Properties

Label 34-177e17-1.1-c7e17-0-0
Degree $34$
Conductor $1.643\times 10^{38}$
Sign $1$
Analytic cond. $4.21967\times 10^{29}$
Root an. cond. $7.43586$
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  + 2·2-s − 459·3-s − 503·4-s − 318·5-s − 918·6-s + 3.14e3·7-s − 309·8-s + 1.11e5·9-s − 636·10-s − 1.76e3·11-s + 2.30e5·12-s + 1.81e4·13-s + 6.29e3·14-s + 1.45e5·15-s + 1.25e5·16-s − 1.55e4·17-s + 2.23e5·18-s + 5.20e4·19-s + 1.59e5·20-s − 1.44e6·21-s − 3.52e3·22-s + 6.38e4·23-s + 1.41e5·24-s − 5.12e5·25-s + 3.63e4·26-s − 1.90e7·27-s − 1.58e6·28-s + ⋯
L(s)  = 1  + 0.176·2-s − 9.81·3-s − 3.92·4-s − 1.13·5-s − 1.73·6-s + 3.46·7-s − 0.213·8-s + 51·9-s − 0.201·10-s − 0.399·11-s + 38.5·12-s + 2.29·13-s + 0.612·14-s + 11.1·15-s + 7.65·16-s − 0.765·17-s + 9.01·18-s + 1.74·19-s + 4.47·20-s − 34.0·21-s − 0.0706·22-s + 1.09·23-s + 2.09·24-s − 6.55·25-s + 0.405·26-s − 186.·27-s − 13.6·28-s + ⋯

Functional equation

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

Invariants

Degree: \(34\)
Conductor: \(3^{17} \cdot 59^{17}\)
Sign: $1$
Analytic conductor: \(4.21967\times 10^{29}\)
Root analytic conductor: \(7.43586\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{177} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((34,\ 3^{17} \cdot 59^{17} ,\ ( \ : [7/2]^{17} ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(0.02106206568\)
\(L(\frac12)\) \(\approx\) \(0.02106206568\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + p^{3} T )^{17} \)
59 \( ( 1 + p^{3} T )^{17} \)
good2 \( 1 - p T + 507 T^{2} - 1711 T^{3} + 33107 p^{2} T^{4} - 506371 T^{5} + 5444619 p^{2} T^{6} - 5182067 p^{4} T^{7} + 148316755 p^{4} T^{8} - 51373675 p^{7} T^{9} + 2900106549 p^{6} T^{10} - 189145853 p^{7} T^{11} + 83644053581 p^{8} T^{12} + 132986999079 p^{8} T^{13} + 1263692235687 p^{12} T^{14} - 1021963748489 p^{11} T^{15} + 62311201249131 p^{14} T^{16} - 26113024812171 p^{15} T^{17} + 62311201249131 p^{21} T^{18} - 1021963748489 p^{25} T^{19} + 1263692235687 p^{33} T^{20} + 132986999079 p^{36} T^{21} + 83644053581 p^{43} T^{22} - 189145853 p^{49} T^{23} + 2900106549 p^{55} T^{24} - 51373675 p^{63} T^{25} + 148316755 p^{67} T^{26} - 5182067 p^{74} T^{27} + 5444619 p^{79} T^{28} - 506371 p^{84} T^{29} + 33107 p^{93} T^{30} - 1711 p^{98} T^{31} + 507 p^{105} T^{32} - p^{113} T^{33} + p^{119} T^{34} \)
