|L'/L(1,chi)|(29)  = 0.210487287471705555026387870947
min_{chi}|L'/L(1,chi)|(29)  = 0.186466418002260383262831736558
min_{chi even}|L'/L(1,chi)|(31)  = 0.156365159195612900544888732701
min_{chi odd}|L'/L(1,chi)|(31)  = 0.226531400579609328321637507195
min_{chi}|L'/L(1,chi)|(31)  = 0.156365159195612900544888732701
min_{chi even}|L'/L(1,chi)|(37)  = 0.0845822972107739170894042193210
min_{chi odd}|L'/L(1,chi)|(37)  = 0.448804269886543733544163714137
min_{chi}|L'/L(1,chi)|(37)  = 0.0845822972107739170894042193210
min_{chi even}|L'/L(1,chi)|(41)  = 0.0384910485315000734390454511958
min_{chi odd}|L'/L(1,chi)|(41)  = 0.307373737855214954072653837311
min_{chi}|L'/L(1,chi)|(41)  = 0.0384910485315000734390454511958
min_{chi even}|L'/L(1,chi)|(43)  = 0.137995302770293343459953078899
min_{chi odd}|L'/L(1,chi)|(43)  = 0.249482327072494912530578645574
min_{chi}|L'/L(1,chi)|(43)  = 0.137995302770293343459953078899
min_{chi even}|L'/L(1,chi)|(47)  = 0.0357460126243181113240627751991
min_{chi odd}|L'/L(1,chi)|(47)  = 0.421012499637038607090027535291
min_{chi}|L'/L(1,chi)|(47)  = 0.0357460126243181113240627751991
min_{chi even}|L'/L(1,chi)|(53)  = 0.0793454526366054701446266191193
min_{chi odd}|L'/L(1,chi)|(53)  = 0.199585619393762501278007891708
min_{chi}|L'/L(1,chi)|(53)  = 0.0793454526366054701446266191193
min_{chi even}|L'/L(1,chi)|(59)  = 0.0708148084822218033521349700161
min_{chi odd}|L'/L(1,chi)|(59)  = 0.347478043033565586287929330489
min_{chi}|L'/L(1,chi)|(59)  = 0.0708148084822218033521349700161
min_{chi even}|L'/L(1,chi)|(61)  = 0.00442474213920035518177134199910
min_{chi odd}|L'/L(1,chi)|(61)  = 0.529500540871080086694083882215
min_{chi}|L'/L(1,chi)|(61)  = 0.00442474213920035518177134199910
min_{chi even}|L'/L(1,chi)|(67)  = 0.101724238410284799512624760672
min_{chi odd}|L'/L(1,chi)|(67)  = 0.167717594161878463840779005453
min_{chi}|L'/L(1,chi)|(67)  = 0.101724238410284799512624760672
min_{chi even}|L'/L(1,chi)|(71)  = 0.0836770191842498469691851889386
min_{chi odd}|L'/L(1,chi)|(71)  = 0.224126384699479652358168455221
min_{chi}|L'/L(1,chi)|(71)  = 0.0836770191842498469691851889386
min_{chi even}|L'/L(1,chi)|(73)  = 0.0378141956295635250974224780597
min_{chi odd}|L'/L(1,chi)|(73)  = 0.249007239167146280288258256043
min_{chi}|L'/L(1,chi)|(73)  = 0.0378141956295635250974224780597
min_{chi even}|L'/L(1,chi)|(79)  = 0.0667166297023534383411396761343
min_{chi odd}|L'/L(1,chi)|(79)  = 0.284436911248792198104678557001
min_{chi}|L'/L(1,chi)|(79)  = 0.0667166297023534383411396761343
min_{chi even}|L'/L(1,chi)|(83)  = 0.106137806076174129263066902611
min_{chi odd}|L'/L(1,chi)|(83)  = 0.400685071263987184113390425249
min_{chi}|L'/L(1,chi)|(83)  = 0.106137806076174129263066902611
min_{chi even}|L'/L(1,chi)|(89)  = 0.0914541225417151405532456786380
min_{chi odd}|L'/L(1,chi)|(89)  = 0.242419268003889865542346646494
min_{chi}|L'/L(1,chi)|(89)  = 0.0914541225417151405532456786380
min_{chi even}|L'/L(1,chi)|(97)  = 0.100225166851282154186793751382
min_{chi odd}|L'/L(1,chi)|(97)  = 0.181291676360141918232652491889
min_{chi}|L'/L(1,chi)|(97)  = 0.100225166851282154186793751382
min_{chi even}|L'/L(1,chi)|(101)  = 0.0885329550880521674632941004982
min_{chi odd}|L'/L(1,chi)|(101)  = 0.260871974921412198868030017144
min_{chi}|L'/L(1,chi)|(101)  = 0.0885329550880521674632941004982
min_{chi even}|L'/L(1,chi)|(103)  = 0.0670895905177661877619451826530
min_{chi odd}|L'/L(1,chi)|(103)  = 0.154895679992609463505472441731
min_{chi}|L'/L(1,chi)|(103)  = 0.0670895905177661877619451826530
min_{chi even}|L'/L(1,chi)|(107)  = 0.139712944593061380336324862879
min_{chi odd}|L'/L(1,chi)|(107)  = 0.0728423751168310389181649987470
min_{chi}|L'/L(1,chi)|(107)  = 0.0728423751168310389181649987470
min_{chi even}|L'/L(1,chi)|(109)  = 0.0507696923478970293805943698020
min_{chi odd}|L'/L(1,chi)|(109)  = 0.106004437344195864163148574909
min_{chi}|L'/L(1,chi)|(109)  = 0.0507696923478970293805943698020
min_{chi even}|L'/L(1,chi)|(113)  = 0.167205084932062785213399111802
min_{chi odd}|L'/L(1,chi)|(113)  = 0.137803959714882644119589746949
min_{chi}|L'/L(1,chi)|(113)  = 0.137803959714882644119589746949
min_{chi even}|L'/L(1,chi)|(127)  = 0.0407368364724548616418909422615
min_{chi odd}|L'/L(1,chi)|(127)  = 0.225329609079045297177821664287
min_{chi}|L'/L(1,chi)|(127)  = 0.0407368364724548616418909422615
min_{chi even}|L'/L(1,chi)|(131)  = 0.0348399009057285258336111065956
min_{chi odd}|L'/L(1,chi)|(131)  = 0.178222032535568050865847626454
min_{chi}|L'/L(1,chi)|(131)  = 0.0348399009057285258336111065956
min_{chi even}|L'/L(1,chi)|(137)  = 0.173847605183310967783545356651
min_{chi odd}|L'/L(1,chi)|(137)  = 0.191960495680177864998749836966
min_{chi}|L'/L(1,chi)|(137)  = 0.173847605183310967783545356651
min_{chi even}|L'/L(1,chi)|(139)  = 0.147142652999229150227145151427
min_{chi odd}|L'/L(1,chi)|(139)  = 0.0648753769395312623381306959380
min_{chi}|L'/L(1,chi)|(139)  = 0.0648753769395312623381306959380
min_{chi even}|L'/L(1,chi)|(149)  = 0.0155848488200922115251563107109
min_{chi odd}|L'/L(1,chi)|(149)  = 0.349193501682225651186688176493
min_{chi}|L'/L(1,chi)|(149)  = 0.0155848488200922115251563107109
min_{chi even}|L'/L(1,chi)|(151)  = 0.0764919860022148235712866802980
min_{chi odd}|L'/L(1,chi)|(151)  = 0.113736890028550119601116057121
min_{chi}|L'/L(1,chi)|(151)  = 0.0764919860022148235712866802980
min_{chi even}|L'/L(1,chi)|(157)  = 0.0890563130365299239584814564780
