Properties

Label 34-531e17-1.1-c7e17-0-1
Degree $34$
Conductor $2.121\times 10^{46}$
Sign $1$
Analytic cond. $5.44929\times 10^{37}$
Root an. cond. $12.8793$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 32·2-s + 7·4-s + 1.07e3·5-s − 2.40e3·7-s − 9.84e3·8-s + 3.43e4·10-s + 8.88e3·11-s − 1.27e4·13-s − 7.70e4·14-s − 6.26e4·16-s + 3.61e4·17-s − 7.10e4·19-s + 7.50e3·20-s + 2.84e5·22-s + 2.69e5·23-s − 4.08e4·25-s − 4.06e5·26-s − 1.68e4·28-s + 5.43e5·29-s − 6.33e5·31-s + 1.52e6·32-s + 1.15e6·34-s − 2.58e6·35-s − 8.67e5·37-s − 2.27e6·38-s − 1.05e7·40-s + 1.42e6·41-s + ⋯
L(s)  = 1  + 2.82·2-s + 0.0546·4-s + 3.83·5-s − 2.65·7-s − 6.80·8-s + 10.8·10-s + 2.01·11-s − 1.60·13-s − 7.50·14-s − 3.82·16-s + 1.78·17-s − 2.37·19-s + 0.209·20-s + 5.69·22-s + 4.62·23-s − 0.522·25-s − 4.53·26-s − 0.145·28-s + 4.14·29-s − 3.81·31-s + 8.22·32-s + 5.04·34-s − 10.1·35-s − 2.81·37-s − 6.72·38-s − 26.0·40-s + 3.23·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{34} \cdot 59^{17}\right)^{s/2} \, \Gamma_{\C}(s)^{17} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{34} \cdot 59^{17}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{17} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(34\)
Conductor: \(3^{34} \cdot 59^{17}\)
Sign: $1$
Analytic conductor: \(5.44929\times 10^{37}\)
Root analytic conductor: \(12.8793\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{531} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((34,\ 3^{34} \cdot 59^{17} ,\ ( \ : [7/2]^{17} ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(11.00918266\)
\(L(\frac12)\) \(\approx\) \(11.00918266\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( ( 1 - p^{3} T )^{17} \)
good2 \( 1 - p^{5} T + 1017 T^{2} - 22471 T^{3} + 229729 p T^{4} - 8059581 T^{5} + 32886979 p^{2} T^{6} - 488571353 p^{2} T^{7} + 846834165 p^{5} T^{8} - 10877670135 p^{5} T^{9} + 32566719527 p^{7} T^{10} - 362403433013 p^{7} T^{11} + 937664491643 p^{9} T^{12} - 17960538945881 p^{8} T^{13} + 40232872218243 p^{10} T^{14} - 339017790928233 p^{10} T^{15} + 370928698840307 p^{13} T^{16} - 1830203416589157 p^{14} T^{17} + 370928698840307 p^{20} T^{18} - 339017790928233 p^{24} T^{19} + 40232872218243 p^{31} T^{20} - 17960538945881 p^{36} T^{21} + 937664491643 p^{44} T^{22} - 362403433013 p^{49} T^{23} + 32566719527 p^{56} T^{24} - 10877670135 p^{61} T^{25} + 846834165 p^{68} T^{26} - 488571353 p^{72} T^{27} + 32886979 p^{79} T^{28} - 8059581 p^{84} T^{29} + 229729 p^{92} T^{30} - 22471 p^{98} T^{31} + 1017 p^{105} T^{32} - p^{117} T^{33} + p^{119} T^{34} \)
5 \( 1 - 1072 T + 237998 p T^{2} - 868967544 T^{3} + 599313233967 T^{4} - 342862008634604 T^{5} + 183692783241656127 T^{6} - 87926796113903596504 T^{7} + \)\(39\!\cdots\!06\)\( T^{8} - \)\(32\!\cdots\!92\)\( p T^{9} + \)\(52\!\cdots\!97\)\( p^{3} T^{10} - \)\(38\!\cdots\!96\)\( p^{4} T^{11} + \)\(13\!\cdots\!02\)\( p^{4} T^{12} - \)\(91\!\cdots\!88\)\( p^{5} T^{13} + \)\(59\!\cdots\!64\)\( p^{6} T^{14} - \)\(36\!\cdots\!24\)\( p^{7} T^{15} + \)\(43\!\cdots\!62\)\( p^{9} T^{16} - \)\(24\!\cdots\!76\)\( p^{10} T^{17} + \)\(43\!\cdots\!62\)\( p^{16} T^{18} - \)\(36\!\cdots\!24\)\( p^{21} T^{19} + \)\(59\!\cdots\!64\)\( p^{27} T^{20} - \)\(91\!\cdots\!88\)\( p^{33} T^{21} + \)\(13\!\cdots\!02\)\( p^{39} T^{22} - \)\(38\!\cdots\!96\)\( p^{46} T^{23} + \)\(52\!\cdots\!97\)\( p^{52} T^{24} - \)\(32\!\cdots\!92\)\( p^{57} T^{25} + \)\(39\!\cdots\!06\)\( p^{63} T^{26} - 87926796113903596504 p^{70} T^{27} + 183692783241656127 p^{77} T^{28} - 342862008634604 p^{84} T^{29} + 599313233967 p^{91} T^{30} - 868967544 p^{98} T^{31} + 237998 p^{106} T^{32} - 1072 p^{112} T^{33} + p^{119} T^{34} \)
