/*
    Copyright (C) 2019 Alessandro Languasco
*/

/****************   	A. LANGUASCO  		******************** 
************* COMPUTATION OF THE max_chi |L(1,chi)| MOD q(PRIME) *******/ 

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

\\ Global variables:

  
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(1,chi)| MOD q(PRIME) ******* 
*************                   FOR r1<=q<=r2                        *************
*************          Using the loggamma-function       ********************/

  
{minL_logGamma(r1,r2,defaultprecision)=local(minutes, millisec, seconds, rows,
columns, elaptimefinalcomp, elaptimeprecomp, M, Lvalue, evenresult,oddresult,
qminusone, qminustwo, correctionLvalue, logtwopi, qminusonehalf, qminusonehalfminusone,
twoPi, twoPiI, u, conjv, chi, aux, 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(1,chi)|**********");
 print("******* FOR q IN ONE INTERVAL ********");
 print("********* with the logGamma-FUNCTION   **********");
 
\\ 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; 


gettime(); 
print("Performing precomputations of log(Gamma(a/q)");
print(" for this interval");

precomplogGamma(r1,r2);

elaptimeprecomp=gettime();

twoPi=2*Pi;
logtwopi=log(twoPi);
twoPiI=twoPi*I;
 
MinLfile = fileopen("minL-values.tex", "w");
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;
	correctionLvalue= 2/sqrt(q);
 	
	u=twoPiI/qminusone;  
	
	M=10^(10000);Lvalue=0;
	\\ even characters
	for (m=1, qminusonehalfminusone, 
					numer=0; 
					aux=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,qminustwo,   \\chi=v^(-k) built via subsequent products 					 
 						aux += chi * log_gamma_matrix[i,k+1]; \\log(gamma(g^k%q/q))
 						chi= chi * conjv;
						); 
					Lvalue=abs(correctionLvalue*aux);
					if (Lvalue<M, M=Lvalue);
			);
	evenresult= M;
	\\if(evenresult==10^(10000), print("no even characters mod ",q),
	\\print("min_{chi even}|L(1,chi)|(",q")  = ",   evenresult)); 
	
	M=10^(10000);Lvalue=0;
	\\ odd characters
	correctionLvalue= Pi/(q^(1.5)); 
	for(m=1, qminusonehalf, 
					numer=0; 
					aux=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
						aux += chi * ak_matrix[i,k+1]; \\ a= g^k%q
						chi= chi * conjv;
						\\print(numer); print(aux);
	 					); 
					Lvalue=abs(correctionLvalue*aux);
					if (Lvalue<M, M=Lvalue);
			);
	oddresult=M;
	\\if(oddresult==10^(10000), print("no odd characters mod ",q),
	\\print("min_{chi odd}|L(1,chi)|(",q")  = ", oddresult)
	\\); 
	\\print("min_{chi}|L(1,chi)|(",q")  = ", min(oddresult,evenresult)); 
	
	filewrite1(MinLfile, "$");
	filewrite1(MinLfile,q);
	filewrite1(MinLfile,"$ & $");
	filewrite1(MinLfile, min(oddresult,evenresult));
	filewrite(MinLfile,"\\dotsc$ \\\\");

	print("$",q,"$ & $", min(oddresult,evenresult), "\\dotsc$ \\\\");
 
);

fileclose(MinLfile);  
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 max 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   ********"); 
}
 
 
 

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

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

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;  
			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));	 
				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
				);

		);
 

}

 
 
/**** RESULTS ****

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

*******************
 gp 2.11.2  on macmini
 
languasc-macmini:gp_scripts languasc$ gp2c-run -pmy_ -g -W  MinL-loggamma-final.gp

Reading GPRC: /Users/languasc/.gprc ...Done.

GP/PARI CALCULATOR Version 2.11.2 (released)
i386 running darwin (x86-64/GMP-6.1.2 kernel) 64-bit version
compiled: Aug  3 2019, Apple LLVM version 10.0.1 (clang-1001.0.46.4)
threading engine: single
(readline v8.0 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.

parisizemax = 400003072, primelimit = 500000

? init_MinL_loggamma_final();minL_logGamma(3,1000,30)
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L(1,chi)|**********
******* FOR q IN ONE INTERVAL ********
********* with the logGamma-FUNCTION   **********
Interval=[3,997]
Performing precomputations of log(Gamma(a/q)
 for this interval
 