min_{chi odd}|L'/L(1,chi)|(157)  = 0.228766933189576696917428039474
min_{chi}|L'/L(1,chi)|(157)  = 0.0890563130365299239584814564780
min_{chi even}|L'/L(1,chi)|(163)  = 0.0511179444167989450784945776361
min_{chi odd}|L'/L(1,chi)|(163)  = 0.00751515379218384362186458320241
min_{chi}|L'/L(1,chi)|(163)  = 0.00751515379218384362186458320241
min_{chi even}|L'/L(1,chi)|(167)  = 0.178152889325652423484065979974
min_{chi odd}|L'/L(1,chi)|(167)  = 0.0314475163524144129547023433163
min_{chi}|L'/L(1,chi)|(167)  = 0.0314475163524144129547023433163
min_{chi even}|L'/L(1,chi)|(173)  = 0.0459014569168100411318884222821
min_{chi odd}|L'/L(1,chi)|(173)  = 0.138432987046720457789015636256
min_{chi}|L'/L(1,chi)|(173)  = 0.0459014569168100411318884222821
min_{chi even}|L'/L(1,chi)|(179)  = 0.160094788150849933458046549985
min_{chi odd}|L'/L(1,chi)|(179)  = 0.0792099176027585599027606987206
min_{chi}|L'/L(1,chi)|(179)  = 0.0792099176027585599027606987206
min_{chi even}|L'/L(1,chi)|(181)  = 0.105259956225068695509834874200
min_{chi odd}|L'/L(1,chi)|(181)  = 0.0839766581363991520612309092188
min_{chi}|L'/L(1,chi)|(181)  = 0.0839766581363991520612309092188
min_{chi even}|L'/L(1,chi)|(191)  = 0.0789308222787531407296850176101
min_{chi odd}|L'/L(1,chi)|(191)  = 0.0908480638248994765247896140318
min_{chi}|L'/L(1,chi)|(191)  = 0.0789308222787531407296850176101
min_{chi even}|L'/L(1,chi)|(193)  = 0.0707040141791806123853699547421
min_{chi odd}|L'/L(1,chi)|(193)  = 0.125200310796495150295943860756
min_{chi}|L'/L(1,chi)|(193)  = 0.0707040141791806123853699547421
min_{chi even}|L'/L(1,chi)|(197)  = 0.0730728299465142920128373077968
min_{chi odd}|L'/L(1,chi)|(197)  = 0.0441995027985420425440248602339
min_{chi}|L'/L(1,chi)|(197)  = 0.0441995027985420425440248602339
min_{chi even}|L'/L(1,chi)|(199)  = 0.126082784303363107778654204938
min_{chi odd}|L'/L(1,chi)|(199)  = 0.251108717956335161241813681777
min_{chi}|L'/L(1,chi)|(199)  = 0.126082784303363107778654204938
min_{chi even}|L'/L(1,chi)|(211)  = 0.0881135264911339824805535145257
min_{chi odd}|L'/L(1,chi)|(211)  = 0.152441426437981620319601314590
min_{chi}|L'/L(1,chi)|(211)  = 0.0881135264911339824805535145257
min_{chi even}|L'/L(1,chi)|(223)  = 0.138902667140733393971537692885
min_{chi odd}|L'/L(1,chi)|(223)  = 0.0578983659206501109813005602793
min_{chi}|L'/L(1,chi)|(223)  = 0.0578983659206501109813005602793
min_{chi even}|L'/L(1,chi)|(227)  = 0.0534596708638658958298574816818
min_{chi odd}|L'/L(1,chi)|(227)  = 0.113248820526700043734404572068
min_{chi}|L'/L(1,chi)|(227)  = 0.0534596708638658958298574816818
min_{chi even}|L'/L(1,chi)|(229)  = 0.0593718950909137857907447819285
min_{chi odd}|L'/L(1,chi)|(229)  = 0.120360678211094937830318288869
min_{chi}|L'/L(1,chi)|(229)  = 0.0593718950909137857907447819285
min_{chi even}|L'/L(1,chi)|(233)  = 0.0583264070453445696593827443199
min_{chi odd}|L'/L(1,chi)|(233)  = 0.0415614908915708656704099689393
min_{chi}|L'/L(1,chi)|(233)  = 0.0415614908915708656704099689393
min_{chi even}|L'/L(1,chi)|(239)  = 0.0658201417803652300140917751233
min_{chi odd}|L'/L(1,chi)|(239)  = 0.126235879085340713587545898438
min_{chi}|L'/L(1,chi)|(239)  = 0.0658201417803652300140917751233
min_{chi even}|L'/L(1,chi)|(241)  = 0.116241207361035151243341970645
min_{chi odd}|L'/L(1,chi)|(241)  = 0.110515313722982307702505567111
min_{chi}|L'/L(1,chi)|(241)  = 0.110515313722982307702505567111
min_{chi even}|L'/L(1,chi)|(251)  = 0.132796526520476545800099262378
min_{chi odd}|L'/L(1,chi)|(251)  = 0.0988693645622321180211978795860
min_{chi}|L'/L(1,chi)|(251)  = 0.0988693645622321180211978795860
min_{chi even}|L'/L(1,chi)|(257)  = 0.0262896149810497935321916653038
min_{chi odd}|L'/L(1,chi)|(257)  = 0.113107731259099124760493884385
min_{chi}|L'/L(1,chi)|(257)  = 0.0262896149810497935321916653038
min_{chi even}|L'/L(1,chi)|(263)  = 0.0478871723232749724670955338105
min_{chi odd}|L'/L(1,chi)|(263)  = 0.210961394880472667916901844087
min_{chi}|L'/L(1,chi)|(263)  = 0.0478871723232749724670955338105
min_{chi even}|L'/L(1,chi)|(269)  = 0.0436427857963467137372788213553
min_{chi odd}|L'/L(1,chi)|(269)  = 0.0619489191182276334654215261932
min_{chi}|L'/L(1,chi)|(269)  = 0.0436427857963467137372788213553
min_{chi even}|L'/L(1,chi)|(271)  = 0.0728159582275104070923470243041
min_{chi odd}|L'/L(1,chi)|(271)  = 0.154172680686767172149993665224
min_{chi}|L'/L(1,chi)|(271)  = 0.0728159582275104070923470243041
min_{chi even}|L'/L(1,chi)|(277)  = 0.0799679700983308521326444578191
min_{chi odd}|L'/L(1,chi)|(277)  = 0.136158690682032503833258642474
min_{chi}|L'/L(1,chi)|(277)  = 0.0799679700983308521326444578191
min_{chi even}|L'/L(1,chi)|(281)  = 0.0895193939492523991206651186617
min_{chi odd}|L'/L(1,chi)|(281)  = 0.0600621685887541570482558244628
min_{chi}|L'/L(1,chi)|(281)  = 0.0600621685887541570482558244628
min_{chi even}|L'/L(1,chi)|(283)  = 0.0358249270068699385555870272088
min_{chi odd}|L'/L(1,chi)|(283)  = 0.00315066809414823334453446070044
min_{chi}|L'/L(1,chi)|(283)  = 0.00315066809414823334453446070044
min_{chi even}|L'/L(1,chi)|(293)  = 0.135103364633442165799200521286
min_{chi odd}|L'/L(1,chi)|(293)  = 0.217191862924918726477613375060
min_{chi}|L'/L(1,chi)|(293)  = 0.135103364633442165799200521286
min_{chi even}|L'/L(1,chi)|(307)  = 0.0886078568196548075127375171815
min_{chi odd}|L'/L(1,chi)|(307)  = 0.0803619248031226099760039226426
min_{chi}|L'/L(1,chi)|(307)  = 0.0803619248031226099760039226426