7 \( 1 + 2407 T + 7721571 T^{2} + 13486192583 T^{3} + 26781825968668 T^{4} + 38848500997255261 T^{5} + 60770212709730449602 T^{6} + \)\(78\!\cdots\!86\)\( T^{7} + \)\(15\!\cdots\!82\)\( p T^{8} + \)\(12\!\cdots\!53\)\( T^{9} + \)\(14\!\cdots\!18\)\( T^{10} + \)\(16\!\cdots\!81\)\( T^{11} + \)\(18\!\cdots\!39\)\( T^{12} + \)\(18\!\cdots\!17\)\( T^{13} + \)\(19\!\cdots\!27\)\( T^{14} + \)\(18\!\cdots\!74\)\( T^{15} + \)\(25\!\cdots\!64\)\( p T^{16} + \)\(16\!\cdots\!84\)\( T^{17} + \)\(25\!\cdots\!64\)\( p^{8} T^{18} + \)\(18\!\cdots\!74\)\( p^{14} T^{19} + \)\(19\!\cdots\!27\)\( p^{21} T^{20} + \)\(18\!\cdots\!17\)\( p^{28} T^{21} + \)\(18\!\cdots\!39\)\( p^{35} T^{22} + \)\(16\!\cdots\!81\)\( p^{42} T^{23} + \)\(14\!\cdots\!18\)\( p^{49} T^{24} + \)\(12\!\cdots\!53\)\( p^{56} T^{25} + \)\(15\!\cdots\!82\)\( p^{64} T^{26} + \)\(78\!\cdots\!86\)\( p^{70} T^{27} + 60770212709730449602 p^{77} T^{28} + 38848500997255261 p^{84} T^{29} + 26781825968668 p^{91} T^{30} + 13486192583 p^{98} T^{31} + 7721571 p^{105} T^{32} + 2407 p^{112} T^{33} + p^{119} T^{34} \)
11 \( 1 - 808 p T + 182353361 T^{2} - 1431678907464 T^{3} + 1514284076573407 p T^{4} - \)\(11\!\cdots\!32\)\( T^{5} + \)\(10\!\cdots\!43\)\( T^{6} - \)\(62\!\cdots\!04\)\( T^{7} + \)\(44\!\cdots\!63\)\( T^{8} - \)\(25\!\cdots\!36\)\( T^{9} + \)\(15\!\cdots\!79\)\( T^{10} - \)\(81\!\cdots\!04\)\( T^{11} + \)\(45\!\cdots\!94\)\( T^{12} - \)\(22\!\cdots\!12\)\( T^{13} + \)\(11\!\cdots\!26\)\( T^{14} - \)\(51\!\cdots\!68\)\( T^{15} + \)\(24\!\cdots\!94\)\( T^{16} - \)\(96\!\cdots\!24\)\( p T^{17} + \)\(24\!\cdots\!94\)\( p^{7} T^{18} - \)\(51\!\cdots\!68\)\( p^{14} T^{19} + \)\(11\!\cdots\!26\)\( p^{21} T^{20} - \)\(22\!\cdots\!12\)\( p^{28} T^{21} + \)\(45\!\cdots\!94\)\( p^{35} T^{22} - \)\(81\!\cdots\!04\)\( p^{42} T^{23} + \)\(15\!\cdots\!79\)\( p^{49} T^{24} - \)\(25\!\cdots\!36\)\( p^{56} T^{25} + \)\(44\!\cdots\!63\)\( p^{63} T^{26} - \)\(62\!\cdots\!04\)\( p^{70} T^{27} + \)\(10\!\cdots\!43\)\( p^{77} T^{28} - \)\(11\!\cdots\!32\)\( p^{84} T^{29} + 1514284076573407 p^{92} T^{30} - 1431678907464 p^{98} T^{31} + 182353361 p^{105} T^{32} - 808 p^{113} T^{33} + p^{119} T^{34} \)
13 \( 1 + 12702 T + 569033035 T^{2} + 6739417747646 T^{3} + 162855847734232859 T^{4} + \)\(17\!\cdots\!44\)\( T^{5} + \)\(31\!\cdots\!69\)\( T^{6} + \)\(32\!\cdots\!50\)\( T^{7} + \)\(35\!\cdots\!29\)\( p T^{8} + \)\(44\!\cdots\!50\)\( T^{9} + \)\(53\!\cdots\!79\)\( T^{10} + \)\(49\!\cdots\!72\)\( T^{11} + \)\(52\!\cdots\!76\)\( T^{12} + \)\(45\!\cdots\!40\)\( T^{13} + \)\(44\!\cdots\!68\)\( T^{14} + \)\(35\!\cdots\!96\)\( T^{15} + \)\(31\!\cdots\!52\)\( T^{16} + \)\(24\!\cdots\!84\)\( T^{17} + \)\(31\!\cdots\!52\)\( p^{7} T^{18} + \)\(35\!\cdots\!96\)\( p^{14} T^{19} + \)\(44\!\cdots\!68\)\( p^{21} T^{20} + \)\(45\!\cdots\!40\)\( p^{28} T^{21} + \)\(52\!\cdots\!76\)\( p^{35} T^{22} + \)\(49\!\cdots\!72\)\( p^{42} T^{23} + \)\(53\!\cdots\!79\)\( p^{49} T^{24} + \)\(44\!\cdots\!50\)\( p^{56} T^{25} + \)\(35\!\cdots\!29\)\( p^{64} T^{26} + \)\(32\!\cdots\!50\)\( p^{70} T^{27} + \)\(31\!\cdots\!69\)\( p^{77} T^{28} + \)\(17\!\cdots\!44\)\( p^{84} T^{29} + 162855847734232859 p^{91} T^{30} + 6739417747646 p^{98} T^{31} + 569033035 p^{105} T^{32} + 12702 p^{112} T^{33} + p^{119} T^{34} \)