Starting computation of min_q
no even characters mod 3
min_{chi odd}|L(1,chi)|(3)  = 0.604599788078072616864692752547
min_{chi}|L(1,chi)|(3)  = 0.604599788078072616864692752547
min_{chi even}|L(1,chi)|(5)  = 0.430408940964004038889433232951
min_{chi odd}|L(1,chi)|(5)  = 0.888576587631673249403176198012
min_{chi}|L(1,chi)|(5)  = 0.430408940964004038889433232951
min_{chi even}|L(1,chi)|(7)  = 0.547959686797993973084485988763
min_{chi odd}|L(1,chi)|(7)  = 0.897597901025655210989326680937
min_{chi}|L(1,chi)|(7)  = 0.547959686797993973084485988763
min_{chi even}|L(1,chi)|(11)  = 0.618351934876807874060419662662
min_{chi odd}|L(1,chi)|(11)  = 0.758626668111486078431089616627
min_{chi}|L(1,chi)|(11)  = 0.618351934876807874060419662662
min_{chi even}|L(1,chi)|(13)  = 0.648085907789387629121568128662
min_{chi odd}|L(1,chi)|(13)  = 0.598987497945465758207499250242
min_{chi}|L(1,chi)|(13)  = 0.598987497945465758207499250242
min_{chi even}|L(1,chi)|(17)  = 0.623600052291150279799369494609
min_{chi odd}|L(1,chi)|(17)  = 0.453546340908733659287599108349
min_{chi}|L(1,chi)|(17)  = 0.453546340908733659287599108349
min_{chi even}|L(1,chi)|(19)  = 0.641077496999726757619728146767
min_{chi odd}|L(1,chi)|(19)  = 0.413193436540565451244291268589
min_{chi}|L(1,chi)|(19)  = 0.413193436540565451244291268589
min_{chi even}|L(1,chi)|(23)  = 0.552304916713058385866569568830
min_{chi odd}|L(1,chi)|(23)  = 0.572952712849875065845465007087
min_{chi}|L(1,chi)|(23)  = 0.552304916713058385866569568830
min_{chi even}|L(1,chi)|(29)  = 0.597794677224574154950308072386
min_{chi odd}|L(1,chi)|(29)  = 0.451093787499735920935796126723
min_{chi}|L(1,chi)|(29)  = 0.451093787499735920935796126723
min_{chi even}|L(1,chi)|(31)  = 0.599459567825350227295820629918
min_{chi odd}|L(1,chi)|(31)  = 0.440122223433962808617040155019
min_{chi}|L(1,chi)|(31)  = 0.440122223433962808617040155019
min_{chi even}|L(1,chi)|(37)  = 0.547762092618530407475448933411
min_{chi odd}|L(1,chi)|(37)  = 0.420012687371836528710987165573
min_{chi}|L(1,chi)|(37)  = 0.420012687371836528710987165573
min_{chi even}|L(1,chi)|(41)  = 0.563249976119887917298592261027
min_{chi odd}|L(1,chi)|(41)  = 0.531058786094205975539953409040
min_{chi}|L(1,chi)|(41)  = 0.531058786094205975539953409040
min_{chi even}|L(1,chi)|(43)  = 0.480143280802223299192745364975
min_{chi odd}|L(1,chi)|(43)  = 0.479088388239857211764493892920
min_{chi}|L(1,chi)|(43)  = 0.479088388239857211764493892920
min_{chi even}|L(1,chi)|(47)  = 0.367129807516487530311860512131
min_{chi odd}|L(1,chi)|(47)  = 0.486605568012058360780792284723
min_{chi}|L(1,chi)|(47)  = 0.367129807516487530311860512131
min_{chi even}|L(1,chi)|(53)  = 0.540024945102558214767618788153
min_{chi odd}|L(1,chi)|(53)  = 0.413967522107356708589184273274
min_{chi}|L(1,chi)|(53)  = 0.413967522107356708589184273274
min_{chi even}|L(1,chi)|(59)  = 0.332420580251333196195200170353
min_{chi odd}|L(1,chi)|(59)  = 0.386580311492533923974376850483
min_{chi}|L(1,chi)|(59)  = 0.332420580251333196195200170353
min_{chi even}|L(1,chi)|(61)  = 0.365767556044524607545195023959
min_{chi odd}|L(1,chi)|(61)  = 0.416429202365449470740481401595
min_{chi}|L(1,chi)|(61)  = 0.365767556044524607545195023959
min_{chi even}|L(1,chi)|(67)  = 0.413857743415213923670419930982
min_{chi odd}|L(1,chi)|(67)  = 0.383806628882915516388036615854
min_{chi}|L(1,chi)|(67)  = 0.383806628882915516388036615854
min_{chi even}|L(1,chi)|(71)  = 0.445439715279504151396753278541
min_{chi odd}|L(1,chi)|(71)  = 0.468043545110666513379704866504
min_{chi}|L(1,chi)|(71)  = 0.445439715279504151396753278541
min_{chi even}|L(1,chi)|(73)  = 0.332816572422086433683238786444
min_{chi odd}|L(1,chi)|(73)  = 0.455547911543412560405190037579
min_{chi}|L(1,chi)|(73)  = 0.332816572422086433683238786444
min_{chi even}|L(1,chi)|(79)  = 0.487699074808110616078499610657
min_{chi odd}|L(1,chi)|(79)  = 0.428163851610403317805757592040
min_{chi}|L(1,chi)|(79)  = 0.428163851610403317805757592040
min_{chi even}|L(1,chi)|(83)  = 0.462387538865549865563618907321
min_{chi odd}|L(1,chi)|(83)  = 0.480192178221007036621752328332
min_{chi}|L(1,chi)|(83)  = 0.462387538865549865563618907321
min_{chi even}|L(1,chi)|(89)  = 0.412201340454147183311378976779
min_{chi odd}|L(1,chi)|(89)  = 0.389534102336872450885689912904
min_{chi}|L(1,chi)|(89)  = 0.389534102336872450885689912904
min_{chi even}|L(1,chi)|(97)  = 0.459789127276729866953178410104
min_{chi odd}|L(1,chi)|(97)  = 0.389092591237496211100095150861
min_{chi}|L(1,chi)|(97)  = 0.389092591237496211100095150861
min_{chi even}|L(1,chi)|(101)  = 0.356508197108894602900482691212
min_{chi odd}|L(1,chi)|(101)  = 0.536804134316850168768311103306
min_{chi}|L(1,chi)|(101)  = 0.356508197108894602900482691212
min_{chi even}|L(1,chi)|(103)  = 0.397853727338476109339913228255
min_{chi odd}|L(1,chi)|(103)  = 0.410165188628622667426673564749
min_{chi}|L(1,chi)|(103)  = 0.397853727338476109339913228255
min_{chi even}|L(1,chi)|(107)  = 0.404936054291382959432040324919
min_{chi odd}|L(1,chi)|(107)  = 0.416271112498618418911924355121
min_{chi}|L(1,chi)|(107)  = 0.404936054291382959432040324919
min_{chi even}|L(1,chi)|(109)  = 0.437942799378047241735313240686
min_{chi odd}|L(1,chi)|(109)  = 0.395439291752596225073889921584
min_{chi}|L(1,chi)|(109)  = 0.395439291752596225073889921584
min_{chi even}|L(1,chi)|(113)  = 0.430408362362510906005195624421