5 \( 1 + 318 T + 613548 T^{2} + 201978956 T^{3} + 39851404977 p T^{4} + 13143095502816 p T^{5} + 45514592136451157 T^{6} + 14561716969709899908 T^{7} + \)\(16\!\cdots\!48\)\( p T^{8} + \)\(19\!\cdots\!94\)\( p^{3} T^{9} + \)\(94\!\cdots\!49\)\( p^{3} T^{10} + \)\(54\!\cdots\!68\)\( p^{4} T^{11} + \)\(45\!\cdots\!66\)\( p^{5} T^{12} + \)\(24\!\cdots\!88\)\( p^{6} T^{13} + \)\(19\!\cdots\!56\)\( p^{7} T^{14} + \)\(97\!\cdots\!96\)\( p^{8} T^{15} + \)\(13\!\cdots\!42\)\( p^{10} T^{16} + \)\(32\!\cdots\!52\)\( p^{10} T^{17} + \)\(13\!\cdots\!42\)\( p^{17} T^{18} + \)\(97\!\cdots\!96\)\( p^{22} T^{19} + \)\(19\!\cdots\!56\)\( p^{28} T^{20} + \)\(24\!\cdots\!88\)\( p^{34} T^{21} + \)\(45\!\cdots\!66\)\( p^{40} T^{22} + \)\(54\!\cdots\!68\)\( p^{46} T^{23} + \)\(94\!\cdots\!49\)\( p^{52} T^{24} + \)\(19\!\cdots\!94\)\( p^{59} T^{25} + \)\(16\!\cdots\!48\)\( p^{64} T^{26} + 14561716969709899908 p^{70} T^{27} + 45514592136451157 p^{77} T^{28} + 13143095502816 p^{85} T^{29} + 39851404977 p^{92} T^{30} + 201978956 p^{98} T^{31} + 613548 p^{105} T^{32} + 318 p^{112} T^{33} + p^{119} T^{34} \)
7 \( 1 - 3145 T + 10998087 T^{2} - 24027636985 T^{3} + 52596107155668 T^{4} - 91208847043334611 T^{5} + \)\(15\!\cdots\!10\)\( T^{6} - \)\(22\!\cdots\!06\)\( T^{7} + \)\(33\!\cdots\!10\)\( T^{8} - \)\(42\!\cdots\!15\)\( T^{9} + \)\(54\!\cdots\!74\)\( T^{10} - \)\(64\!\cdots\!11\)\( T^{11} + \)\(73\!\cdots\!67\)\( T^{12} - \)\(78\!\cdots\!91\)\( T^{13} + \)\(83\!\cdots\!95\)\( T^{14} - \)\(11\!\cdots\!06\)\( p T^{15} + \)\(80\!\cdots\!00\)\( T^{16} - \)\(73\!\cdots\!44\)\( T^{17} + \)\(80\!\cdots\!00\)\( p^{7} T^{18} - \)\(11\!\cdots\!06\)\( p^{15} T^{19} + \)\(83\!\cdots\!95\)\( p^{21} T^{20} - \)\(78\!\cdots\!91\)\( p^{28} T^{21} + \)\(73\!\cdots\!67\)\( p^{35} T^{22} - \)\(64\!\cdots\!11\)\( p^{42} T^{23} + \)\(54\!\cdots\!74\)\( p^{49} T^{24} - \)\(42\!\cdots\!15\)\( p^{56} T^{25} + \)\(33\!\cdots\!10\)\( p^{63} T^{26} - \)\(22\!\cdots\!06\)\( p^{70} T^{27} + \)\(15\!\cdots\!10\)\( p^{77} T^{28} - 91208847043334611 p^{84} T^{29} + 52596107155668 p^{91} T^{30} - 24027636985 p^{98} T^{31} + 10998087 p^{105} T^{32} - 3145 p^{112} T^{33} + p^{119} T^{34} \)
11 \( 1 + 1764 T + 169177313 T^{2} + 162361613596 T^{3} + 1298350627847479 p T^{4} + 3513789552790827976 T^{5} + \)\(80\!\cdots\!31\)\( T^{6} - \)\(31\!\cdots\!08\)\( T^{7} + \)\(31\!\cdots\!13\)\( p T^{8} - \)\(32\!\cdots\!32\)\( T^{9} + \)\(12\!\cdots\!63\)\( T^{10} - \)\(16\!\cdots\!72\)\( T^{11} + \)\(35\!\cdots\!42\)\( T^{12} - \)\(59\!\cdots\!24\)\( T^{13} + \)\(88\!\cdots\!78\)\( T^{14} - \)\(16\!\cdots\!76\)\( T^{15} + \)\(19\!\cdots\!66\)\( T^{16} - \)\(35\!