17 \( 1 - 36167 T + 4920728411 T^{2} - 136316984330091 T^{3} + 10911695330906804354 T^{4} - \)\(23\!\cdots\!03\)\( T^{5} + \)\(88\!\cdots\!70\)\( p T^{6} - \)\(26\!\cdots\!18\)\( T^{7} + \)\(14\!\cdots\!02\)\( T^{8} - \)\(20\!\cdots\!91\)\( T^{9} + \)\(11\!\cdots\!80\)\( T^{10} - \)\(12\!\cdots\!29\)\( T^{11} + \)\(68\!\cdots\!61\)\( T^{12} - \)\(59\!\cdots\!85\)\( T^{13} + \)\(35\!\cdots\!93\)\( T^{14} - \)\(25\!\cdots\!78\)\( T^{15} + \)\(16\!\cdots\!92\)\( T^{16} - \)\(10\!\cdots\!64\)\( T^{17} + \)\(16\!\cdots\!92\)\( p^{7} T^{18} - \)\(25\!\cdots\!78\)\( p^{14} T^{19} + \)\(35\!\cdots\!93\)\( p^{21} T^{20} - \)\(59\!\cdots\!85\)\( p^{28} T^{21} + \)\(68\!\cdots\!61\)\( p^{35} T^{22} - \)\(12\!\cdots\!29\)\( p^{42} T^{23} + \)\(11\!\cdots\!80\)\( p^{49} T^{24} - \)\(20\!\cdots\!91\)\( p^{56} T^{25} + \)\(14\!\cdots\!02\)\( p^{63} T^{26} - \)\(26\!\cdots\!18\)\( p^{70} T^{27} + \)\(88\!\cdots\!70\)\( p^{78} T^{28} - \)\(23\!\cdots\!03\)\( p^{84} T^{29} + 10911695330906804354 p^{91} T^{30} - 136316984330091 p^{98} T^{31} + 4920728411 p^{105} T^{32} - 36167 p^{112} T^{33} + p^{119} T^{34} \)
19 \( 1 + 71037 T + 10447686728 T^{2} + 564768891389822 T^{3} + 48592142606271223662 T^{4} + \)\(21\!\cdots\!31\)\( T^{5} + \)\(14\!\cdots\!51\)\( T^{6} + \)\(56\!\cdots\!60\)\( T^{7} + \)\(30\!\cdots\!69\)\( T^{8} + \)\(11\!\cdots\!91\)\( T^{9} + \)\(53\!\cdots\!44\)\( T^{10} + \)\(17\!\cdots\!12\)\( T^{11} + \)\(75\!\cdots\!52\)\( T^{12} + \)\(23\!\cdots\!45\)\( T^{13} + \)\(90\!\cdots\!39\)\( T^{14} + \)\(26\!\cdots\!50\)\( T^{15} + \)\(94\!\cdots\!30\)\( T^{16} + \)\(25\!\cdots\!84\)\( T^{17} + \)\(94\!\cdots\!30\)\( p^{7} T^{18} + \)\(26\!\cdots\!50\)\( p^{14} T^{19} + \)\(90\!\cdots\!39\)\( p^{21} T^{20} + \)\(23\!\cdots\!45\)\( p^{28} T^{21} + \)\(75\!\cdots\!52\)\( p^{35} T^{22} + \)\(17\!\cdots\!12\)\( p^{42} T^{23} + \)\(53\!\cdots\!44\)\( p^{49} T^{24} + \)\(11\!\cdots\!91\)\( p^{56} T^{25} + \)\(30\!\cdots\!69\)\( p^{63} T^{26} + \)\(56\!\cdots\!60\)\( p^{70} T^{27} + \)\(14\!\cdots\!51\)\( p^{77} T^{28} + \)\(21\!\cdots\!31\)\( p^{84} T^{29} + 48592142606271223662 p^{91} T^{30} + 564768891389822 p^{98} T^{31} + 10447686728 p^{105} T^{32} + 71037 p^{112} T^{33} + p^{119} T^{34} \)
23 \( 1 - 269995 T + 64241187606 T^{2} - 10307174581250450 T^{3} + \)\(14\!\cdots\!04\)\( T^{4} - \)\(17\!\cdots\!25\)\( T^{5} + \)\(19\!\cdots\!27\)\( T^{6} - \)\(19\!\cdots\!84\)\( T^{7} + \)\(17\!\cdots\!41\)\( T^{8} - \)\(15\!\cdots\!89\)\( T^{9} + \)\(12\!\cdots\!22\)\( T^{10} - \)\(95\!\cdots\!16\)\( T^{11} + \)\(70\!\cdots\!66\)\( T^{12} - \)\(49\!\cdots\!35\)\( T^{13} + \)\(33\!\cdots\!91\)\( T^{14} - \)\(21\!\cdots\!86\)\( T^{15} + \)\(13\!\cdots\!18\)\( T^{16} - \)\(78\!\cdots\!36\)\( T^{17} + \)\(13\!\cdots\!18\)\( p^{7} T^{18} - \)\(21\!\cdots\!86\)\( p^{14} T^{19} + \)\(33\!\cdots\!91\)\( p^{21} T^{20} - \)\(49\!\cdots\!35\)\( p^{28} T^{21} + \)\(70\!\cdots\!66\)\( p^{35} T^{22} - \)\(95\!\cdots\!16\)\( p^{42} T^{23} + \)\(12\!\cdots\!22\)\( p^{49} T^{24} - \)\(15\!\cdots\!89\)\( p^{56} T^{25} + \)\(17\!\cdots\!41\)\( p^{63} T^{26} - \)\(19\!\cdots\!84\)\( p^{70} T^{27} + \)\(19\!\cdots\!27\)\( p^{77} T^{28} - \)\(17\!\cdots\!25\)\( p^{84} T^{29} + \)\(14\!\cdots\!04\)\( p^{91} T^{30} - 10307174581250450 p^{98} T^{31} + 64241187606 p^{105} T^{32} - 269995 p^{112} T^{33} + p^{119} T^{34} \)
29 \( 1 - 543825 T + 238671437776 T^{2} - 69165454438559300 T^{3} + \)\(18\!\cdots\!24\)\( T^{4} - \)\(38\!\cdots\!19\)\( T^{5} + \)\(27\!\cdots\!51\)\( p T^{6} - \)\(14\!\cdots\!12\)\( T^{7} + \)\(26\!\cdots\!67\)\( T^{8} - \)\(44\!\cdots\!43\)\( T^{9} + \)\(76\!\cdots\!90\)\( T^{10} - \)\(11\!\cdots\!78\)\( T^{11} + \)\(19\!\cdots\!72\)\( T^{12} - \)\(95\!\cdots\!89\)\( p T^{13} + \)\(41\!\cdots\!17\)\( T^{14} - \)\(56\!\cdots\!02\)\( T^{15} + \)\(80\!\cdots\!78\)\( T^{16} - \)\(10\!\cdots\!60\)\( T^{17} + \)\(80\!\cdots\!78\)\( p^{7} T^{18} - \)\(56\!\cdots\!02\)\( p^{14} T^{19} + \)\(41\!\cdots\!17\)\( p^{21} T^{20} - \)\(95\!\cdots\!89\)\( p^{29} T^{21} + \)\(19\!\cdots\!72\)\( p^{35} T^{22} - \)\(11\!\cdots\!78\)\( p^{42} T^{23} + \)\(76\!\cdots\!90\)\( p^{49} T^{24} - \)\(44\!\cdots\!43\)\( p^{56} T^{25} + \)\(26\!\cdots\!67\)\( p^{63} T^{26} - \)\(14\!\cdots\!12\)\( p^{70} T^{27} + \)\(27\!\cdots\!51\)\( p^{78} T^{28} - \)\(38\!\cdots\!19\)\( p^{84} T^{29} + \)\(18\!\cdots\!24\)\( p^{91} T^{30} - 69165454438559300 p^{98} T^{31} + 238671437776 p^{105} T^{32} - 543825 p^{112} T^{33} + p^{119} T^{34} \)