min_{chi odd}|L(1,chi)|(113)  = 0.429318876983891967381982481536
min_{chi}|L(1,chi)|(113)  = 0.429318876983891967381982481536
min_{chi even}|L(1,chi)|(127)  = 0.370583958563266768017674320198
min_{chi odd}|L(1,chi)|(127)  = 0.427575025368673386789992839487
min_{chi}|L(1,chi)|(127)  = 0.370583958563266768017674320198
min_{chi even}|L(1,chi)|(131)  = 0.461080734046890989699680441649
min_{chi odd}|L(1,chi)|(131)  = 0.352086298168602566529507073368
min_{chi}|L(1,chi)|(131)  = 0.352086298168602566529507073368
min_{chi even}|L(1,chi)|(137)  = 0.417505824794320762685433840755
min_{chi odd}|L(1,chi)|(137)  = 0.418212229052647323642619498345
min_{chi}|L(1,chi)|(137)  = 0.417505824794320762685433840755
min_{chi even}|L(1,chi)|(139)  = 0.416862458745916605478311229554
min_{chi odd}|L(1,chi)|(139)  = 0.462011941714540530203350276388
min_{chi}|L(1,chi)|(139)  = 0.416862458745916605478311229554
min_{chi even}|L(1,chi)|(149)  = 0.383901386511619291505439140039
min_{chi odd}|L(1,chi)|(149)  = 0.414024040950331421945878886347
min_{chi}|L(1,chi)|(149)  = 0.383901386511619291505439140039
min_{chi even}|L(1,chi)|(151)  = 0.375982979158935926530947639675
min_{chi odd}|L(1,chi)|(151)  = 0.414277700043992838787973954031
min_{chi}|L(1,chi)|(151)  = 0.375982979158935926530947639675
min_{chi even}|L(1,chi)|(157)  = 0.396626546050889600653705298913
min_{chi odd}|L(1,chi)|(157)  = 0.369949970311229129154525082455
min_{chi}|L(1,chi)|(157)  = 0.369949970311229129154525082455
min_{chi even}|L(1,chi)|(163)  = 0.454409867055008666526111163161
min_{chi odd}|L(1,chi)|(163)  = 0.246068527552960243897853273760
min_{chi}|L(1,chi)|(163)  = 0.246068527552960243897853273760
min_{chi even}|L(1,chi)|(167)  = 0.350631724723697517489360519141
min_{chi odd}|L(1,chi)|(167)  = 0.435597948784802671082827357539
min_{chi}|L(1,chi)|(167)  = 0.350631724723697517489360519141
min_{chi even}|L(1,chi)|(173)  = 0.390910841189933659394315522450
min_{chi odd}|L(1,chi)|(173)  = 0.382663127794428634867406502671
min_{chi}|L(1,chi)|(173)  = 0.382663127794428634867406502671
min_{chi even}|L(1,chi)|(179)  = 0.368730092513171929365571640440
min_{chi odd}|L(1,chi)|(179)  = 0.341096922920489890524209012663
min_{chi}|L(1,chi)|(179)  = 0.341096922920489890524209012663
min_{chi even}|L(1,chi)|(181)  = 0.421643886059876656322640343183
min_{chi odd}|L(1,chi)|(181)  = 0.400130740444375343545002983965
min_{chi}|L(1,chi)|(181)  = 0.400130740444375343545002983965
min_{chi even}|L(1,chi)|(191)  = 0.369946585080866896719707254253
min_{chi odd}|L(1,chi)|(191)  = 0.384239384835200745116317694728
min_{chi}|L(1,chi)|(191)  = 0.369946585080866896719707254253
min_{chi even}|L(1,chi)|(193)  = 0.362378327133943195753957944776
min_{chi odd}|L(1,chi)|(193)  = 0.413218300137075816577402977006
min_{chi}|L(1,chi)|(193)  = 0.362378327133943195753957944776
min_{chi even}|L(1,chi)|(197)  = 0.373837653864628774889214137854
min_{chi odd}|L(1,chi)|(197)  = 0.392927474574867768422548064651
min_{chi}|L(1,chi)|(197)  = 0.373837653864628774889214137854
min_{chi even}|L(1,chi)|(199)  = 0.407431952492485762271173360682
min_{chi odd}|L(1,chi)|(199)  = 0.414669940662630997056285309864
min_{chi}|L(1,chi)|(199)  = 0.407431952492485762271173360682
min_{chi even}|L(1,chi)|(211)  = 0.389585855312613155930182619225
min_{chi odd}|L(1,chi)|(211)  = 0.352658667943486049139179190376
min_{chi}|L(1,chi)|(211)  = 0.352658667943486049139179190376
min_{chi even}|L(1,chi)|(223)  = 0.442341994813507335744506695744
min_{chi odd}|L(1,chi)|(223)  = 0.430673753410189075458236898964
min_{chi}|L(1,chi)|(223)  = 0.430673753410189075458236898964
min_{chi even}|L(1,chi)|(227)  = 0.411608321651195443923787559820
min_{chi odd}|L(1,chi)|(227)  = 0.381439782634825445136263631484
min_{chi}|L(1,chi)|(227)  = 0.381439782634825445136263631484
min_{chi even}|L(1,chi)|(229)  = 0.414131358265927305561673753257
min_{chi odd}|L(1,chi)|(229)  = 0.383729157778237776053663569946
min_{chi}|L(1,chi)|(229)  = 0.383729157778237776053663569946
min_{chi even}|L(1,chi)|(233)  = 0.397286498219736148468940688107
min_{chi odd}|L(1,chi)|(233)  = 0.365137320866318953249895485603
min_{chi}|L(1,chi)|(233)  = 0.365137320866318953249895485603
min_{chi even}|L(1,chi)|(239)  = 0.392821720721540457023716162704
min_{chi odd}|L(1,chi)|(239)  = 0.361752701714412120902910350331
min_{chi}|L(1,chi)|(239)  = 0.361752701714412120902910350331
min_{chi even}|L(1,chi)|(241)  = 0.352065536871065140512991463133
min_{chi odd}|L(1,chi)|(241)  = 0.409434233741757049545770376333
min_{chi}|L(1,chi)|(241)  = 0.352065536871065140512991463133
min_{chi even}|L(1,chi)|(251)  = 0.443073955328902500235378328986
min_{chi odd}|L(1,chi)|(251)  = 0.423961989644231760650605953707
min_{chi}|L(1,chi)|(251)  = 0.423961989644231760650605953707
min_{chi even}|L(1,chi)|(257)  = 0.388887492169207487425676370870
min_{chi odd}|L(1,chi)|(257)  = 0.452873208121726387559340232721
min_{chi}|L(1,chi)|(257)  = 0.388887492169207487425676370870
min_{chi even}|L(1,chi)|(263)  = 0.348202135404538171971904967311
min_{chi odd}|L(1,chi)|(263)  = 0.404805016930669018047907785832
min_{chi}|L(1,chi)|(263)  = 0.348202135404538171971904967311
min_{chi even}|L(1,chi)|(269)  = 0.404844287635172668787118855557
min_{chi odd}|L(1,chi)|(269)  = 0.408153134620960925251979949491
min_{chi}|L(1,chi)|(269)  = 0.404844287635172668787118855557
min_{chi even}|L(1,chi)|(271)  = 0.388417908172449738891567966036