\cdots\!80\)\( T^{17} + \)\(19\!\cdots\!66\)\( p^{7} T^{18} - \)\(16\!\cdots\!76\)\( p^{14} T^{19} + \)\(88\!\cdots\!78\)\( p^{21} T^{20} - \)\(59\!\cdots\!24\)\( p^{28} T^{21} + \)\(35\!\cdots\!42\)\( p^{35} T^{22} - \)\(16\!\cdots\!72\)\( p^{42} T^{23} + \)\(12\!\cdots\!63\)\( p^{49} T^{24} - \)\(32\!\cdots\!32\)\( p^{56} T^{25} + \)\(31\!\cdots\!13\)\( p^{64} T^{26} - \)\(31\!\cdots\!08\)\( p^{70} T^{27} + \)\(80\!\cdots\!31\)\( p^{77} T^{28} + 3513789552790827976 p^{84} T^{29} + 1298350627847479 p^{92} T^{30} + 162361613596 p^{98} T^{31} + 169177313 p^{105} T^{32} + 1764 p^{112} T^{33} + p^{119} T^{34} \)
13 \( 1 - 18192 T + 622149345 T^{2} - 8467021061036 T^{3} + 169600365007363819 T^{4} - \)\(18\!\cdots\!44\)\( T^{5} + \)\(22\!\cdots\!33\)\( p T^{6} - \)\(27\!\cdots\!00\)\( T^{7} + \)\(36\!\cdots\!47\)\( T^{8} - \)\(31\!\cdots\!40\)\( T^{9} + \)\(29\!\cdots\!43\)\( p T^{10} - \)\(18\!\cdots\!88\)\( p^{2} T^{11} + \)\(35\!\cdots\!56\)\( T^{12} - \)\(26\!\cdots\!40\)\( T^{13} + \)\(28\!\cdots\!68\)\( T^{14} - \)\(19\!\cdots\!96\)\( T^{15} + \)\(19\!\cdots\!72\)\( T^{16} - \)\(12\!\cdots\!24\)\( T^{17} + \)\(19\!\cdots\!72\)\( p^{7} T^{18} - \)\(19\!\cdots\!96\)\( p^{14} T^{19} + \)\(28\!\cdots\!68\)\( p^{21} T^{20} - \)\(26\!\cdots\!40\)\( p^{28} T^{21} + \)\(35\!\cdots\!56\)\( p^{35} T^{22} - \)\(18\!\cdots\!88\)\( p^{44} T^{23} + \)\(29\!\cdots\!43\)\( p^{50} T^{24} - \)\(31\!\cdots\!40\)\( p^{56} T^{25} + \)\(36\!\cdots\!47\)\( p^{63} T^{26} - \)\(27\!\cdots\!00\)\( p^{70} T^{27} + \)\(22\!\cdots\!33\)\( p^{78} T^{28} - \)\(18\!\cdots\!44\)\( p^{84} T^{29} + 169600365007363819 p^{91} T^{30} - 8467021061036 p^{98} T^{31} + 622149345 p^{105} T^{32} - 18192 p^{112} T^{33} + p^{119} T^{34} \)
17 \( 1 + 15507 T + 1880005649 T^{2} + 48932663290079 T^{3} + 2142551039520689390 T^{4} + \)\(60\!\cdots\!87\)\( T^{5} + \)\(20\!\cdots\!52\)\( T^{6} + \)\(51\!\cdots\!14\)\( T^{7} + \)\(15\!\cdots\!92\)\( T^{8} + \)\(36\!\cdots\!07\)\( T^{9} + \)\(94\!\cdots\!20\)\( T^{10} + \)\(21\!\cdots\!45\)\( T^{11} + \)\(51\!\cdots\!11\)\( T^{12} + \)\(11\!\cdots\!49\)\( T^{13} + \)\(25\!\cdots\!13\)\( T^{14} + \)\(51\!\cdots\!70\)\( T^{15} + \)\(11\!\cdots\!16\)\( T^{16} + \)\(22\!\cdots\!52\)\( T^{17} + \)\(11\!\cdots\!16\)\( p^{7} T^{18} + \)\(51\!\cdots\!70\)\( p^{14} T^{19} + \)\(25\!\cdots\!13\)\( p^{21} T^{20} + \)\(11\!\cdots\!49\)\( p^{28} T^{21} + \)\(51\!\cdots\!11\)\( p^{35} T^{22} + \)\(21\!\cdots\!45\)\( p^{42} T^{23} + \)\(94\!\cdots\!20\)\( p^{49} T^{24} + \)\(36\!\cdots\!07\)\( p^{56} T^{25} + \)\(15\!\cdots\!92\)\( p^{63} T^{26} + \)\(51\!\cdots\!14\)\( p^{70} T^{27} + \)\(20\!\cdots\!52\)\( p^{77} T^{28} + \)\(60\!\cdots\!87\)\( p^{84} T^{29} + 2142551039520689390 p^{91} T^{30} + 48932663290079 p^{98} T^{31} + 1880005649 p^{105} T^{32} + 15507 p^{112} T^{33} + p^{119} T^{34} \)