31 \( 1 + 633109 T + 310861263310 T^{2} + 104984082738087870 T^{3} + \)\(30\!\cdots\!58\)\( T^{4} + \)\(70\!\cdots\!59\)\( T^{5} + \)\(14\!\cdots\!17\)\( T^{6} + \)\(25\!\cdots\!04\)\( T^{7} + \)\(43\!\cdots\!47\)\( T^{8} + \)\(65\!\cdots\!39\)\( T^{9} + \)\(10\!\cdots\!32\)\( T^{10} + \)\(15\!\cdots\!04\)\( T^{11} + \)\(23\!\cdots\!10\)\( T^{12} + \)\(27\!\cdots\!89\)\( T^{13} + \)\(26\!\cdots\!39\)\( T^{14} - \)\(93\!\cdots\!06\)\( T^{15} - \)\(64\!\cdots\!94\)\( T^{16} - \)\(16\!\cdots\!36\)\( T^{17} - \)\(64\!\cdots\!94\)\( p^{7} T^{18} - \)\(93\!\cdots\!06\)\( p^{14} T^{19} + \)\(26\!\cdots\!39\)\( p^{21} T^{20} + \)\(27\!\cdots\!89\)\( p^{28} T^{21} + \)\(23\!\cdots\!10\)\( p^{35} T^{22} + \)\(15\!\cdots\!04\)\( p^{42} T^{23} + \)\(10\!\cdots\!32\)\( p^{49} T^{24} + \)\(65\!\cdots\!39\)\( p^{56} T^{25} + \)\(43\!\cdots\!47\)\( p^{63} T^{26} + \)\(25\!\cdots\!04\)\( p^{70} T^{27} + \)\(14\!\cdots\!17\)\( p^{77} T^{28} + \)\(70\!\cdots\!59\)\( p^{84} T^{29} + \)\(30\!\cdots\!58\)\( p^{91} T^{30} + 104984082738087870 p^{98} T^{31} + 310861263310 p^{105} T^{32} + 633109 p^{112} T^{33} + p^{119} T^{34} \)
37 \( 1 + 867607 T + 1228914462787 T^{2} + 808845470307200853 T^{3} + \)\(66\!\cdots\!18\)\( T^{4} + \)\(35\!\cdots\!11\)\( T^{5} + \)\(21\!\cdots\!40\)\( T^{6} + \)\(98\!\cdots\!54\)\( T^{7} + \)\(49\!\cdots\!04\)\( T^{8} + \)\(19\!\cdots\!43\)\( T^{9} + \)\(83\!\cdots\!48\)\( T^{10} + \)\(28\!\cdots\!19\)\( T^{11} + \)\(10\!\cdots\!45\)\( T^{12} + \)\(34\!\cdots\!33\)\( T^{13} + \)\(12\!\cdots\!85\)\( T^{14} + \)\(34\!\cdots\!78\)\( T^{15} + \)\(11\!\cdots\!88\)\( T^{16} + \)\(33\!\cdots\!64\)\( T^{17} + \)\(11\!\cdots\!88\)\( p^{7} T^{18} + \)\(34\!\cdots\!78\)\( p^{14} T^{19} + \)\(12\!\cdots\!85\)\( p^{21} T^{20} + \)\(34\!\cdots\!33\)\( p^{28} T^{21} + \)\(10\!\cdots\!45\)\( p^{35} T^{22} + \)\(28\!\cdots\!19\)\( p^{42} T^{23} + \)\(83\!\cdots\!48\)\( p^{49} T^{24} + \)\(19\!\cdots\!43\)\( p^{56} T^{25} + \)\(49\!\cdots\!04\)\( p^{63} T^{26} + \)\(98\!\cdots\!54\)\( p^{70} T^{27} + \)\(21\!\cdots\!40\)\( p^{77} T^{28} + \)\(35\!\cdots\!11\)\( p^{84} T^{29} + \)\(66\!\cdots\!18\)\( p^{91} T^{30} + 808845470307200853 p^{98} T^{31} + 1228914462787 p^{105} T^{32} + 867607 p^{112} T^{33} + p^{119} T^{34} \)
41 \( 1 - 1428939 T + 1699440549923 T^{2} - 1288469922310536263 T^{3} + \)\(95\!\cdots\!70\)\( T^{4} - \)\(13\!\cdots\!35\)\( p T^{5} + \)\(34\!\cdots\!94\)\( T^{6} - \)\(16\!\cdots\!46\)\( T^{7} + \)\(88\!\cdots\!42\)\( T^{8} - \)\(35\!\cdots\!83\)\( T^{9} + \)\(15\!\cdots\!20\)\( T^{10} - \)\(44\!\cdots\!65\)\( T^{11} + \)\(15\!\cdots\!41\)\( T^{12} - \)\(24\!\cdots\!93\)\( T^{13} - \)\(67\!\cdots\!35\)\( T^{14} + \)\(15\!\cdots\!98\)\( T^{15} - \)\(64\!\cdots\!76\)\( T^{16} + \)\(42\!\cdots\!68\)\( T^{17} - \)\(64\!\cdots\!76\)\( p^{7} T^{18} + \)\(15\!\cdots\!98\)\( p^{14} T^{19} - \)\(67\!\cdots\!35\)\( p^{21} T^{20} - \)\(24\!\cdots\!93\)\( p^{28} T^{21} + \)\(15\!\cdots\!41\)\( p^{35} T^{22} - \)\(44\!\cdots\!65\)\( p^{42} T^{23} + \)\(15\!\cdots\!20\)\( p^{49} T^{24} - \)\(35\!\cdots\!83\)\( p^{56} T^{25} + \)\(88\!\cdots\!42\)\( p^{63} T^{26} - \)\(16\!\cdots\!46\)\( p^{70} T^{27} + \)\(34\!\cdots\!94\)\( p^{77} T^{28} - \)\(13\!\cdots\!35\)\( p^{85} T^{29} + \)\(95\!\cdots\!70\)\( p^{91} T^{30} - 1288469922310536263 p^{98} T^{31} + 1699440549923 p^{105} T^{32} - 1428939 p^{112} T^{33} + p^{119} T^{34} \)