min_{chi odd}|L(1,chi)|(271)  = 0.381676325868210243355238004400
min_{chi}|L(1,chi)|(271)  = 0.381676325868210243355238004400
min_{chi even}|L(1,chi)|(277)  = 0.367772078360884977669216925848
min_{chi odd}|L(1,chi)|(277)  = 0.386690194712454166908695602258
min_{chi}|L(1,chi)|(277)  = 0.367772078360884977669216925848
min_{chi even}|L(1,chi)|(281)  = 0.394911093147797458730423538938
min_{chi odd}|L(1,chi)|(281)  = 0.367654214871298109838839266841
min_{chi}|L(1,chi)|(281)  = 0.367654214871298109838839266841
min_{chi even}|L(1,chi)|(283)  = 0.360060056162870590740646688749
min_{chi odd}|L(1,chi)|(283)  = 0.379532431367231148721509445012
min_{chi}|L(1,chi)|(283)  = 0.360060056162870590740646688749
min_{chi even}|L(1,chi)|(293)  = 0.331438394291933793807551984375
min_{chi odd}|L(1,chi)|(293)  = 0.405431392453914998787434891820
min_{chi}|L(1,chi)|(293)  = 0.331438394291933793807551984375
min_{chi even}|L(1,chi)|(307)  = 0.377587985636141516739672519455
min_{chi odd}|L(1,chi)|(307)  = 0.338116784161146375199978675010
min_{chi}|L(1,chi)|(307)  = 0.338116784161146375199978675010
min_{chi even}|L(1,chi)|(311)  = 0.367525774725253937340225560338
min_{chi odd}|L(1,chi)|(311)  = 0.364786872431841488117413342623
min_{chi}|L(1,chi)|(311)  = 0.364786872431841488117413342623
min_{chi even}|L(1,chi)|(313)  = 0.391360680749734937346541615066
min_{chi odd}|L(1,chi)|(313)  = 0.348283081331744875759624637812
min_{chi}|L(1,chi)|(313)  = 0.348283081331744875759624637812
min_{chi even}|L(1,chi)|(317)  = 0.353920323300651284267391675261
min_{chi odd}|L(1,chi)|(317)  = 0.367792154804458735451066273817
min_{chi}|L(1,chi)|(317)  = 0.353920323300651284267391675261
min_{chi even}|L(1,chi)|(331)  = 0.384662955943923874879364794182
min_{chi odd}|L(1,chi)|(331)  = 0.408312325379740293829711494845
min_{chi}|L(1,chi)|(331)  = 0.384662955943923874879364794182
min_{chi even}|L(1,chi)|(337)  = 0.438966054265195649056806986086
min_{chi odd}|L(1,chi)|(337)  = 0.395558761496812443683554845556
min_{chi}|L(1,chi)|(337)  = 0.395558761496812443683554845556
min_{chi even}|L(1,chi)|(347)  = 0.377945303983869233549712938160
min_{chi odd}|L(1,chi)|(347)  = 0.388837745736212493762584658523
min_{chi}|L(1,chi)|(347)  = 0.377945303983869233549712938160
min_{chi even}|L(1,chi)|(349)  = 0.361133254391610034773517750960
min_{chi odd}|L(1,chi)|(349)  = 0.346392157694090105389329641835
min_{chi}|L(1,chi)|(349)  = 0.346392157694090105389329641835
min_{chi even}|L(1,chi)|(353)  = 0.392050917804241245253197847943
min_{chi odd}|L(1,chi)|(353)  = 0.395808872295397123068894475046
min_{chi}|L(1,chi)|(353)  = 0.392050917804241245253197847943
min_{chi even}|L(1,chi)|(359)  = 0.372107247747624127090483414376
min_{chi odd}|L(1,chi)|(359)  = 0.383034197073906207735408984777
min_{chi}|L(1,chi)|(359)  = 0.372107247747624127090483414376
min_{chi even}|L(1,chi)|(367)  = 0.395093008663268448130153769232
min_{chi odd}|L(1,chi)|(367)  = 0.292520324658371008714768864693
min_{chi}|L(1,chi)|(367)  = 0.292520324658371008714768864693
min_{chi even}|L(1,chi)|(373)  = 0.412449831490017072211483595160
min_{chi odd}|L(1,chi)|(373)  = 0.365107572487765521944814562236
min_{chi}|L(1,chi)|(373)  = 0.365107572487765521944814562236
min_{chi even}|L(1,chi)|(379)  = 0.399517304428683104760905282923
min_{chi odd}|L(1,chi)|(379)  = 0.344529506663803122648629099168
min_{chi}|L(1,chi)|(379)  = 0.344529506663803122648629099168
min_{chi even}|L(1,chi)|(383)  = 0.369414454892205779888789721588
min_{chi odd}|L(1,chi)|(383)  = 0.373013785763289172794059952276
min_{chi}|L(1,chi)|(383)  = 0.369414454892205779888789721588
min_{chi even}|L(1,chi)|(389)  = 0.334596609797765346511075221154
min_{chi odd}|L(1,chi)|(389)  = 0.309685712040159744098308002254
min_{chi}|L(1,chi)|(389)  = 0.309685712040159744098308002254
min_{chi even}|L(1,chi)|(397)  = 0.345438689174522431354681542746
min_{chi odd}|L(1,chi)|(397)  = 0.366364359268040406221466699242
min_{chi}|L(1,chi)|(397)  = 0.345438689174522431354681542746
min_{chi even}|L(1,chi)|(401)  = 0.374008182395715337562657042856
min_{chi odd}|L(1,chi)|(401)  = 0.382072635349745452663988745319
min_{chi}|L(1,chi)|(401)  = 0.374008182395715337562657042856
min_{chi even}|L(1,chi)|(409)  = 0.410228555255367861167353389885
min_{chi odd}|L(1,chi)|(409)  = 0.343912321305157763624260753609
min_{chi}|L(1,chi)|(409)  = 0.343912321305157763624260753609
min_{chi even}|L(1,chi)|(419)  = 0.415228071025741054175102379753
min_{chi odd}|L(1,chi)|(419)  = 0.398860743207736694380625064238
min_{chi}|L(1,chi)|(419)  = 0.398860743207736694380625064238
min_{chi even}|L(1,chi)|(421)  = 0.391799970954197886250931208039
min_{chi odd}|L(1,chi)|(421)  = 0.355558244246456895812338230951
min_{chi}|L(1,chi)|(421)  = 0.355558244246456895812338230951
min_{chi even}|L(1,chi)|(431)  = 0.368928618466358156443766961930
min_{chi odd}|L(1,chi)|(431)  = 0.304249924284011002888004890483
min_{chi}|L(1,chi)|(431)  = 0.304249924284011002888004890483
min_{chi even}|L(1,chi)|(433)  = 0.374437191561992583380814554550
min_{chi odd}|L(1,chi)|(433)  = 0.395422740556217750744403993601
min_{chi}|L(1,chi)|(433)  = 0.374437191561992583380814554550
min_{chi even}|L(1,chi)|(439)  = 0.387411447493331647486199330004
min_{chi odd}|L(1,chi)|(439)  = 0.385201582268506040100915711884
min_{chi}|L(1,chi)|(439)  = 0.385201582268506040100915711884
min_{chi even}|L(1,chi)|(443)  = 0.371745438979544263616425761364