19 \( 1 - 52083 T + 9321462028 T^{2} - 357804508845938 T^{3} + 39080364906460910122 T^{4} - \)\(11\!\cdots\!69\)\( T^{5} + \)\(10\!\cdots\!91\)\( T^{6} - \)\(21\!\cdots\!80\)\( T^{7} + \)\(18\!\cdots\!69\)\( T^{8} - \)\(25\!\cdots\!69\)\( T^{9} + \)\(25\!\cdots\!64\)\( T^{10} - \)\(14\!\cdots\!08\)\( T^{11} + \)\(28\!\cdots\!92\)\( T^{12} + \)\(99\!\cdots\!45\)\( T^{13} + \)\(25\!\cdots\!99\)\( T^{14} + \)\(33\!\cdots\!10\)\( T^{15} + \)\(22\!\cdots\!70\)\( T^{16} + \)\(39\!\cdots\!44\)\( T^{17} + \)\(22\!\cdots\!70\)\( p^{7} T^{18} + \)\(33\!\cdots\!10\)\( p^{14} T^{19} + \)\(25\!\cdots\!99\)\( p^{21} T^{20} + \)\(99\!\cdots\!45\)\( p^{28} T^{21} + \)\(28\!\cdots\!92\)\( p^{35} T^{22} - \)\(14\!\cdots\!08\)\( p^{42} T^{23} + \)\(25\!\cdots\!64\)\( p^{49} T^{24} - \)\(25\!\cdots\!69\)\( p^{56} T^{25} + \)\(18\!\cdots\!69\)\( p^{63} T^{26} - \)\(21\!\cdots\!80\)\( p^{70} T^{27} + \)\(10\!\cdots\!91\)\( p^{77} T^{28} - \)\(11\!\cdots\!69\)\( p^{84} T^{29} + 39080364906460910122 p^{91} T^{30} - 357804508845938 p^{98} T^{31} + 9321462028 p^{105} T^{32} - 52083 p^{112} T^{33} + p^{119} T^{34} \)
23 \( 1 - 63823 T + 30946140198 T^{2} - 1616117332719058 T^{3} + \)\(45\!\cdots\!84\)\( T^{4} - \)\(19\!\cdots\!53\)\( T^{5} + \)\(42\!\cdots\!75\)\( T^{6} - \)\(16\!\cdots\!60\)\( T^{7} + \)\(29\!\cdots\!37\)\( T^{8} - \)\(10\!\cdots\!17\)\( T^{9} + \)\(16\!\cdots\!98\)\( T^{10} - \)\(57\!\cdots\!32\)\( T^{11} + \)\(80\!\cdots\!94\)\( T^{12} - \)\(27\!\cdots\!35\)\( T^{13} + \)\(33\!\cdots\!67\)\( T^{14} - \)\(11\!\cdots\!54\)\( T^{15} + \)\(12\!\cdots\!58\)\( T^{16} - \)\(18\!\cdots\!20\)\( p T^{17} + \)\(12\!\cdots\!58\)\( p^{7} T^{18} - \)\(11\!\cdots\!54\)\( p^{14} T^{19} + \)\(33\!\cdots\!67\)\( p^{21} T^{20} - \)\(27\!\cdots\!35\)\( p^{28} T^{21} + \)\(80\!\cdots\!94\)\( p^{35} T^{22} - \)\(57\!\cdots\!32\)\( p^{42} T^{23} + \)\(16\!\cdots\!98\)\( p^{49} T^{24} - \)\(10\!\cdots\!17\)\( p^{56} T^{25} + \)\(29\!\cdots\!37\)\( p^{63} T^{26} - \)\(16\!\cdots\!60\)\( p^{70} T^{27} + \)\(42\!\cdots\!75\)\( p^{77} T^{28} - \)\(19\!\cdots\!53\)\( p^{84} T^{29} + \)\(45\!\cdots\!84\)\( p^{91} T^{30} - 1616117332719058 p^{98} T^{31} + 30946140198 p^{105} T^{32} - 63823 p^{112} T^{33} + p^{119} T^{34} \)