min_{chi even}|L'/L(1,chi)|(311)  = 0.112087080686900716685580606376
min_{chi odd}|L'/L(1,chi)|(311)  = 0.0208091388398358503756073033607
min_{chi}|L'/L(1,chi)|(311)  = 0.0208091388398358503756073033607
min_{chi even}|L'/L(1,chi)|(313)  = 0.0381524330307066067454338637721
min_{chi odd}|L'/L(1,chi)|(313)  = 0.148092404316728524551307545551
min_{chi}|L'/L(1,chi)|(313)  = 0.0381524330307066067454338637721
min_{chi even}|L'/L(1,chi)|(317)  = 0.120310482217012060618479876460
min_{chi odd}|L'/L(1,chi)|(317)  = 0.138652736885612272747038662875
min_{chi}|L'/L(1,chi)|(317)  = 0.120310482217012060618479876460
min_{chi even}|L'/L(1,chi)|(331)  = 0.0901702204089663732775787193637
min_{chi odd}|L'/L(1,chi)|(331)  = 0.0930083055101366424056377499147
min_{chi}|L'/L(1,chi)|(331)  = 0.0901702204089663732775787193637
min_{chi even}|L'/L(1,chi)|(337)  = 0.0879603589052307662710686590643
min_{chi odd}|L'/L(1,chi)|(337)  = 0.0756197676365422814687006359451
min_{chi}|L'/L(1,chi)|(337)  = 0.0756197676365422814687006359451
min_{chi even}|L'/L(1,chi)|(347)  = 0.0615625279807070291740854076773
min_{chi odd}|L'/L(1,chi)|(347)  = 0.116838139862345965351188817132
min_{chi}|L'/L(1,chi)|(347)  = 0.0615625279807070291740854076773
min_{chi even}|L'/L(1,chi)|(349)  = 0.0246598003990320650094025324427
min_{chi odd}|L'/L(1,chi)|(349)  = 0.150363394560074574613850499767
min_{chi}|L'/L(1,chi)|(349)  = 0.0246598003990320650094025324427
min_{chi even}|L'/L(1,chi)|(353)  = 0.202589799981679949514960505613
min_{chi odd}|L'/L(1,chi)|(353)  = 0.0970606724165674900522082317723
min_{chi}|L'/L(1,chi)|(353)  = 0.0970606724165674900522082317723
min_{chi even}|L'/L(1,chi)|(359)  = 0.163668524725891410859341772072
min_{chi odd}|L'/L(1,chi)|(359)  = 0.159565297851633355627166678429
min_{chi}|L'/L(1,chi)|(359)  = 0.159565297851633355627166678429
min_{chi even}|L'/L(1,chi)|(367)  = 0.120752854906502520642266950975
min_{chi odd}|L'/L(1,chi)|(367)  = 0.146459299782921263699113633565
min_{chi}|L'/L(1,chi)|(367)  = 0.120752854906502520642266950975
min_{chi even}|L'/L(1,chi)|(373)  = 0.0366843739980697037487017563103
min_{chi odd}|L'/L(1,chi)|(373)  = 0.0559668190338237003539474203368
min_{chi}|L'/L(1,chi)|(373)  = 0.0366843739980697037487017563103
min_{chi even}|L'/L(1,chi)|(379)  = 0.0908821064770797858180543818598
min_{chi odd}|L'/L(1,chi)|(379)  = 0.0625127837981578358882932191837
min_{chi}|L'/L(1,chi)|(379)  = 0.0625127837981578358882932191837
min_{chi even}|L'/L(1,chi)|(383)  = 0.0724669290591699014875235814606
min_{chi odd}|L'/L(1,chi)|(383)  = 0.180337598645753736995867393076
min_{chi}|L'/L(1,chi)|(383)  = 0.0724669290591699014875235814606
min_{chi even}|L'/L(1,chi)|(389)  = 0.0359870707732210299498607805889
min_{chi odd}|L'/L(1,chi)|(389)  = 0.0809033163345613421644718963498
min_{chi}|L'/L(1,chi)|(389)  = 0.0359870707732210299498607805889
min_{chi even}|L'/L(1,chi)|(397)  = 0.0272279520150464588427338529174
min_{chi odd}|L'/L(1,chi)|(397)  = 0.0670783958038133657392180846686
min_{chi}|L'/L(1,chi)|(397)  = 0.0272279520150464588427338529174
min_{chi even}|L'/L(1,chi)|(401)  = 0.140496356355224624752955679609
min_{chi odd}|L'/L(1,chi)|(401)  = 0.104399057003481324855914310851
min_{chi}|L'/L(1,chi)|(401)  = 0.104399057003481324855914310851
min_{chi even}|L'/L(1,chi)|(409)  = 0.0461943225163586218848450500915
min_{chi odd}|L'/L(1,chi)|(409)  = 0.118514891820679301534400709586
min_{chi}|L'/L(1,chi)|(409)  = 0.0461943225163586218848450500915
min_{chi even}|L'/L(1,chi)|(419)  = 0.0392262407939465708970825954936
min_{chi odd}|L'/L(1,chi)|(419)  = 0.0358771334266051485411310588993
min_{chi}|L'/L(1,chi)|(419)  = 0.0358771334266051485411310588993
min_{chi even}|L'/L(1,chi)|(421)  = 0.0926420811050822859446356383386
min_{chi odd}|L'/L(1,chi)|(421)  = 0.152712307407457935098062709469
min_{chi}|L'/L(1,chi)|(421)  = 0.0926420811050822859446356383386
min_{chi even}|L'/L(1,chi)|(431)  = 0.0631985869835045176315719835299
min_{chi odd}|L'/L(1,chi)|(431)  = 0.0606108727038685622330922958284
min_{chi}|L'/L(1,chi)|(431)  = 0.0606108727038685622330922958284
min_{chi even}|L'/L(1,chi)|(433)  = 0.0457785295179136540690802694874
min_{chi odd}|L'/L(1,chi)|(433)  = 0.131245620616203266142936294299
min_{chi}|L'/L(1,chi)|(433)  = 0.0457785295179136540690802694874
min_{chi even}|L'/L(1,chi)|(439)  = 0.108439335102642353657462341324
min_{chi odd}|L'/L(1,chi)|(439)  = 0.147023938494258383973185086208
min_{chi}|L'/L(1,chi)|(439)  = 0.108439335102642353657462341324
min_{chi even}|L'/L(1,chi)|(443)  = 0.111339328725617085715177812263
min_{chi odd}|L'/L(1,chi)|(443)  = 0.0349071682700416299413423346821
min_{chi}|L'/L(1,chi)|(443)  = 0.0349071682700416299413423346821
min_{chi even}|L'/L(1,chi)|(449)  = 0.0682360658772905687894082775490
min_{chi odd}|L'/L(1,chi)|(449)  = 0.0949496461256034496349485184886
min_{chi}|L'/L(1,chi)|(449)  = 0.0682360658772905687894082775490
min_{chi even}|L'/L(1,chi)|(457)  = 0.111566405402641079895678728654
min_{chi odd}|L'/L(1,chi)|(457)  = 0.145703776221462050239125998605
min_{chi}|L'/L(1,chi)|(457)  = 0.111566405402641079895678728654
min_{chi even}|L'/L(1,chi)|(461)  = 0.153501927886347063791799161768
min_{chi odd}|L'/L(1,chi)|(461)  = 0.0591455968943122753912953232655
min_{chi}|L'/L(1,chi)|(461)  = 0.0591455968943122753912953232655
min_{chi even}|L'/L(1,chi)|(463)  = 0.0779391295890192267502349036658