min_{chi odd}|L(1,chi)|(443)  = 0.348072727822271656582475517939
min_{chi}|L(1,chi)|(443)  = 0.348072727822271656582475517939
min_{chi even}|L(1,chi)|(449)  = 0.392414811758698533586486382190
min_{chi odd}|L(1,chi)|(449)  = 0.387387308651078457138994462088
min_{chi}|L(1,chi)|(449)  = 0.387387308651078457138994462088
min_{chi even}|L(1,chi)|(457)  = 0.344128199358590128646593703075
min_{chi odd}|L(1,chi)|(457)  = 0.396940410911565584170384990992
min_{chi}|L(1,chi)|(457)  = 0.344128199358590128646593703075
min_{chi even}|L(1,chi)|(461)  = 0.355756131758284758004388441801
min_{chi odd}|L(1,chi)|(461)  = 0.342282439127743493973555398134
min_{chi}|L(1,chi)|(461)  = 0.342282439127743493973555398134
min_{chi even}|L(1,chi)|(463)  = 0.322810266925933801000377491409
min_{chi odd}|L(1,chi)|(463)  = 0.382723525882736750419305133049
min_{chi}|L(1,chi)|(463)  = 0.322810266925933801000377491409
min_{chi even}|L(1,chi)|(467)  = 0.346262914614287495219714965304
min_{chi odd}|L(1,chi)|(467)  = 0.397352896321351771417705638422
min_{chi}|L(1,chi)|(467)  = 0.346262914614287495219714965304
min_{chi even}|L(1,chi)|(479)  = 0.369908761992829266539160581984
min_{chi odd}|L(1,chi)|(479)  = 0.358217517993363694169353276634
min_{chi}|L(1,chi)|(479)  = 0.358217517993363694169353276634
min_{chi even}|L(1,chi)|(487)  = 0.370815523551482197487199339298
min_{chi odd}|L(1,chi)|(487)  = 0.347056625920168854172557770589
min_{chi}|L(1,chi)|(487)  = 0.347056625920168854172557770589
min_{chi even}|L(1,chi)|(491)  = 0.354518317406980698673697228286
min_{chi odd}|L(1,chi)|(491)  = 0.328216811995353442798930033747
min_{chi}|L(1,chi)|(491)  = 0.328216811995353442798930033747
min_{chi even}|L(1,chi)|(499)  = 0.343916679837668925712980759633
min_{chi odd}|L(1,chi)|(499)  = 0.392745666699937013609534544232
min_{chi}|L(1,chi)|(499)  = 0.343916679837668925712980759633
min_{chi even}|L(1,chi)|(503)  = 0.387076354926718101134754218726
min_{chi odd}|L(1,chi)|(503)  = 0.305270080648936874982633449014
min_{chi}|L(1,chi)|(503)  = 0.305270080648936874982633449014
min_{chi even}|L(1,chi)|(509)  = 0.358400496109432533323994071825
min_{chi odd}|L(1,chi)|(509)  = 0.368606923733423227614352809879
min_{chi}|L(1,chi)|(509)  = 0.358400496109432533323994071825
min_{chi even}|L(1,chi)|(521)  = 0.375505057981653062694067212965
min_{chi odd}|L(1,chi)|(521)  = 0.326639577620790076266875239423
min_{chi}|L(1,chi)|(521)  = 0.326639577620790076266875239423
min_{chi even}|L(1,chi)|(523)  = 0.370263699010744524473643538398
min_{chi odd}|L(1,chi)|(523)  = 0.371868883015216947128179398535
min_{chi}|L(1,chi)|(523)  = 0.370263699010744524473643538398
min_{chi even}|L(1,chi)|(541)  = 0.371841857227428197484414254772
min_{chi odd}|L(1,chi)|(541)  = 0.374865895273528158123362809498
min_{chi}|L(1,chi)|(541)  = 0.371841857227428197484414254772
min_{chi even}|L(1,chi)|(547)  = 0.344451554730668674041547075694
min_{chi odd}|L(1,chi)|(547)  = 0.393427065676548240507060209544
min_{chi}|L(1,chi)|(547)  = 0.344451554730668674041547075694
min_{chi even}|L(1,chi)|(557)  = 0.381164313406877279086486884738
min_{chi odd}|L(1,chi)|(557)  = 0.331277738629462528622134186640
min_{chi}|L(1,chi)|(557)  = 0.331277738629462528622134186640
min_{chi even}|L(1,chi)|(563)  = 0.388184812853218534945156718789
min_{chi odd}|L(1,chi)|(563)  = 0.336288953230595448646092763257
min_{chi}|L(1,chi)|(563)  = 0.336288953230595448646092763257
min_{chi even}|L(1,chi)|(569)  = 0.355458621282047158720345039619
min_{chi odd}|L(1,chi)|(569)  = 0.362268438323865259885798216737
min_{chi}|L(1,chi)|(569)  = 0.355458621282047158720345039619
min_{chi even}|L(1,chi)|(571)  = 0.329684299342773187729572222989
min_{chi odd}|L(1,chi)|(571)  = 0.308093921260701873971115764321
min_{chi}|L(1,chi)|(571)  = 0.308093921260701873971115764321
min_{chi even}|L(1,chi)|(577)  = 0.351768032853537169914900076369
min_{chi odd}|L(1,chi)|(577)  = 0.360526758873597870387720469187
min_{chi}|L(1,chi)|(577)  = 0.351768032853537169914900076369
min_{chi even}|L(1,chi)|(587)  = 0.365788601438852255648675003233
min_{chi odd}|L(1,chi)|(587)  = 0.364218277748308523039650017021
min_{chi}|L(1,chi)|(587)  = 0.364218277748308523039650017021
min_{chi even}|L(1,chi)|(593)  = 0.340193487475883167256908171940
min_{chi odd}|L(1,chi)|(593)  = 0.339509407115695414229878807307
min_{chi}|L(1,chi)|(593)  = 0.339509407115695414229878807307
min_{chi even}|L(1,chi)|(599)  = 0.352374411311513496889433605686
min_{chi odd}|L(1,chi)|(599)  = 0.359856741538461912426433357778
min_{chi}|L(1,chi)|(599)  = 0.352374411311513496889433605686
min_{chi even}|L(1,chi)|(601)  = 0.375819933078454379732588214679
min_{chi odd}|L(1,chi)|(601)  = 0.350992936007857504857261176829
min_{chi}|L(1,chi)|(601)  = 0.350992936007857504857261176829
min_{chi even}|L(1,chi)|(607)  = 0.329981981645739789066558958285
min_{chi odd}|L(1,chi)|(607)  = 0.367869002210405867167043847184
min_{chi}|L(1,chi)|(607)  = 0.329981981645739789066558958285
min_{chi even}|L(1,chi)|(613)  = 0.365348500177310090256111540624
min_{chi odd}|L(1,chi)|(613)  = 0.323436732518113096063105608197
min_{chi}|L(1,chi)|(613)  = 0.323436732518113096063105608197
min_{chi even}|L(1,chi)|(617)  = 0.333733731209755325952269458510
min_{chi odd}|L(1,chi)|(617)  = 0.372260296855890177105611502200
min_{chi}|L(1,chi)|(617)  = 0.333733731209755325952269458510
min_{chi even}|L(1,chi)|(619)  = 0.363853513253924249520317635224