43 \( 1 + 477060 T + 2167763722989 T^{2} + 1125207623026716140 T^{3} + \)\(25\!\cdots\!21\)\( T^{4} + \)\(13\!\cdots\!96\)\( T^{5} + \)\(20\!\cdots\!79\)\( T^{6} + \)\(11\!\cdots\!20\)\( T^{7} + \)\(13\!\cdots\!91\)\( T^{8} + \)\(70\!\cdots\!80\)\( T^{9} + \)\(68\!\cdots\!31\)\( T^{10} + \)\(34\!\cdots\!84\)\( T^{11} + \)\(29\!\cdots\!78\)\( T^{12} + \)\(14\!\cdots\!28\)\( T^{13} + \)\(10\!\cdots\!30\)\( T^{14} + \)\(49\!\cdots\!88\)\( T^{15} + \)\(34\!\cdots\!98\)\( T^{16} + \)\(14\!\cdots\!36\)\( T^{17} + \)\(34\!\cdots\!98\)\( p^{7} T^{18} + \)\(49\!\cdots\!88\)\( p^{14} T^{19} + \)\(10\!\cdots\!30\)\( p^{21} T^{20} + \)\(14\!\cdots\!28\)\( p^{28} T^{21} + \)\(29\!\cdots\!78\)\( p^{35} T^{22} + \)\(34\!\cdots\!84\)\( p^{42} T^{23} + \)\(68\!\cdots\!31\)\( p^{49} T^{24} + \)\(70\!\cdots\!80\)\( p^{56} T^{25} + \)\(13\!\cdots\!91\)\( p^{63} T^{26} + \)\(11\!\cdots\!20\)\( p^{70} T^{27} + \)\(20\!\cdots\!79\)\( p^{77} T^{28} + \)\(13\!\cdots\!96\)\( p^{84} T^{29} + \)\(25\!\cdots\!21\)\( p^{91} T^{30} + 1125207623026716140 p^{98} T^{31} + 2167763722989 p^{105} T^{32} + 477060 p^{112} T^{33} + p^{119} T^{34} \)
47 \( 1 - 1217849 T + 4871711251396 T^{2} - 4938762490337898806 T^{3} + \)\(10\!\cdots\!22\)\( T^{4} - \)\(93\!\cdots\!35\)\( T^{5} + \)\(14\!\cdots\!31\)\( T^{6} - \)\(11\!\cdots\!92\)\( T^{7} + \)\(13\!\cdots\!81\)\( T^{8} - \)\(10\!\cdots\!47\)\( T^{9} + \)\(10\!\cdots\!12\)\( T^{10} - \)\(78\!\cdots\!32\)\( T^{11} + \)\(70\!\cdots\!44\)\( T^{12} - \)\(51\!\cdots\!33\)\( T^{13} + \)\(41\!\cdots\!11\)\( T^{14} - \)\(64\!\cdots\!74\)\( p T^{15} + \)\(22\!\cdots\!70\)\( T^{16} - \)\(15\!\cdots\!56\)\( T^{17} + \)\(22\!\cdots\!70\)\( p^{7} T^{18} - \)\(64\!\cdots\!74\)\( p^{15} T^{19} + \)\(41\!\cdots\!11\)\( p^{21} T^{20} - \)\(51\!\cdots\!33\)\( p^{28} T^{21} + \)\(70\!\cdots\!44\)\( p^{35} T^{22} - \)\(78\!\cdots\!32\)\( p^{42} T^{23} + \)\(10\!\cdots\!12\)\( p^{49} T^{24} - \)\(10\!\cdots\!47\)\( p^{56} T^{25} + \)\(13\!\cdots\!81\)\( p^{63} T^{26} - \)\(11\!\cdots\!92\)\( p^{70} T^{27} + \)\(14\!\cdots\!31\)\( p^{77} T^{28} - \)\(93\!\cdots\!35\)\( p^{84} T^{29} + \)\(10\!\cdots\!22\)\( p^{91} T^{30} - 4938762490337898806 p^{98} T^{31} + 4871711251396 p^{105} T^{32} - 1217849 p^{112} T^{33} + p^{119} T^{34} \)
53 \( 1 - 3487068 T + 16296207161346 T^{2} - 42007763167447733424 T^{3} + \)\(11\!\cdots\!95\)\( T^{4} - \)\(24\!\cdots\!28\)\( T^{5} + \)\(53\!\cdots\!75\)\( T^{6} - \)\(96\!\cdots\!28\)\( T^{7} + \)\(17\!\cdots\!58\)\( T^{8} - \)\(27\!\cdots\!16\)\( T^{9} + \)\(43\!\cdots\!81\)\( T^{10} - \)\(62\!\cdots\!52\)\( T^{11} + \)\(88\!\cdots\!70\)\( T^{12} - \)\(11\!\cdots\!44\)\( T^{13} + \)\(14\!\cdots\!76\)\( T^{14} - \)\(17\!\cdots\!84\)\( T^{15} + \)\(20\!\cdots\!34\)\( T^{16} - \)\(22\!\cdots\!04\)\( T^{17} + \)\(20\!\cdots\!34\)\( p^{7} T^{18} - \)\(17\!\cdots\!84\)\( p^{14} T^{19} + \)\(14\!\cdots\!76\)\( p^{21} T^{20} - \)\(11\!\cdots\!44\)\( p^{28} T^{21} + \)\(88\!\cdots\!70\)\( p^{35} T^{22} - \)\(62\!\cdots\!52\)\( p^{42} T^{23} + \)\(43\!\cdots\!81\)\( p^{49} T^{24} - \)\(27\!\cdots\!16\)\( p^{56} T^{25} + \)\(17\!\cdots\!58\)\( p^{63} T^{26} - \)\(96\!\cdots\!28\)\( p^{70} T^{27} + \)\(53\!\cdots\!75\)\( p^{77} T^{28} - \)\(24\!\cdots\!28\)\( p^{84} T^{29} + \)\(11\!\cdots\!95\)\( p^{91} T^{30} - 42007763167447733424 p^{98} T^{31} + 16296207161346 p^{105} T^{32} - 3487068 p^{112} T^{33} + p^{119} T^{34} \)