min_{chi odd}|L'/L(1,chi)|(463)  = 0.0536411219752542868820265449210
min_{chi}|L'/L(1,chi)|(463)  = 0.0536411219752542868820265449210
min_{chi even}|L'/L(1,chi)|(467)  = 0.0827471742138066919008490700144
min_{chi odd}|L'/L(1,chi)|(467)  = 0.174083373334657724628471806149
min_{chi}|L'/L(1,chi)|(467)  = 0.0827471742138066919008490700144
min_{chi even}|L'/L(1,chi)|(479)  = 0.162816188128040174230952310623
min_{chi odd}|L'/L(1,chi)|(479)  = 0.0531922683431111592528767826083
min_{chi}|L'/L(1,chi)|(479)  = 0.0531922683431111592528767826083
min_{chi even}|L'/L(1,chi)|(487)  = 0.0884614874488209776565402399549
min_{chi odd}|L'/L(1,chi)|(487)  = 0.0720152423097935074305577406746
min_{chi}|L'/L(1,chi)|(487)  = 0.0720152423097935074305577406746
min_{chi even}|L'/L(1,chi)|(491)  = 0.123882790404982921414934803157
min_{chi odd}|L'/L(1,chi)|(491)  = 0.0888594449466553640104256764925
min_{chi}|L'/L(1,chi)|(491)  = 0.0888594449466553640104256764925
min_{chi even}|L'/L(1,chi)|(499)  = 0.0360514825889189527434165024749
min_{chi odd}|L'/L(1,chi)|(499)  = 0.0806732965711188585729574918688
min_{chi}|L'/L(1,chi)|(499)  = 0.0360514825889189527434165024749
min_{chi even}|L'/L(1,chi)|(503)  = 0.121647033819317929681731372456
min_{chi odd}|L'/L(1,chi)|(503)  = 0.101930324454999846629303155693
min_{chi}|L'/L(1,chi)|(503)  = 0.101930324454999846629303155693
min_{chi even}|L'/L(1,chi)|(509)  = 0.0466410171757205564272644665801
min_{chi odd}|L'/L(1,chi)|(509)  = 0.0708196250412458847706153567496
min_{chi}|L'/L(1,chi)|(509)  = 0.0466410171757205564272644665801
min_{chi even}|L'/L(1,chi)|(521)  = 0.0978049019206338163086602894989
min_{chi odd}|L'/L(1,chi)|(521)  = 0.0649687233432153693123937686333
min_{chi}|L'/L(1,chi)|(521)  = 0.0649687233432153693123937686333
min_{chi even}|L'/L(1,chi)|(523)  = 0.0494900199839317719732783479456
min_{chi odd}|L'/L(1,chi)|(523)  = 0.0948953748610086680006180282525
min_{chi}|L'/L(1,chi)|(523)  = 0.0494900199839317719732783479456
min_{chi even}|L'/L(1,chi)|(541)  = 0.103583990568823834134348721897
min_{chi odd}|L'/L(1,chi)|(541)  = 0.0782018025725783017599510222291
min_{chi}|L'/L(1,chi)|(541)  = 0.0782018025725783017599510222291
min_{chi even}|L'/L(1,chi)|(547)  = 0.0954208413819494678362282878956
min_{chi odd}|L'/L(1,chi)|(547)  = 0.0114460726031084518838333520856
min_{chi}|L'/L(1,chi)|(547)  = 0.0114460726031084518838333520856
min_{chi even}|L'/L(1,chi)|(557)  = 0.0326494298637704894970737658164
min_{chi odd}|L'/L(1,chi)|(557)  = 0.121448412083855210947000169944
min_{chi}|L'/L(1,chi)|(557)  = 0.0326494298637704894970737658164
min_{chi even}|L'/L(1,chi)|(563)  = 0.0400127908759677927484679937960
min_{chi odd}|L'/L(1,chi)|(563)  = 0.102420309358750245000502208582
min_{chi}|L'/L(1,chi)|(563)  = 0.0400127908759677927484679937960
min_{chi even}|L'/L(1,chi)|(569)  = 0.117576774970007675000451898870
min_{chi odd}|L'/L(1,chi)|(569)  = 0.0541043180752370669031737315059
min_{chi}|L'/L(1,chi)|(569)  = 0.0541043180752370669031737315059
min_{chi even}|L'/L(1,chi)|(571)  = 0.168236761659457334841723873681
min_{chi odd}|L'/L(1,chi)|(571)  = 0.0988002749261318536425665834177
min_{chi}|L'/L(1,chi)|(571)  = 0.0988002749261318536425665834177
min_{chi even}|L'/L(1,chi)|(577)  = 0.146705277464044953953211684347
min_{chi odd}|L'/L(1,chi)|(577)  = 0.0576162300491405990071397825123
min_{chi}|L'/L(1,chi)|(577)  = 0.0576162300491405990071397825123
min_{chi even}|L'/L(1,chi)|(587)  = 0.112150265731164471581019634313
min_{chi odd}|L'/L(1,chi)|(587)  = 0.0284532211398846524698332677249
min_{chi}|L'/L(1,chi)|(587)  = 0.0284532211398846524698332677249
min_{chi even}|L'/L(1,chi)|(593)  = 0.124651126850235416934289235072
min_{chi odd}|L'/L(1,chi)|(593)  = 0.0406330709723297465927924146507
min_{chi}|L'/L(1,chi)|(593)  = 0.0406330709723297465927924146507
min_{chi even}|L'/L(1,chi)|(599)  = 0.0672677314115971851495524575809
min_{chi odd}|L'/L(1,chi)|(599)  = 0.0408061200005478020017125417510
min_{chi}|L'/L(1,chi)|(599)  = 0.0408061200005478020017125417510
min_{chi even}|L'/L(1,chi)|(601)  = 0.0464795606407484912387546894774
min_{chi odd}|L'/L(1,chi)|(601)  = 0.215015081993123412285671810014
min_{chi}|L'/L(1,chi)|(601)  = 0.0464795606407484912387546894774
min_{chi even}|L'/L(1,chi)|(607)  = 0.0646562627541483888205544142289
min_{chi odd}|L'/L(1,chi)|(607)  = 0.0534313218234699255988846674806
min_{chi}|L'/L(1,chi)|(607)  = 0.0534313218234699255988846674806
min_{chi even}|L'/L(1,chi)|(613)  = 0.0751414314687747258019437688155
min_{chi odd}|L'/L(1,chi)|(613)  = 0.0269793980515011639613098756616
min_{chi}|L'/L(1,chi)|(613)  = 0.0269793980515011639613098756616
min_{chi even}|L'/L(1,chi)|(617)  = 0.0974372880607116942888973393896
min_{chi odd}|L'/L(1,chi)|(617)  = 0.0864556082443851738086323076375
min_{chi}|L'/L(1,chi)|(617)  = 0.0864556082443851738086323076375
min_{chi even}|L'/L(1,chi)|(619)  = 0.0876035194621054927692287021891
min_{chi odd}|L'/L(1,chi)|(619)  = 0.0368090018771068661803359155593
min_{chi}|L'/L(1,chi)|(619)  = 0.0368090018771068661803359155593
min_{chi even}|L'/L(1,chi)|(631)  = 0.0217406558769722478351170561432
min_{chi odd}|L'/L(1,chi)|(631)  = 0.0905594647947913788585778845214
min_{chi}|L'/L(1,chi)|(631)  = 0.0217406558769722478351170561432
min_{chi even}|L'/L(1,chi)|(641)  = 0.0520397827921640186305838750444
min_{chi odd}|L'/L(1,chi)|(641)  = 0.0946126989726791534662581280068
min_{chi}|L'/L(1,chi)|(641)  = 0.0520397827921640186305838750444