29 \( 1 + 502955 T + 9021163774 p T^{2} + 89549671281564820 T^{3} + \)\(29\!\cdots\!74\)\( T^{4} + \)\(78\!\cdots\!89\)\( T^{5} + \)\(19\!\cdots\!39\)\( T^{6} + \)\(44\!\cdots\!12\)\( T^{7} + \)\(95\!\cdots\!47\)\( T^{8} + \)\(18\!\cdots\!73\)\( T^{9} + \)\(35\!\cdots\!00\)\( T^{10} + \)\(61\!\cdots\!78\)\( T^{11} + \)\(10\!\cdots\!42\)\( T^{12} + \)\(16\!\cdots\!31\)\( T^{13} + \)\(25\!\cdots\!97\)\( T^{14} + \)\(36\!\cdots\!02\)\( T^{15} + \)\(51\!\cdots\!38\)\( T^{16} + \)\(69\!\cdots\!60\)\( T^{17} + \)\(51\!\cdots\!38\)\( p^{7} T^{18} + \)\(36\!\cdots\!02\)\( p^{14} T^{19} + \)\(25\!\cdots\!97\)\( p^{21} T^{20} + \)\(16\!\cdots\!31\)\( p^{28} T^{21} + \)\(10\!\cdots\!42\)\( p^{35} T^{22} + \)\(61\!\cdots\!78\)\( p^{42} T^{23} + \)\(35\!\cdots\!00\)\( p^{49} T^{24} + \)\(18\!\cdots\!73\)\( p^{56} T^{25} + \)\(95\!\cdots\!47\)\( p^{63} T^{26} + \)\(44\!\cdots\!12\)\( p^{70} T^{27} + \)\(19\!\cdots\!39\)\( p^{77} T^{28} + \)\(78\!\cdots\!89\)\( p^{84} T^{29} + \)\(29\!\cdots\!74\)\( p^{91} T^{30} + 89549671281564820 p^{98} T^{31} + 9021163774 p^{106} T^{32} + 502955 p^{112} T^{33} + p^{119} T^{34} \)
31 \( 1 - 347531 T + 228385119580 T^{2} - 57720245587264410 T^{3} + \)\(22\!\cdots\!48\)\( T^{4} - \)\(45\!\cdots\!61\)\( T^{5} + \)\(14\!\cdots\!37\)\( T^{6} - \)\(23\!\cdots\!36\)\( T^{7} + \)\(66\!\cdots\!67\)\( T^{8} - \)\(93\!\cdots\!61\)\( T^{9} + \)\(26\!\cdots\!22\)\( T^{10} - \)\(32\!\cdots\!76\)\( T^{11} + \)\(92\!\cdots\!40\)\( T^{12} - \)\(10\!\cdots\!51\)\( T^{13} + \)\(29\!\cdots\!79\)\( T^{14} - \)\(30\!\cdots\!06\)\( T^{15} + \)\(86\!\cdots\!26\)\( T^{16} - \)\(84\!\cdots\!76\)\( T^{17} + \)\(86\!\cdots\!26\)\( p^{7} T^{18} - \)\(30\!\cdots\!06\)\( p^{14} T^{19} + \)\(29\!\cdots\!79\)\( p^{21} T^{20} - \)\(10\!\cdots\!51\)\( p^{28} T^{21} + \)\(92\!\cdots\!40\)\( p^{35} T^{22} - \)\(32\!\cdots\!76\)\( p^{42} T^{23} + \)\(26\!\cdots\!22\)\( p^{49} T^{24} - \)\(93\!\cdots\!61\)\( p^{56} T^{25} + \)\(66\!\cdots\!67\)\( p^{63} T^{26} - \)\(23\!\cdots\!36\)\( p^{70} T^{27} + \)\(14\!\cdots\!37\)\( p^{77} T^{28} - \)\(45\!\cdots\!61\)\( p^{84} T^{29} + \)\(22\!\cdots\!48\)\( p^{91} T^{30} - 57720245587264410 p^{98} T^{31} + 228385119580 p^{105} T^{32} - 347531 p^{112} T^{33} + p^{119} T^{34} \)