61 \( 1 - 998917 T + 19311022999118 T^{2} - 16323849703981171336 T^{3} + \)\(18\!\cdots\!78\)\( T^{4} - \)\(16\!\cdots\!23\)\( T^{5} + \)\(12\!\cdots\!19\)\( T^{6} - \)\(14\!\cdots\!12\)\( T^{7} + \)\(71\!\cdots\!39\)\( T^{8} - \)\(94\!\cdots\!11\)\( T^{9} + \)\(33\!\cdots\!00\)\( T^{10} - \)\(51\!\cdots\!94\)\( T^{11} + \)\(13\!\cdots\!66\)\( T^{12} - \)\(24\!\cdots\!77\)\( T^{13} + \)\(49\!\cdots\!41\)\( T^{14} - \)\(95\!\cdots\!66\)\( T^{15} + \)\(16\!\cdots\!06\)\( T^{16} - \)\(32\!\cdots\!84\)\( T^{17} + \)\(16\!\cdots\!06\)\( p^{7} T^{18} - \)\(95\!\cdots\!66\)\( p^{14} T^{19} + \)\(49\!\cdots\!41\)\( p^{21} T^{20} - \)\(24\!\cdots\!77\)\( p^{28} T^{21} + \)\(13\!\cdots\!66\)\( p^{35} T^{22} - \)\(51\!\cdots\!94\)\( p^{42} T^{23} + \)\(33\!\cdots\!00\)\( p^{49} T^{24} - \)\(94\!\cdots\!11\)\( p^{56} T^{25} + \)\(71\!\cdots\!39\)\( p^{63} T^{26} - \)\(14\!\cdots\!12\)\( p^{70} T^{27} + \)\(12\!\cdots\!19\)\( p^{77} T^{28} - \)\(16\!\cdots\!23\)\( p^{84} T^{29} + \)\(18\!\cdots\!78\)\( p^{91} T^{30} - 16323849703981171336 p^{98} T^{31} + 19311022999118 p^{105} T^{32} - 998917 p^{112} T^{33} + p^{119} T^{34} \)
67 \( 1 + 356026 T + 57416266052296 T^{2} + 34301320789216132396 T^{3} + \)\(16\!\cdots\!35\)\( T^{4} + \)\(14\!\cdots\!72\)\( T^{5} + \)\(31\!\cdots\!29\)\( T^{6} + \)\(33\!\cdots\!56\)\( T^{7} + \)\(45\!\cdots\!92\)\( T^{8} + \)\(56\!\cdots\!86\)\( T^{9} + \)\(52\!\cdots\!63\)\( T^{10} + \)\(70\!\cdots\!28\)\( T^{11} + \)\(50\!\cdots\!98\)\( T^{12} + \)\(68\!\cdots\!56\)\( T^{13} + \)\(41\!\cdots\!88\)\( T^{14} + \)\(55\!\cdots\!56\)\( T^{15} + \)\(29\!\cdots\!26\)\( T^{16} + \)\(36\!\cdots\!52\)\( T^{17} + \)\(29\!\cdots\!26\)\( p^{7} T^{18} + \)\(55\!\cdots\!56\)\( p^{14} T^{19} + \)\(41\!\cdots\!88\)\( p^{21} T^{20} + \)\(68\!\cdots\!56\)\( p^{28} T^{21} + \)\(50\!\cdots\!98\)\( p^{35} T^{22} + \)\(70\!\cdots\!28\)\( p^{42} T^{23} + \)\(52\!\cdots\!63\)\( p^{49} T^{24} + \)\(56\!\cdots\!86\)\( p^{56} T^{25} + \)\(45\!\cdots\!92\)\( p^{63} T^{26} + \)\(33\!\cdots\!56\)\( p^{70} T^{27} + \)\(31\!\cdots\!29\)\( p^{77} T^{28} + \)\(14\!\cdots\!72\)\( p^{84} T^{29} + \)\(16\!\cdots\!35\)\( p^{91} T^{30} + 34301320789216132396 p^{98} T^{31} + 57416266052296 p^{105} T^{32} + 356026 p^{112} T^{33} + p^{119} T^{34} \)
71 \( 1 - 12879428 T + 185963061699457 T^{2} - \)\(16\!\cdots\!94\)\( T^{3} + \)\(14\!\cdots\!95\)\( T^{4} - \)\(96\!\cdots\!90\)\( T^{5} + \)\(63\!\cdots\!31\)\( T^{6} - \)\(35\!\cdots\!50\)\( T^{7} + \)\(19\!\cdots\!93\)\( T^{8} - \)\(93\!\cdots\!68\)\( T^{9} + \)\(43\!\cdots\!85\)\( T^{10} - \)\(18\!\cdots\!04\)\( T^{11} + \)\(10\!\cdots\!72\)\( p T^{12} - \)\(28\!\cdots\!92\)\( T^{13} + \)\(10\!\cdots\!36\)\( T^{14} - \)\(35\!\cdots\!56\)\( T^{15} + \)\(11\!\cdots\!52\)\( T^{16} - \)\(35\!\cdots\!16\)\( T^{17} + \)\(11\!\cdots\!52\)\( p^{7} T^{18} - \)\(35\!\cdots\!56\)\( p^{14} T^{19} + \)\(10\!\cdots\!36\)\( p^{21} T^{20} - \)\(28\!\cdots\!92\)\( p^{28} T^{21} + \)\(10\!\cdots\!72\)\( p^{36} T^{22} - \)\(18\!\cdots\!04\)\( p^{42} T^{23} + \)\(43\!\cdots\!85\)\( p^{49} T^{24} - \)\(93\!\cdots\!68\)\( p^{56} T^{25} + \)\(19\!\cdots\!93\)\( p^{63} T^{26} - \)\(35\!\cdots\!50\)\( p^{70} T^{27} + \)\(63\!\cdots\!31\)\( p^{77} T^{28} - \)\(96\!\cdots\!90\)\( p^{84} T^{29} + \)\(14\!\cdots\!95\)\( p^{91} T^{30} - \)\(16\!\cdots\!94\)\( p^{98} T^{31} + 185963061699457 p^{105} T^{32} - 12879428 p^{112} T^{33} + p^{119} T^{34} \)