min_{chi even}|L'/L(1,chi)|(643)  = 0.0947309579100757960069957978729
min_{chi odd}|L'/L(1,chi)|(643)  = 0.112320099849165376107603843485
min_{chi}|L'/L(1,chi)|(643)  = 0.0947309579100757960069957978729
min_{chi even}|L'/L(1,chi)|(647)  = 0.0480596613492117633035810861515
min_{chi odd}|L'/L(1,chi)|(647)  = 0.0227075323476812053173396489940
min_{chi}|L'/L(1,chi)|(647)  = 0.0227075323476812053173396489940
min_{chi even}|L'/L(1,chi)|(653)  = 0.0109793068790313592199275735187
min_{chi odd}|L'/L(1,chi)|(653)  = 0.0130092523477821547811290857633
min_{chi}|L'/L(1,chi)|(653)  = 0.0109793068790313592199275735187
min_{chi even}|L'/L(1,chi)|(659)  = 0.0673467468805846164031566160699
min_{chi odd}|L'/L(1,chi)|(659)  = 0.0463080155951653860463944675221
min_{chi}|L'/L(1,chi)|(659)  = 0.0463080155951653860463944675221
min_{chi even}|L'/L(1,chi)|(661)  = 0.103916589731771042952256481790
min_{chi odd}|L'/L(1,chi)|(661)  = 0.115535507102578886482404836241
min_{chi}|L'/L(1,chi)|(661)  = 0.103916589731771042952256481790
min_{chi even}|L'/L(1,chi)|(673)  = 0.0497552637760596574836057179794
min_{chi odd}|L'/L(1,chi)|(673)  = 0.0642564306899299716923435626276
min_{chi}|L'/L(1,chi)|(673)  = 0.0497552637760596574836057179794
min_{chi even}|L'/L(1,chi)|(677)  = 0.142755400255780122377155553810
min_{chi odd}|L'/L(1,chi)|(677)  = 0.0576931809100598759672258810057
min_{chi}|L'/L(1,chi)|(677)  = 0.0576931809100598759672258810057
min_{chi even}|L'/L(1,chi)|(683)  = 0.0451975918242860779775765123461
min_{chi odd}|L'/L(1,chi)|(683)  = 0.0982171175583886135301894856473
min_{chi}|L'/L(1,chi)|(683)  = 0.0451975918242860779775765123461
min_{chi even}|L'/L(1,chi)|(691)  = 0.0697441182546344904286995522049
min_{chi odd}|L'/L(1,chi)|(691)  = 0.0273897731331396791316661083337
min_{chi}|L'/L(1,chi)|(691)  = 0.0273897731331396791316661083337
min_{chi even}|L'/L(1,chi)|(701)  = 0.0257920633811751607317354420544
min_{chi odd}|L'/L(1,chi)|(701)  = 0.0524596243039008894818450657435
min_{chi}|L'/L(1,chi)|(701)  = 0.0257920633811751607317354420544
min_{chi even}|L'/L(1,chi)|(709)  = 0.0589441056653326364691691299178
min_{chi odd}|L'/L(1,chi)|(709)  = 0.0348496138007138382959494883397
min_{chi}|L'/L(1,chi)|(709)  = 0.0348496138007138382959494883397
min_{chi even}|L'/L(1,chi)|(719)  = 0.0753918701734695450098300447582
min_{chi odd}|L'/L(1,chi)|(719)  = 0.0320148286993998192803754727459
min_{chi}|L'/L(1,chi)|(719)  = 0.0320148286993998192803754727459
min_{chi even}|L'/L(1,chi)|(727)  = 0.0465486186679150431029073400898
min_{chi odd}|L'/L(1,chi)|(727)  = 0.0569622315219072911363329207941
min_{chi}|L'/L(1,chi)|(727)  = 0.0465486186679150431029073400898
min_{chi even}|L'/L(1,chi)|(733)  = 0.0421398314615777503362090459572
min_{chi odd}|L'/L(1,chi)|(733)  = 0.147284434774134105578737556210
min_{chi}|L'/L(1,chi)|(733)  = 0.0421398314615777503362090459572
min_{chi even}|L'/L(1,chi)|(739)  = 0.0157089578697360124402697243768
min_{chi odd}|L'/L(1,chi)|(739)  = 0.0783526455866317758135699699311
min_{chi}|L'/L(1,chi)|(739)  = 0.0157089578697360124402697243768
min_{chi even}|L'/L(1,chi)|(743)  = 0.156596671609721063459927314644
min_{chi odd}|L'/L(1,chi)|(743)  = 0.116540873617071834980829355749
min_{chi}|L'/L(1,chi)|(743)  = 0.116540873617071834980829355749
min_{chi even}|L'/L(1,chi)|(751)  = 0.164841415030964520948711337362
min_{chi odd}|L'/L(1,chi)|(751)  = 0.152061136399012561767846170124
min_{chi}|L'/L(1,chi)|(751)  = 0.152061136399012561767846170124
min_{chi even}|L'/L(1,chi)|(757)  = 0.0569756816649977327061391530507
min_{chi odd}|L'/L(1,chi)|(757)  = 0.0590584260759905012232238403282
min_{chi}|L'/L(1,chi)|(757)  = 0.0569756816649977327061391530507
min_{chi even}|L'/L(1,chi)|(761)  = 0.0867987092239932912709837046197
min_{chi odd}|L'/L(1,chi)|(761)  = 0.0752610452494586069157710176441
min_{chi}|L'/L(1,chi)|(761)  = 0.0752610452494586069157710176441
min_{chi even}|L'/L(1,chi)|(769)  = 0.0459009157407703403073832277099
min_{chi odd}|L'/L(1,chi)|(769)  = 0.0131531108000294491259390803693
min_{chi}|L'/L(1,chi)|(769)  = 0.0131531108000294491259390803693
min_{chi even}|L'/L(1,chi)|(773)  = 0.0354266675820331759031910687056
min_{chi odd}|L'/L(1,chi)|(773)  = 0.0306513597923025551748529398059
min_{chi}|L'/L(1,chi)|(773)  = 0.0306513597923025551748529398059
min_{chi even}|L'/L(1,chi)|(787)  = 0.102752757847727200331723581045
min_{chi odd}|L'/L(1,chi)|(787)  = 0.0190382442043225184221273346143
min_{chi}|L'/L(1,chi)|(787)  = 0.0190382442043225184221273346143
min_{chi even}|L'/L(1,chi)|(797)  = 0.0854783988682297117191132887568
min_{chi odd}|L'/L(1,chi)|(797)  = 0.0215678917742577236724694542150
min_{chi}|L'/L(1,chi)|(797)  = 0.0215678917742577236724694542150
min_{chi even}|L'/L(1,chi)|(809)  = 0.133059025164725399605246354258
min_{chi odd}|L'/L(1,chi)|(809)  = 0.0637752182255411178465817976085
min_{chi}|L'/L(1,chi)|(809)  = 0.0637752182255411178465817976085
min_{chi even}|L'/L(1,chi)|(811)  = 0.0922264428433055323982128499685
min_{chi odd}|L'/L(1,chi)|(811)  = 0.114293245038664880171077917669
min_{chi}|L'/L(1,chi)|(811)  = 0.0922264428433055323982128499685
min_{chi even}|L'/L(1,chi)|(821)  = 0.0718299182791731584249653115678
min_{chi odd}|L'/L(1,chi)|(821)  = 0.0635669574425181420385993176934
min_{chi}|L'/L(1,chi)|(821)  = 0.0635669574425181420385993176934
min_{chi even}|L'/L(1,chi)|(823)  = 0.0472893707922568985941844901830
min_{chi odd}|L'/L(1,chi)|(823)  = 0.0588329718358916588092467457907