min_{chi odd}|L(1,chi)|(619)  = 0.371103217386749497290471672650
min_{chi}|L(1,chi)|(619)  = 0.363853513253924249520317635224
min_{chi even}|L(1,chi)|(631)  = 0.393123164922613281908265980175
min_{chi odd}|L(1,chi)|(631)  = 0.325338069145847294024656423120
min_{chi}|L(1,chi)|(631)  = 0.325338069145847294024656423120
min_{chi even}|L(1,chi)|(641)  = 0.352147403815645998874657459363
min_{chi odd}|L(1,chi)|(641)  = 0.367241556076360273152030750772
min_{chi}|L(1,chi)|(641)  = 0.352147403815645998874657459363
min_{chi even}|L(1,chi)|(643)  = 0.378305907687446807004515537379
min_{chi odd}|L(1,chi)|(643)  = 0.337445372236761198609110503052
min_{chi}|L(1,chi)|(643)  = 0.337445372236761198609110503052
min_{chi even}|L(1,chi)|(647)  = 0.399168452202860469434121989532
min_{chi odd}|L(1,chi)|(647)  = 0.361173357971713913910561639944
min_{chi}|L(1,chi)|(647)  = 0.361173357971713913910561639944
min_{chi even}|L(1,chi)|(653)  = 0.344528546854302142286423852753
min_{chi odd}|L(1,chi)|(653)  = 0.346948021377954965872672311713
min_{chi}|L(1,chi)|(653)  = 0.344528546854302142286423852753
min_{chi even}|L(1,chi)|(659)  = 0.367219898132189339873602019767
min_{chi odd}|L(1,chi)|(659)  = 0.338132160088261589245693055585
min_{chi}|L(1,chi)|(659)  = 0.338132160088261589245693055585
min_{chi even}|L(1,chi)|(661)  = 0.386988652475998476424076094777
min_{chi odd}|L(1,chi)|(661)  = 0.360619845188122885583816505945
min_{chi}|L(1,chi)|(661)  = 0.360619845188122885583816505945
min_{chi even}|L(1,chi)|(673)  = 0.379941662062213040744027321466
min_{chi odd}|L(1,chi)|(673)  = 0.376129394432057443687895596361
min_{chi}|L(1,chi)|(673)  = 0.376129394432057443687895596361
min_{chi even}|L(1,chi)|(677)  = 0.303745675684779321753979753781
min_{chi odd}|L(1,chi)|(677)  = 0.378350515979641718209131975495
min_{chi}|L(1,chi)|(677)  = 0.303745675684779321753979753781
min_{chi even}|L(1,chi)|(683)  = 0.333368382358596313476605760567
min_{chi odd}|L(1,chi)|(683)  = 0.374334099604314713542268711800
min_{chi}|L(1,chi)|(683)  = 0.333368382358596313476605760567
min_{chi even}|L(1,chi)|(691)  = 0.383305396571474295606978030935
min_{chi odd}|L(1,chi)|(691)  = 0.400059045468505676239829180249
min_{chi}|L(1,chi)|(691)  = 0.383305396571474295606978030935
min_{chi even}|L(1,chi)|(701)  = 0.384964862737165928875785141935
min_{chi odd}|L(1,chi)|(701)  = 0.357535298673103247072275901603
min_{chi}|L(1,chi)|(701)  = 0.357535298673103247072275901603
min_{chi even}|L(1,chi)|(709)  = 0.358127149918438747730782304062
min_{chi odd}|L(1,chi)|(709)  = 0.365137555213240130484500848341
min_{chi}|L(1,chi)|(709)  = 0.358127149918438747730782304062
min_{chi even}|L(1,chi)|(719)  = 0.376328861407967050637150521231
min_{chi odd}|L(1,chi)|(719)  = 0.370805494541737091847758990324
min_{chi}|L(1,chi)|(719)  = 0.370805494541737091847758990324
min_{chi even}|L(1,chi)|(727)  = 0.372898383226526400669880140310
min_{chi odd}|L(1,chi)|(727)  = 0.390475985802664454294691709625
min_{chi}|L(1,chi)|(727)  = 0.372898383226526400669880140310
min_{chi even}|L(1,chi)|(733)  = 0.370320929844056734836902906246
min_{chi odd}|L(1,chi)|(733)  = 0.350732615201881873478746719218
min_{chi}|L(1,chi)|(733)  = 0.350732615201881873478746719218
min_{chi even}|L(1,chi)|(739)  = 0.345737740482078554707235643741
min_{chi odd}|L(1,chi)|(739)  = 0.304897212128635209913817219538
min_{chi}|L(1,chi)|(739)  = 0.304897212128635209913817219538
min_{chi even}|L(1,chi)|(743)  = 0.352336063056891753760152372944
min_{chi odd}|L(1,chi)|(743)  = 0.354875879534802078701167017234
min_{chi}|L(1,chi)|(743)  = 0.352336063056891753760152372944
min_{chi even}|L(1,chi)|(751)  = 0.365201801672573303930768081451
min_{chi odd}|L(1,chi)|(751)  = 0.328140563247974273370269669938
min_{chi}|L(1,chi)|(751)  = 0.328140563247974273370269669938
min_{chi even}|L(1,chi)|(757)  = 0.340246974123488236004380740209
min_{chi odd}|L(1,chi)|(757)  = 0.319411975837870470401241096947
min_{chi}|L(1,chi)|(757)  = 0.319411975837870470401241096947
min_{chi even}|L(1,chi)|(761)  = 0.321973143068848452624799526258
min_{chi odd}|L(1,chi)|(761)  = 0.355127488130494746743050326432
min_{chi}|L(1,chi)|(761)  = 0.321973143068848452624799526258
min_{chi even}|L(1,chi)|(769)  = 0.338178333435205826998970942276
min_{chi odd}|L(1,chi)|(769)  = 0.350048912350642141446639044748
min_{chi}|L(1,chi)|(769)  = 0.338178333435205826998970942276
min_{chi even}|L(1,chi)|(773)  = 0.354965105866232352811142020422
min_{chi odd}|L(1,chi)|(773)  = 0.365903350740214278565099589999
min_{chi}|L(1,chi)|(773)  = 0.354965105866232352811142020422
min_{chi even}|L(1,chi)|(787)  = 0.360604040933363437146725654039
min_{chi odd}|L(1,chi)|(787)  = 0.324266393608051759348322750934
min_{chi}|L(1,chi)|(787)  = 0.324266393608051759348322750934
min_{chi even}|L(1,chi)|(797)  = 0.350515221653708975482852748559
min_{chi odd}|L(1,chi)|(797)  = 0.359227808468086662641664921362
min_{chi}|L(1,chi)|(797)  = 0.350515221653708975482852748559
min_{chi even}|L(1,chi)|(809)  = 0.326434531678769569498090895809
min_{chi odd}|L(1,chi)|(809)  = 0.368387676033860365830398203651
min_{chi}|L(1,chi)|(809)  = 0.326434531678769569498090895809
min_{chi even}|L(1,chi)|(811)  = 0.366973169968077571731993646166
min_{chi odd}|L(1,chi)|(811)  = 0.308510351209093190875996531501
min_{chi}|L(1,chi)|(811)  = 0.308510351209093190875996531501