73 \( 1 + 6176157 T + 92298138937078 T^{2} + \)\(54\!\cdots\!76\)\( T^{3} + \)\(45\!\cdots\!32\)\( T^{4} + \)\(24\!\cdots\!63\)\( T^{5} + \)\(15\!\cdots\!21\)\( T^{6} + \)\(75\!\cdots\!48\)\( T^{7} + \)\(40\!\cdots\!33\)\( T^{8} + \)\(17\!\cdots\!11\)\( T^{9} + \)\(82\!\cdots\!22\)\( T^{10} + \)\(33\!\cdots\!18\)\( T^{11} + \)\(14\!\cdots\!54\)\( T^{12} + \)\(52\!\cdots\!65\)\( T^{13} + \)\(20\!\cdots\!09\)\( T^{14} + \)\(71\!\cdots\!14\)\( T^{15} + \)\(25\!\cdots\!50\)\( T^{16} + \)\(83\!\cdots\!32\)\( T^{17} + \)\(25\!\cdots\!50\)\( p^{7} T^{18} + \)\(71\!\cdots\!14\)\( p^{14} T^{19} + \)\(20\!\cdots\!09\)\( p^{21} T^{20} + \)\(52\!\cdots\!65\)\( p^{28} T^{21} + \)\(14\!\cdots\!54\)\( p^{35} T^{22} + \)\(33\!\cdots\!18\)\( p^{42} T^{23} + \)\(82\!\cdots\!22\)\( p^{49} T^{24} + \)\(17\!\cdots\!11\)\( p^{56} T^{25} + \)\(40\!\cdots\!33\)\( p^{63} T^{26} + \)\(75\!\cdots\!48\)\( p^{70} T^{27} + \)\(15\!\cdots\!21\)\( p^{77} T^{28} + \)\(24\!\cdots\!63\)\( p^{84} T^{29} + \)\(45\!\cdots\!32\)\( p^{91} T^{30} + \)\(54\!\cdots\!76\)\( p^{98} T^{31} + 92298138937078 p^{105} T^{32} + 6176157 p^{112} T^{33} + p^{119} T^{34} \)
79 \( 1 + 18886490 T + 351681335371575 T^{2} + \)\(44\!\cdots\!50\)\( T^{3} + \)\(51\!\cdots\!01\)\( T^{4} + \)\(50\!\cdots\!12\)\( T^{5} + \)\(45\!\cdots\!71\)\( T^{6} + \)\(37\!\cdots\!70\)\( T^{7} + \)\(28\!\cdots\!25\)\( T^{8} + \)\(19\!\cdots\!86\)\( T^{9} + \)\(13\!\cdots\!39\)\( T^{10} + \)\(80\!\cdots\!88\)\( T^{11} + \)\(47\!\cdots\!42\)\( T^{12} + \)\(26\!\cdots\!68\)\( T^{13} + \)\(13\!\cdots\!26\)\( T^{14} + \)\(67\!\cdots\!72\)\( T^{15} + \)\(32\!\cdots\!02\)\( T^{16} + \)\(14\!\cdots\!60\)\( T^{17} + \)\(32\!\cdots\!02\)\( p^{7} T^{18} + \)\(67\!\cdots\!72\)\( p^{14} T^{19} + \)\(13\!\cdots\!26\)\( p^{21} T^{20} + \)\(26\!\cdots\!68\)\( p^{28} T^{21} + \)\(47\!\cdots\!42\)\( p^{35} T^{22} + \)\(80\!\cdots\!88\)\( p^{42} T^{23} + \)\(13\!\cdots\!39\)\( p^{49} T^{24} + \)\(19\!\cdots\!86\)\( p^{56} T^{25} + \)\(28\!\cdots\!25\)\( p^{63} T^{26} + \)\(37\!\cdots\!70\)\( p^{70} T^{27} + \)\(45\!\cdots\!71\)\( p^{77} T^{28} + \)\(50\!\cdots\!12\)\( p^{84} T^{29} + \)\(51\!\cdots\!01\)\( p^{91} T^{30} + \)\(44\!\cdots\!50\)\( p^{98} T^{31} + 351681335371575 p^{105} T^{32} + 18886490 p^{112} T^{33} + p^{119} T^{34} \)
83 \( 1 - 22824893 T + 461242237823497 T^{2} - \)\(60\!\cdots\!59\)\( T^{3} + \)\(73\!\cdots\!26\)\( T^{4} - \)\(70\!\cdots\!87\)\( T^{5} + \)\(64\!\cdots\!14\)\( T^{6} - \)\(49\!\cdots\!42\)\( T^{7} + \)\(37\!\cdots\!30\)\( T^{8} - \)\(24\!\cdots\!47\)\( T^{9} + \)\(16\!\cdots\!84\)\( T^{10} - \)\(96\!\cdots\!61\)\( T^{11} + \)\(59\!\cdots\!05\)\( T^{12} - \)\(33\!\cdots\!15\)\( T^{13} + \)\(19\!\cdots\!19\)\( T^{14} - \)\(10\!\cdots\!30\)\( T^{15} + \)\(57\!\cdots\!40\)\( T^{16} - \)\(29\!\cdots\!20\)\( T^{17} + \)\(57\!\cdots\!40\)\( p^{7} T^{18} - \)\(10\!\cdots\!30\)\( p^{14} T^{19} + \)\(19\!\cdots\!19\)\( p^{21} T^{20} - \)\(33\!\cdots\!15\)\( p^{28} T^{21} + \)\(59\!\cdots\!05\)\( p^{35} T^{22} - \)\(96\!\cdots\!61\)\( p^{42} T^{23} + \)\(16\!\cdots\!84\)\( p^{49} T^{24} - \)\(24\!\cdots\!47\)\( p^{56} T^{25} + \)\(37\!\cdots\!30\)\( p^{63} T^{26} - \)\(49\!\cdots\!42\)\( p^{70} T^{27} + \)\(64\!\cdots\!14\)\( p^{77} T^{28} - \)\(70\!\cdots\!87\)\( p^{84} T^{29} + \)\(73\!\cdots\!26\)\( p^{91} T^{30} - \)\(60\!\cdots\!59\)\( p^{98} T^{31} + 461242237823497 p^{105} T^{32} - 22824893 p^{112} T^{33} + p^{119} T^{34} \)