min_{chi}|L'/L(1,chi)|(823)  = 0.0472893707922568985941844901830
min_{chi even}|L'/L(1,chi)|(827)  = 0.128983723826953003895285279725
min_{chi odd}|L'/L(1,chi)|(827)  = 0.0127495972799939813783787670123
min_{chi}|L'/L(1,chi)|(827)  = 0.0127495972799939813783787670123
min_{chi even}|L'/L(1,chi)|(829)  = 0.0677303056027600183718524666199
min_{chi odd}|L'/L(1,chi)|(829)  = 0.0259980315671650451708544855267
min_{chi}|L'/L(1,chi)|(829)  = 0.0259980315671650451708544855267
min_{chi even}|L'/L(1,chi)|(839)  = 0.105301750406131847863250273311
min_{chi odd}|L'/L(1,chi)|(839)  = 0.0844362921098061369937540709097
min_{chi}|L'/L(1,chi)|(839)  = 0.0844362921098061369937540709097
min_{chi even}|L'/L(1,chi)|(853)  = 0.0582876741109979172351053034500
min_{chi odd}|L'/L(1,chi)|(853)  = 0.0944435165305002540489991217024
min_{chi}|L'/L(1,chi)|(853)  = 0.0582876741109979172351053034500
min_{chi even}|L'/L(1,chi)|(857)  = 0.0724250462911102985330239434326
min_{chi odd}|L'/L(1,chi)|(857)  = 0.0805248146573942062698251023591
min_{chi}|L'/L(1,chi)|(857)  = 0.0724250462911102985330239434326
min_{chi even}|L'/L(1,chi)|(859)  = 0.0356788651318577458261510757666
min_{chi odd}|L'/L(1,chi)|(859)  = 0.0693804595880382152965300485216
min_{chi}|L'/L(1,chi)|(859)  = 0.0356788651318577458261510757666
min_{chi even}|L'/L(1,chi)|(863)  = 0.0467701444423371000554024187807
min_{chi odd}|L'/L(1,chi)|(863)  = 0.136809675009194499343039572969
min_{chi}|L'/L(1,chi)|(863)  = 0.0467701444423371000554024187807
min_{chi even}|L'/L(1,chi)|(877)  = 0.0731953404291962924338846914546
min_{chi odd}|L'/L(1,chi)|(877)  = 0.0352938656941859369263980814898
min_{chi}|L'/L(1,chi)|(877)  = 0.0352938656941859369263980814898
min_{chi even}|L'/L(1,chi)|(881)  = 0.0546341834252807424220517429792
min_{chi odd}|L'/L(1,chi)|(881)  = 0.0793005675820382216497042002593
min_{chi}|L'/L(1,chi)|(881)  = 0.0546341834252807424220517429792
min_{chi even}|L'/L(1,chi)|(883)  = 0.0376332428510107502755540252376
min_{chi odd}|L'/L(1,chi)|(883)  = 0.0654173085598632193054525878480
min_{chi}|L'/L(1,chi)|(883)  = 0.0376332428510107502755540252376
min_{chi even}|L'/L(1,chi)|(887)  = 0.0533666284287108655281103826791
min_{chi odd}|L'/L(1,chi)|(887)  = 0.0225542694386274870499259378220
min_{chi}|L'/L(1,chi)|(887)  = 0.0225542694386274870499259378220
min_{chi even}|L'/L(1,chi)|(907)  = 0.0526432954981171403649195649800
min_{chi odd}|L'/L(1,chi)|(907)  = 0.130331388122801885012056037672
min_{chi}|L'/L(1,chi)|(907)  = 0.0526432954981171403649195649800
min_{chi even}|L'/L(1,chi)|(911)  = 0.0780283853610266319847452247298
min_{chi odd}|L'/L(1,chi)|(911)  = 0.0354634927085568205606591488028
min_{chi}|L'/L(1,chi)|(911)  = 0.0354634927085568205606591488028
min_{chi even}|L'/L(1,chi)|(919)  = 0.0620886099809973288758206702800
min_{chi odd}|L'/L(1,chi)|(919)  = 0.0146733603009505601455304221611
min_{chi}|L'/L(1,chi)|(919)  = 0.0146733603009505601455304221611
min_{chi even}|L'/L(1,chi)|(929)  = 0.0304448052051389188239876828455
min_{chi odd}|L'/L(1,chi)|(929)  = 0.0443489932037781248770860129337
min_{chi}|L'/L(1,chi)|(929)  = 0.0304448052051389188239876828455
min_{chi even}|L'/L(1,chi)|(937)  = 0.0360779119311899976537079665819
min_{chi odd}|L'/L(1,chi)|(937)  = 0.0731878433205571532714961159243
min_{chi}|L'/L(1,chi)|(937)  = 0.0360779119311899976537079665819
min_{chi even}|L'/L(1,chi)|(941)  = 0.0111095524628427711753997093064
min_{chi odd}|L'/L(1,chi)|(941)  = 0.0199060400034997447392768047594
min_{chi}|L'/L(1,chi)|(941)  = 0.0111095524628427711753997093064
min_{chi even}|L'/L(1,chi)|(947)  = 0.0744052843549941163477816034746
min_{chi odd}|L'/L(1,chi)|(947)  = 0.0641851342397302622517628190343
min_{chi}|L'/L(1,chi)|(947)  = 0.0641851342397302622517628190343
min_{chi even}|L'/L(1,chi)|(953)  = 0.0469959056062637481036875607395
min_{chi odd}|L'/L(1,chi)|(953)  = 0.106946984984867689208435399037
min_{chi}|L'/L(1,chi)|(953)  = 0.0469959056062637481036875607395
min_{chi even}|L'/L(1,chi)|(967)  = 0.0239814814636624377847695404904
min_{chi odd}|L'/L(1,chi)|(967)  = 0.0151616042539118365582203683886
min_{chi}|L'/L(1,chi)|(967)  = 0.0151616042539118365582203683886
min_{chi even}|L'/L(1,chi)|(971)  = 0.0129741234283356174227901448522
min_{chi odd}|L'/L(1,chi)|(971)  = 0.0288995758676983192911668009446
min_{chi}|L'/L(1,chi)|(971)  = 0.0129741234283356174227901448522
min_{chi even}|L'/L(1,chi)|(977)  = 0.0328649142631077517403009404113
min_{chi odd}|L'/L(1,chi)|(977)  = 0.0167840913394284338065106473100
min_{chi}|L'/L(1,chi)|(977)  = 0.0167840913394284338065106473100
min_{chi even}|L'/L(1,chi)|(983)  = 0.0413963654269892045778034633428
min_{chi odd}|L'/L(1,chi)|(983)  = 0.0463872142628103935272937708318
min_{chi}|L'/L(1,chi)|(983)  = 0.0413963654269892045778034633428
min_{chi even}|L'/L(1,chi)|(991)  = 0.0308265388088868213057090194549
min_{chi odd}|L'/L(1,chi)|(991)  = 0.0553362584398358779468728304319
min_{chi}|L'/L(1,chi)|(991)  = 0.0308265388088868213057090194549
min_{chi even}|L'/L(1,chi)|(997)  = 0.0328976667624738026812133924121
min_{chi odd}|L'/L(1,chi)|(997)  = 0.0335203058622736060383052385565
min_{chi}|L'/L(1,chi)|(997)  = 0.0328976667624738026812133924121
Precomputation time: 2 min, 49 sec, 359 millisec
Final step min computation time (output time included): 0 min, 24 sec, 224 millisec
Total elapsed time: 3 min, 13 sec, 583 millisec