min_{chi even}|L(1,chi)|(821)  = 0.338533759172131368478596407507
min_{chi odd}|L(1,chi)|(821)  = 0.330244313860677980685649524053
min_{chi}|L(1,chi)|(821)  = 0.330244313860677980685649524053
min_{chi even}|L(1,chi)|(823)  = 0.375223009730851644543900886993
min_{chi odd}|L(1,chi)|(823)  = 0.324283030566918000839563175873
min_{chi}|L(1,chi)|(823)  = 0.324283030566918000839563175873
min_{chi even}|L(1,chi)|(827)  = 0.383528815147561699257131580373
min_{chi odd}|L(1,chi)|(827)  = 0.329051480295616938819583708993
min_{chi}|L(1,chi)|(827)  = 0.329051480295616938819583708993
min_{chi even}|L(1,chi)|(829)  = 0.321222221357102394251381440581
min_{chi odd}|L(1,chi)|(829)  = 0.398339384736577438568175510556
min_{chi}|L(1,chi)|(829)  = 0.321222221357102394251381440581
min_{chi even}|L(1,chi)|(839)  = 0.351875438185464119441010492489
min_{chi odd}|L(1,chi)|(839)  = 0.382348549636708231799551201011
min_{chi}|L(1,chi)|(839)  = 0.351875438185464119441010492489
min_{chi even}|L(1,chi)|(853)  = 0.416150035670110781970760161715
min_{chi odd}|L(1,chi)|(853)  = 0.351818805199888913965741566340
min_{chi}|L(1,chi)|(853)  = 0.351818805199888913965741566340
min_{chi even}|L(1,chi)|(857)  = 0.366066564937826743322512006592
min_{chi odd}|L(1,chi)|(857)  = 0.371766296637931863811115548235
min_{chi}|L(1,chi)|(857)  = 0.366066564937826743322512006592
min_{chi even}|L(1,chi)|(859)  = 0.331002403214058452880530388161
min_{chi odd}|L(1,chi)|(859)  = 0.386900987619169943761868593319
min_{chi}|L(1,chi)|(859)  = 0.331002403214058452880530388161
min_{chi even}|L(1,chi)|(863)  = 0.363812998389375618496007587883
min_{chi odd}|L(1,chi)|(863)  = 0.307797465512278850449583310784
min_{chi}|L(1,chi)|(863)  = 0.307797465512278850449583310784
min_{chi even}|L(1,chi)|(877)  = 0.340810810909356878124676806818
min_{chi odd}|L(1,chi)|(877)  = 0.365619146216705929663089129266
min_{chi}|L(1,chi)|(877)  = 0.340810810909356878124676806818
min_{chi even}|L(1,chi)|(881)  = 0.365522474497724531179329975512
min_{chi odd}|L(1,chi)|(881)  = 0.333426901946204902333090622005
min_{chi}|L(1,chi)|(881)  = 0.333426901946204902333090622005
min_{chi even}|L(1,chi)|(883)  = 0.365100982064518298889725647273
min_{chi odd}|L(1,chi)|(883)  = 0.317169031194076637573267354539
min_{chi}|L(1,chi)|(883)  = 0.317169031194076637573267354539
min_{chi even}|L(1,chi)|(887)  = 0.338488672510289686224158744714
min_{chi odd}|L(1,chi)|(887)  = 0.358463114242689955152405900728
min_{chi}|L(1,chi)|(887)  = 0.338488672510289686224158744714
min_{chi even}|L(1,chi)|(907)  = 0.380544829758497861584032569856
min_{chi odd}|L(1,chi)|(907)  = 0.312944615763902492538994990004
min_{chi}|L(1,chi)|(907)  = 0.312944615763902492538994990004
min_{chi even}|L(1,chi)|(911)  = 0.333229470950277517416975539246
min_{chi odd}|L(1,chi)|(911)  = 0.368338624471254845859678506369
min_{chi}|L(1,chi)|(911)  = 0.333229470950277517416975539246
min_{chi even}|L(1,chi)|(919)  = 0.355302031868156498197335032183
min_{chi odd}|L(1,chi)|(919)  = 0.358555361582633573666505421458
min_{chi}|L(1,chi)|(919)  = 0.355302031868156498197335032183
min_{chi even}|L(1,chi)|(929)  = 0.357591086944711096303847153788
min_{chi odd}|L(1,chi)|(929)  = 0.364823704621872211194704378260
min_{chi}|L(1,chi)|(929)  = 0.357591086944711096303847153788
min_{chi even}|L(1,chi)|(937)  = 0.347008008903403731520713528485
min_{chi odd}|L(1,chi)|(937)  = 0.340988515678272981620106866387
min_{chi}|L(1,chi)|(937)  = 0.340988515678272981620106866387
min_{chi even}|L(1,chi)|(941)  = 0.347914347869332149644718884607
min_{chi odd}|L(1,chi)|(941)  = 0.340561178701543781023570327465
min_{chi}|L(1,chi)|(941)  = 0.340561178701543781023570327465
min_{chi even}|L(1,chi)|(947)  = 0.379684850405948893784190778135
min_{chi odd}|L(1,chi)|(947)  = 0.381541974585030007641416599365
min_{chi}|L(1,chi)|(947)  = 0.379684850405948893784190778135
min_{chi even}|L(1,chi)|(953)  = 0.370761730516982237835157758315
min_{chi odd}|L(1,chi)|(953)  = 0.341656471737723220834223917631
min_{chi}|L(1,chi)|(953)  = 0.341656471737723220834223917631
min_{chi even}|L(1,chi)|(967)  = 0.350795990603971453231229434580
min_{chi odd}|L(1,chi)|(967)  = 0.320793935250234540306919187736
min_{chi}|L(1,chi)|(967)  = 0.320793935250234540306919187736
min_{chi even}|L(1,chi)|(971)  = 0.361084053830653343428457259701
min_{chi odd}|L(1,chi)|(971)  = 0.305685071173978402484536717845
min_{chi}|L(1,chi)|(971)  = 0.305685071173978402484536717845
min_{chi even}|L(1,chi)|(977)  = 0.369913136762492135022760155788
min_{chi odd}|L(1,chi)|(977)  = 0.342232473746414199291917612827
min_{chi}|L(1,chi)|(977)  = 0.342232473746414199291917612827
min_{chi even}|L(1,chi)|(983)  = 0.347747160794603944271769378340
min_{chi odd}|L(1,chi)|(983)  = 0.346431365398324524472675676012
min_{chi}|L(1,chi)|(983)  = 0.346431365398324524472675676012
min_{chi even}|L(1,chi)|(991)  = 0.399187573822782224304166052198
min_{chi odd}|L(1,chi)|(991)  = 0.343527153645248056548459399539
min_{chi}|L(1,chi)|(991)  = 0.343527153645248056548459399539
min_{chi even}|L(1,chi)|(997)  = 0.307684966461541454308715912751
min_{chi odd}|L(1,chi)|(997)  = 0.338239850561923313532354206370
min_{chi}|L(1,chi)|(997)  = 0.307684966461541454308715912751
Precomputation time: 0 min, 0 sec, 995 millisec
Final step max computation time (output time included): 0 min, 15 sec, 822 millisec
Total elapsed time: 0 min, 16 sec, 817 millisec
******  END PROGRAM   ********