89 \( 1 - 30609647 T + 883612835750470 T^{2} - \)\(16\!\cdots\!08\)\( T^{3} + \)\(29\!\cdots\!36\)\( T^{4} - \)\(40\!\cdots\!77\)\( T^{5} + \)\(54\!\cdots\!09\)\( T^{6} - \)\(63\!\cdots\!32\)\( T^{7} + \)\(70\!\cdots\!21\)\( T^{8} - \)\(70\!\cdots\!61\)\( T^{9} + \)\(67\!\cdots\!22\)\( T^{10} - \)\(59\!\cdots\!42\)\( T^{11} + \)\(51\!\cdots\!38\)\( T^{12} - \)\(41\!\cdots\!55\)\( T^{13} + \)\(32\!\cdots\!89\)\( T^{14} - \)\(23\!\cdots\!46\)\( T^{15} + \)\(16\!\cdots\!54\)\( T^{16} - \)\(11\!\cdots\!76\)\( T^{17} + \)\(16\!\cdots\!54\)\( p^{7} T^{18} - \)\(23\!\cdots\!46\)\( p^{14} T^{19} + \)\(32\!\cdots\!89\)\( p^{21} T^{20} - \)\(41\!\cdots\!55\)\( p^{28} T^{21} + \)\(51\!\cdots\!38\)\( p^{35} T^{22} - \)\(59\!\cdots\!42\)\( p^{42} T^{23} + \)\(67\!\cdots\!22\)\( p^{49} T^{24} - \)\(70\!\cdots\!61\)\( p^{56} T^{25} + \)\(70\!\cdots\!21\)\( p^{63} T^{26} - \)\(63\!\cdots\!32\)\( p^{70} T^{27} + \)\(54\!\cdots\!09\)\( p^{77} T^{28} - \)\(40\!\cdots\!77\)\( p^{84} T^{29} + \)\(29\!\cdots\!36\)\( p^{91} T^{30} - \)\(16\!\cdots\!08\)\( p^{98} T^{31} + 883612835750470 p^{105} T^{32} - 30609647 p^{112} T^{33} + p^{119} T^{34} \)
97 \( 1 + 26249806 T + 1041566810058300 T^{2} + \)\(19\!\cdots\!66\)\( T^{3} + \)\(47\!\cdots\!31\)\( T^{4} + \)\(73\!\cdots\!24\)\( T^{5} + \)\(13\!\cdots\!99\)\( T^{6} + \)\(17\!\cdots\!30\)\( T^{7} + \)\(26\!\cdots\!72\)\( T^{8} + \)\(32\!\cdots\!02\)\( T^{9} + \)\(42\!\cdots\!57\)\( T^{10} + \)\(47\!\cdots\!92\)\( T^{11} + \)\(55\!\cdots\!74\)\( T^{12} + \)\(56\!\cdots\!76\)\( T^{13} + \)\(60\!\cdots\!84\)\( T^{14} + \)\(56\!\cdots\!44\)\( T^{15} + \)\(56\!\cdots\!78\)\( T^{16} + \)\(49\!\cdots\!88\)\( T^{17} + \)\(56\!\cdots\!78\)\( p^{7} T^{18} + \)\(56\!\cdots\!44\)\( p^{14} T^{19} + \)\(60\!\cdots\!84\)\( p^{21} T^{20} + \)\(56\!\cdots\!76\)\( p^{28} T^{21} + \)\(55\!\cdots\!74\)\( p^{35} T^{22} + \)\(47\!\cdots\!92\)\( p^{42} T^{23} + \)\(42\!\cdots\!57\)\( p^{49} T^{24} + \)\(32\!\cdots\!02\)\( p^{56} T^{25} + \)\(26\!\cdots\!72\)\( p^{63} T^{26} + \)\(17\!\cdots\!30\)\( p^{70} T^{27} + \)\(13\!\cdots\!99\)\( p^{77} T^{28} + \)\(73\!\cdots\!24\)\( p^{84} T^{29} + \)\(47\!\cdots\!31\)\( p^{91} T^{30} + \)\(19\!\cdots\!66\)\( p^{98} T^{31} + 1041566810058300 p^{105} T^{32} + 26249806 p^{112} T^{33} + p^{119} T^{34} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{34} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.72457213820629353152334023600, −1.69298751127431239938567328625, −1.62388659103032355425682398722, −1.59153133506467045143530716566, −1.55946600107316248983192049630, −1.44886491987207639609741360766, −1.44541693781794139674085200592, −1.31123977586898618859132542878, −1.25394845255747029777790641318, −1.12772725536828263795942129337, −1.04153867641568880216658717569, −0.961940067735263229870880578500, −0.822318140945129499003997876317, −0.69896490452828524142793584359, −0.69670242489744105333993569248, −0.67538919703999010564868657626, −0.65054063078390299829798737826, −0.55254259129790807462441121762, −0.51703408605922632415201744884, −0.41383904490772805646686009437, −0.39648726521804082899851102717, −0.28888081924939630822912019035, −0.18236111938158182420682333919, −0.10164685246247913541882971584, −0.04150207403620255132075027165, 0.04150207403620255132075027165, 0.10164685246247913541882971584, 0.18236111938158182420682333919, 0.28888081924939630822912019035, 0.39648726521804082899851102717, 0.41383904490772805646686009437, 0.51703408605922632415201744884, 0.55254259129790807462441121762, 0.65054063078390299829798737826, 0.67538919703999010564868657626, 0.69670242489744105333993569248, 0.69896490452828524142793584359, 0.822318140945129499003997876317, 0.961940067735263229870880578500, 1.04153867641568880216658717569, 1.12772725536828263795942129337, 1.25394845255747029777790641318, 1.31123977586898618859132542878, 1.44541693781794139674085200592, 1.44886491987207639609741360766, 1.55946600107316248983192049630, 1.59153133506467045143530716566, 1.62388659103032355425682398722, 1.69298751127431239938567328625, 1.72457213820629353152334023600

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.