******  END PROGRAM   ********

 
 

 for(n=1,10,m=nextprime(n*10^3);init_Min_S_final();minS_DIF(m,m,30)) 
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L'/L|**********
******* FOR q IN ONE INTERVAL ********
********* with the S-FUNCTION (decimated in frequency)  **********
Interval=[1009,1009]
Performing precomputations of S(a/q) (decimated in frequency)
and log(Gamma(a/q)) for this interval
Precomputation for q = 1009
Starting computation of min_q
min_{chi even}|L'/L(1,chi)|(1009)  = 0.0466929053328856418794771737216
min_{chi odd}|L'/L(1,chi)|(1009)  = 0.0143948045184337273935779897813
min_{chi}|L'/L(1,chi)|(1009)  = 0.0143948045184337273935779897813
Precomputation time: 0 min, 2 sec, 290 millisec
Final step min computation time (output time included): 0 min, 0 sec, 470 millisec
Total elapsed time: 0 min, 2 sec, 760 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L'/L|**********
******* FOR q IN ONE INTERVAL ********
********* with the S-FUNCTION (decimated in frequency)  **********
Interval=[2003,2003]
Performing precomputations of S(a/q) (decimated in frequency)
and log(Gamma(a/q)) for this interval
Precomputation for q = 2003
Starting computation of min_q
min_{chi even}|L'/L(1,chi)|(2003)  = 0.0299939844097628138626483156936
min_{chi odd}|L'/L(1,chi)|(2003)  = 0.0427038547331235967081257427315
min_{chi}|L'/L(1,chi)|(2003)  = 0.0299939844097628138626483156936
Precomputation time: 0 min, 4 sec, 541 millisec
Final step min computation time (output time included): 0 min, 1 sec, 845 millisec
Total elapsed time: 0 min, 6 sec, 386 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L'/L|**********
******* FOR q IN ONE INTERVAL ********
********* with the S-FUNCTION (decimated in frequency)  **********
Interval=[3001,3001]
Performing precomputations of S(a/q) (decimated in frequency)
and log(Gamma(a/q)) for this interval
Precomputation for q = 3001
Starting computation of min_q
min_{chi even}|L'/L(1,chi)|(3001)  = 0.0106802761713300739799807943293
min_{chi odd}|L'/L(1,chi)|(3001)  = 0.0339240899391414707459706650274
min_{chi}|L'/L(1,chi)|(3001)  = 0.0106802761713300739799807943293
Precomputation time: 0 min, 6 sec, 793 millisec
Final step min computation time (output time included): 0 min, 4 sec, 131 millisec
Total elapsed time: 0 min, 10 sec, 924 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L'/L|**********
******* FOR q IN ONE INTERVAL ********
********* with the S-FUNCTION (decimated in frequency)  **********
Interval=[4001,4001]
Performing precomputations of S(a/q) (decimated in frequency)
and log(Gamma(a/q)) for this interval
Precomputation for q = 4001
Starting computation of min_q
min_{chi even}|L'/L(1,chi)|(4001)  = 0.0440942592975271085108036725913
min_{chi odd}|L'/L(1,chi)|(4001)  = 0.0404500031023693486597721563036
min_{chi}|L'/L(1,chi)|(4001)  = 0.0404500031023693486597721563036
Precomputation time: 0 min, 9 sec, 67 millisec
Final step min computation time (output time included): 0 min, 7 sec, 342 millisec
Total elapsed time: 0 min, 16 sec, 409 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L'/L|**********
******* FOR q IN ONE INTERVAL ********
********* with the S-FUNCTION (decimated in frequency)  **********
Interval=[5003,5003]
Performing precomputations of S(a/q) (decimated in frequency)
and log(Gamma(a/q)) for this interval
Precomputation for q = 5003
Starting computation of min_q
min_{chi even}|L'/L(1,chi)|(5003)  = 0.0356286766445124503978086571161
min_{chi odd}|L'/L(1,chi)|(5003)  = 0.0241898666119341396411837034393
min_{chi}|L'/L(1,chi)|(5003)  = 0.0241898666119341396411837034393
Precomputation time: 0 min, 11 sec, 353 millisec
Final step min computation time (output time included): 0 min, 11 sec, 457 millisec
Total elapsed time: 0 min, 22 sec, 810 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L'/L|**********
******* FOR q IN ONE INTERVAL ********
********* with the S-FUNCTION (decimated in frequency)  **********
Interval=[6007,6007]
Performing precomputations of S(a/q) (decimated in frequency)
and log(Gamma(a/q)) for this interval
Precomputation for q = 6007
Starting computation of min_q
min_{chi even}|L'/L(1,chi)|(6007)  = 0.0113173712562458453955968748439
min_{chi odd}|L'/L(1,chi)|(6007)  = 0.0248941429895498403717449684323
min_{chi}|L'/L(1,chi)|(6007)  = 0.0113173712562458453955968748439
Precomputation time: 0 min, 13 sec, 650 millisec
Final step min computation time (output time included): 0 min, 16 sec, 516 millisec
Total elapsed time: 0 min, 30 sec, 166 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L'/L|**********
******* FOR q IN ONE INTERVAL ********
********* with the S-FUNCTION (decimated in frequency)  **********
Interval=[7001,7001]
Performing precomputations of S(a/q) (decimated in frequency)
and log(Gamma(a/q)) for this interval
Precomputation for q = 7001
Starting computation of min_q
min_{chi even}|L'/L(1,chi)|(7001)  = 0.0176783729941881763614743430234
min_{chi odd}|L'/L(1,chi)|(7001)  = 0.0212420475850115424012444850252
min_{chi}|L'/L(1,chi)|(7001)  = 0.0176783729941881763614743430234
Precomputation time: 0 min, 15 sec, 896 millisec
Final step min computation time (output time included): 0 min, 22 sec, 402 millisec
Total elapsed time: 0 min, 38 sec, 298 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L'/L|**********
******* FOR q IN ONE INTERVAL ********
********* with the S-FUNCTION (decimated in frequency)  **********
Interval=[8009,8009]