 
******  END PROGRAM   ********

? for(n=1,10,m=nextprime(n*10^3);init_MinL_loggamma_final();minL_logGamma(m,m,30))
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L(1,chi)|**********
******* FOR q IN ONE INTERVAL ********
********* with the logGamma-FUNCTION   **********
Interval=[1009,1009]
Performing precomputations of log(Gamma(a/q)
 for this interval
Precomputation for q = 1009
Starting computation of min_q
min_{chi even}|L(1,chi)|(1009)  = 0.370474636922384470878250049906
min_{chi odd}|L(1,chi)|(1009)  = 0.370632531513071710800870895504
min_{chi}|L(1,chi)|(1009)  = 0.370474636922384470878250049906
Precomputation time: 0 min, 0 sec, 13 millisec
Final step max computation time (output time included): 0 min, 0 sec, 304 millisec
Total elapsed time: 0 min, 0 sec, 317 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L(1,chi)|**********
******* FOR q IN ONE INTERVAL ********
********* with the logGamma-FUNCTION   **********
Interval=[2003,2003]
Performing precomputations of log(Gamma(a/q)
 for this interval
Precomputation for q = 2003
Starting computation of min_q
min_{chi even}|L(1,chi)|(2003)  = 0.349767760654230376584731641655
min_{chi odd}|L(1,chi)|(2003)  = 0.346338733321761136174286739106
min_{chi}|L(1,chi)|(2003)  = 0.346338733321761136174286739106
Precomputation time: 0 min, 0 sec, 25 millisec
Final step max computation time (output time included): 0 min, 1 sec, 189 millisec
Total elapsed time: 0 min, 1 sec, 214 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L(1,chi)|**********
******* FOR q IN ONE INTERVAL ********
********* with the logGamma-FUNCTION   **********
Interval=[3001,3001]
Performing precomputations of log(Gamma(a/q)
 for this interval
Precomputation for q = 3001
Starting computation of min_q
min_{chi even}|L(1,chi)|(3001)  = 0.352928689325681764402806009839
min_{chi odd}|L(1,chi)|(3001)  = 0.331064058696032547995590225655
min_{chi}|L(1,chi)|(3001)  = 0.331064058696032547995590225655
Precomputation time: 0 min, 0 sec, 37 millisec
Final step max computation time (output time included): 0 min, 2 sec, 650 millisec
Total elapsed time: 0 min, 2 sec, 687 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L(1,chi)|**********
******* FOR q IN ONE INTERVAL ********
********* with the logGamma-FUNCTION   **********
Interval=[4001,4001]
Performing precomputations of log(Gamma(a/q)
 for this interval
Precomputation for q = 4001
Starting computation of min_q
min_{chi even}|L(1,chi)|(4001)  = 0.295603388543143772722035384706
min_{chi odd}|L(1,chi)|(4001)  = 0.296164088742164463535439389978
min_{chi}|L(1,chi)|(4001)  = 0.295603388543143772722035384706
Precomputation time: 0 min, 0 sec, 49 millisec
Final step max computation time (output time included): 0 min, 4 sec, 751 millisec
Total elapsed time: 0 min, 4 sec, 800 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L(1,chi)|**********
******* FOR q IN ONE INTERVAL ********
********* with the logGamma-FUNCTION   **********
Interval=[5003,5003]
Performing precomputations of log(Gamma(a/q)
 for this interval
Precomputation for q = 5003
Starting computation of min_q
min_{chi even}|L(1,chi)|(5003)  = 0.320135299748006297814394182546
min_{chi odd}|L(1,chi)|(5003)  = 0.345080179403718276218185610046
min_{chi}|L(1,chi)|(5003)  = 0.320135299748006297814394182546
Precomputation time: 0 min, 0 sec, 64 millisec
Final step max computation time (output time included): 0 min, 7 sec, 428 millisec
Total elapsed time: 0 min, 7 sec, 492 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L(1,chi)|**********
******* FOR q IN ONE INTERVAL ********
********* with the logGamma-FUNCTION   **********
Interval=[6007,6007]
Performing precomputations of log(Gamma(a/q)
 for this interval
Precomputation for q = 6007
Starting computation of min_q
min_{chi even}|L(1,chi)|(6007)  = 0.331300130701019007554428083151
min_{chi odd}|L(1,chi)|(6007)  = 0.272307099575062176436498664103
min_{chi}|L(1,chi)|(6007)  = 0.272307099575062176436498664103
Precomputation time: 0 min, 0 sec, 74 millisec
Final step max computation time (output time included): 0 min, 10 sec, 671 millisec
Total elapsed time: 0 min, 10 sec, 745 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L(1,chi)|**********
******* FOR q IN ONE INTERVAL ********
********* with the logGamma-FUNCTION   **********
Interval=[7001,7001]
Performing precomputations of log(Gamma(a/q)
 for this interval
Precomputation for q = 7001
Starting computation of min_q
min_{chi even}|L(1,chi)|(7001)  = 0.312404916546265278749780472055
min_{chi odd}|L(1,chi)|(7001)  = 0.295183872265850511063914147412
min_{chi}|L(1,chi)|(7001)  = 0.295183872265850511063914147412
Precomputation time: 0 min, 0 sec, 86 millisec
Final step max computation time (output time included): 0 min, 14 sec, 495 millisec
Total elapsed time: 0 min, 14 sec, 581 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L(1,chi)|**********
******* FOR q IN ONE INTERVAL ********
********* with the logGamma-FUNCTION   **********
Interval=[8009,8009]
Performing precomputations of log(Gamma(a/q)
 for this interval
Precomputation for q = 8009
Starting computation of min_q
min_{chi even}|L(1,chi)|(8009)  = 0.309715516126026375939107011855
min_{chi odd}|L(1,chi)|(8009)  = 0.311833951181304793177372174202
min_{chi}|L(1,chi)|(8009)  = 0.309715516126026375939107011855
Precomputation time: 0 min, 0 sec, 99 millisec
Final step max computation time (output time included): 0 min, 18 sec, 923 millisec
Total elapsed time: 0 min, 19 sec, 22 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L(1,chi)|**********
******* FOR q IN ONE INTERVAL ********
********* with the logGamma-FUNCTION   **********
Interval=[9001,9001]
Performing precomputations of log(Gamma(a/q)
 for this interval
Precomputation for q = 9001
Starting computation of min_q
min_{chi even}|L(1,chi)|(9001)  = 0.294802028865145458980629118013
min_{chi odd}|L(1,chi)|(9001)  = 0.312567223558619065781685566709
min_{chi}|L(1,chi)|(9001)  = 0.294802028865145458980629118013
Precomputation time: 0 min, 0 sec, 111 millisec
Final step max computation time (output time included): 0 min, 23 sec, 978 millisec
Total elapsed time: 0 min, 24 sec, 89 millisec
******  END PROGRAM   ********
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L(1,chi)|**********
******* FOR q IN ONE INTERVAL ********
********* with the logGamma-FUNCTION   **********
Interval=[10007,10007]
Performing precomputations of log(Gamma(a/q)
 for this interval
Precomputation for q = 10007
Starting computation of min_q
min_{chi even}|L(1,chi)|(10007)  = 0.313054200492943182671828513098
min_{chi odd}|L(1,chi)|(10007)  = 0.301006156549073298592045880662
min_{chi}|L(1,chi)|(10007)  = 0.301006156549073298592045880662
Precomputation time: 0 min, 0 sec, 125 millisec
Final step max computation time (output time included): 0 min, 29 sec, 465 millisec
Total elapsed time: 0 min, 29 sec, 590 millisec
******  END PROGRAM   ********
 
 
 

******************  larger q's **********************
  
? init_MinL_loggamma_final();minL_logGamma(20011,20011,30) 
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L(1,chi)|**********
******* FOR q IN ONE INTERVAL ********
********* with the logGamma-FUNCTION   **********
Interval=[20011,20011]
Performing precomputations of log(Gamma(a/q)
 for this interval
Precomputation for q = 20011
Starting computation of min_q
min_{chi even}|L(1,chi)|(20011)  = 0.303894809033355495605915649367
min_{chi odd}|L(1,chi)|(20011)  = 0.299986479046447933015693324614
min_{chi}|L(1,chi)|(20011)  = 0.299986479046447933015693324614
Precomputation time: 0 min, 0 sec, 249 millisec
Final step max computation time (output time included): 1 min, 58 sec, 379 millisec
Total elapsed time: 1 min, 58 sec, 628 millisec
******  END PROGRAM   ********
 
   
 ? init_MinL_loggamma_final();minL_logGamma(30011,30011,30)
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L(1,chi)|**********
******* FOR q IN ONE INTERVAL ********
********* with the logGamma-FUNCTION   **********
Interval=[30011,30011]
Performing precomputations of log(Gamma(a/q)
 for this interval
Precomputation for q = 30011
Starting computation of min_q
min_{chi even}|L(1,chi)|(30011)  = 0.313189151991153772389978202036
min_{chi odd}|L(1,chi)|(30011)  = 0.294479002574135381458118495466
min_{chi}|L(1,chi)|(30011)  = 0.294479002574135381458118495466
Precomputation time: 0 min, 0 sec, 375 millisec
Final step max computation time (output time included): 4 min, 26 sec, 63 millisec
Total elapsed time: 4 min, 26 sec, 438 millisec
******  END PROGRAM   ********
 
? init_MinL_loggamma_final();minL_logGamma(83089,83089,10)
************ A. LANGUASCO *************
********* COMPUTATION OF THE MIN |L(1,chi)|**********
******* FOR q IN ONE INTERVAL ********
********* with the logGamma-FUNCTION   **********
Interval=[83089,83089]
Performing precomputations of log(Gamma(a/q)
 for this interval
Precomputation for q = 83089
Starting computation of max_q
min_{chi even}|L(1,chi)|(83089)  = 0.2921753145
min_{chi odd}|L(1,chi)|(83089)  = 0.2974893727
min_{chi}|L(1,chi)|(83089)  = 0.2921753145
Precomputation time: 0 min, 0 sec, 778 millisec
Final step max computation time (output time included): 26 min, 24 sec, 100 millisec
Total elapsed time: 26 min, 24 sec, 878 millisec
******  END PROGRAM   ********
 
 
 
 
 
 

 
*******/