Performing precomputations of S(a/q) (decimated in frequency)
and log(Gamma(a/q)) for this interval
Precomputation for q = 8009
Starting computation of min_q
min_{chi even}|L'/L(1,chi)|(8009)  = 0.00903089544183206641524077842858
min_{chi odd}|L'/L(1,chi)|(8009)  = 0.0152719528459262775103148027096
min_{chi}|L'/L(1,chi)|(8009)  = 0.00903089544183206641524077842858
Precomputation time: 0 min, 18 sec, 188 millisec
Final step min computation time (output time included): 0 min, 29 sec, 275 millisec
Total elapsed time: 0 min, 47 sec, 463 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L'/L|**********
******* FOR q IN ONE INTERVAL ********
********* with the S-FUNCTION (decimated in frequency)  **********
Interval=[9001,9001]
Performing precomputations of S(a/q) (decimated in frequency)
and log(Gamma(a/q)) for this interval
Precomputation for q = 9001
Starting computation of min_q
min_{chi even}|L'/L(1,chi)|(9001)  = 0.0223609223083204433409695152692
min_{chi odd}|L'/L(1,chi)|(9001)  = 0.00745179632169385357761500719195
min_{chi}|L'/L(1,chi)|(9001)  = 0.00745179632169385357761500719195
Precomputation time: 0 min, 20 sec, 473 millisec
Final step min computation time (output time included): 0 min, 36 sec, 963 millisec
Total elapsed time: 0 min, 57 sec, 436 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L'/L|**********
******* FOR q IN ONE INTERVAL ********
********* with the S-FUNCTION (decimated in frequency)  **********
Interval=[10007,10007]
Performing precomputations of S(a/q) (decimated in frequency)
and log(Gamma(a/q)) for this interval
Precomputation for q = 10007
Starting computation of min_q
min_{chi even}|L'/L(1,chi)|(10007)  = 0.00485626201541881784182602304571
min_{chi odd}|L'/L(1,chi)|(10007)  = 0.0172669707438994967170880831496
min_{chi}|L'/L(1,chi)|(10007)  = 0.00485626201541881784182602304571
Precomputation time: 0 min, 22 sec, 770 millisec
Final step min computation time (output time included): 0 min, 45 sec, 702 millisec
Total elapsed time: 1 min, 8 sec, 472 millisec
******  END PROGRAM   ********
 

******************  larger q's **********************
  
? init_Min_S_final();minS_DIF(20011,20011,30) 
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L'/L|**********
******* FOR q IN ONE INTERVAL ********
********* with the S-FUNCTION (decimated in frequency)  **********
Interval=[20011,20011]
Performing precomputations of S(a/q) (decimated in frequency)
and log(Gamma(a/q)) for this interval
Precomputation for q = 20011
Starting computation of min_q
min_{chi even}|L'/L(1,chi)|(20011)  = 0.0224444665829929753021593429040
min_{chi odd}|L'/L(1,chi)|(20011)  = 0.00307040130610435606977798155311
min_{chi}|L'/L(1,chi)|(20011)  = 0.00307040130610435606977798155311
Precomputation time: 0 min, 38 sec, 521 millisec
Final step min computation time (output time included): 2 min, 35 sec, 219 millisec
Total elapsed time: 3 min, 13 sec, 740 millisec
******  END PROGRAM   ********

 
? init_Min_S_final();minS_DIF(30011,30011,30)
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L'/L|**********
******* FOR q IN ONE INTERVAL ********
********* with the S-FUNCTION (decimated in frequency)  **********
Interval=[30011,30011]
Performing precomputations of S(a/q) (decimated in frequency)
and log(Gamma(a/q)) for this interval
Precomputation for q = 30011
Starting computation of min_q
min_{chi even}|L'/L(1,chi)|(30011)  = 0.00935666437359357851811817455257
min_{chi odd}|L'/L(1,chi)|(30011)  = 0.00744563949391952495165193581152
min_{chi}|L'/L(1,chi)|(30011)  = 0.00744563949391952495165193581152
Precomputation time: 0 min, 57 sec, 881 millisec
Final step min computation time (output time included): 5 min, 49 sec, 586 millisec
Total elapsed time: 6 min, 47 sec, 467 millisec
******  END PROGRAM   ********
 
 
? init_Min_S_final();minS_DIF(83089,83089,10)
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L'/L|**********
******* FOR q IN ONE INTERVAL ********
********* with the S-FUNCTION (decimated in frequency)  **********
Interval=[83089,83089]
Performing precomputations of S(a/q) (decimated in frequency)
and log(Gamma(a/q)) for this interval
Precomputation for q = 83089
Starting computation of min_q
min_{chi even}|L'/L(1,chi)|(83089)  = 0.01023439905
min_{chi odd}|L'/L(1,chi)|(83089)  = 0.006953552264
min_{chi}|L'/L(1,chi)|(83089)  = 0.006953552264
Precomputation time: 1 min, 16 sec, 115 millisec
Final step min computation time (output time included): 34 min, 18 sec, 481 millisec
Total elapsed time: 35 min, 34 sec, 596 millisec
******  END PROGRAM   ********



------------------------------------------------------
Recomputed for false zeros
  
 ? init_Min_S_final();minS_DIF(89381,89381,10)
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L'/L|**********
******* FOR q IN ONE INTERVAL ********
********* with the S-FUNCTION (decimated in frequency)  **********
Interval=[89381,89381]
Performing precomputations of S(a/q) (decimated in frequency)
and log(Gamma(a/q)) for this interval
Precomputation for q = 89381
Starting computation of min_q
min_{chi even}|L'/L(1,chi)|(89381)  = 0.002770295422
min_{chi odd}|L'/L(1,chi)|(89381)  = 0.002769765011
min_{chi}|L'/L(1,chi)|(89381)  = 0.002769765011
Precomputation time: 1 min, 21 sec, 538 millisec
Final step min computation time (output time included): 40 min, 55 sec, 353 millisec
Total elapsed time: 42 min, 16 sec, 891 millisec
******  END PROGRAM   ********

 


*******/