37 \( 1 - 447615 T + 1016609263873 T^{2} - 447600181670795393 T^{3} + \)\(50\!\cdots\!38\)\( T^{4} - \)\(21\!\cdots\!55\)\( T^{5} + \)\(16\!\cdots\!66\)\( T^{6} - \)\(64\!\cdots\!74\)\( T^{7} + \)\(38\!\cdots\!42\)\( T^{8} - \)\(13\!\cdots\!23\)\( T^{9} + \)\(71\!\cdots\!12\)\( T^{10} - \)\(23\!\cdots\!83\)\( T^{11} + \)\(10\!\cdots\!27\)\( T^{12} - \)\(32\!\cdots\!37\)\( T^{13} + \)\(13\!\cdots\!05\)\( T^{14} - \)\(37\!\cdots\!14\)\( T^{15} + \)\(38\!\cdots\!12\)\( p T^{16} - \)\(37\!\cdots\!44\)\( T^{17} + \)\(38\!\cdots\!12\)\( p^{8} T^{18} - \)\(37\!\cdots\!14\)\( p^{14} T^{19} + \)\(13\!\cdots\!05\)\( p^{21} T^{20} - \)\(32\!\cdots\!37\)\( p^{28} T^{21} + \)\(10\!\cdots\!27\)\( p^{35} T^{22} - \)\(23\!\cdots\!83\)\( p^{42} T^{23} + \)\(71\!\cdots\!12\)\( p^{49} T^{24} - \)\(13\!\cdots\!23\)\( p^{56} T^{25} + \)\(38\!\cdots\!42\)\( p^{63} T^{26} - \)\(64\!\cdots\!74\)\( p^{70} T^{27} + \)\(16\!\cdots\!66\)\( p^{77} T^{28} - \)\(21\!\cdots\!55\)\( p^{84} T^{29} + \)\(50\!\cdots\!38\)\( p^{91} T^{30} - 447600181670795393 p^{98} T^{31} + 1016609263873 p^{105} T^{32} - 447615 p^{112} T^{33} + p^{119} T^{34} \)
41 \( 1 - 22935 p T + 2274390154061 T^{2} - 1679282554736996119 T^{3} + \)\(23\!\cdots\!18\)\( T^{4} - \)\(14\!\cdots\!47\)\( T^{5} + \)\(15\!\cdots\!08\)\( T^{6} - \)\(84\!\cdots\!74\)\( T^{7} + \)\(72\!\cdots\!28\)\( T^{8} - \)\(36\!\cdots\!55\)\( T^{9} + \)\(27\!\cdots\!36\)\( T^{10} - \)\(12\!\cdots\!45\)\( T^{11} + \)\(82\!\cdots\!71\)\( T^{12} - \)\(36\!\cdots\!73\)\( T^{13} + \)\(21\!\cdots\!61\)\( T^{14} - \)\(88\!\cdots\!38\)\( T^{15} + \)\(48\!\cdots\!52\)\( T^{16} - \)\(18\!\cdots\!00\)\( T^{17} + \)\(48\!\cdots\!52\)\( p^{7} T^{18} - \)\(88\!\cdots\!38\)\( p^{14} T^{19} + \)\(21\!\cdots\!61\)\( p^{21} T^{20} - \)\(36\!\cdots\!73\)\( p^{28} T^{21} + \)\(82\!\cdots\!71\)\( p^{35} T^{22} - \)\(12\!\cdots\!45\)\( p^{42} T^{23} + \)\(27\!\cdots\!36\)\( p^{49} T^{24} - \)\(36\!\cdots\!55\)\( p^{56} T^{25} + \)\(72\!\cdots\!28\)\( p^{63} T^{26} - \)\(84\!\cdots\!74\)\( p^{70} T^{27} + \)\(15\!\cdots\!08\)\( p^{77} T^{28} - \)\(14\!\cdots\!47\)\( p^{84} T^{29} + \)\(23\!\cdots\!18\)\( p^{91} T^{30} - 1679282554736996119 p^{98} T^{31} + 2274390154061 p^{105} T^{32} - 22935 p^{113} T^{33} + p^{119} T^{34} \)
43 \( 1 - 478562 T + 2899708225263 T^{2} - 1233303588264367974 T^{3} + \)\(40\!\cdots\!57\)\( T^{4} - \)\(15\!\cdots\!28\)\( T^{5} + \)\(37\!\cdots\!63\)\( T^{6} - \)\(30\!\cdots\!34\)\( p T^{7} + \)\(25\!\cdots\!81\)\( T^{8} - \)\(81\!\cdots\!54\)\( T^{9} + \)\(13\!\cdots\!87\)\( T^{10} - \)\(40\!\cdots\!80\)\( T^{11} + \)\(61\!\cdots\!46\)\( T^{12} - \)\(16\!\cdots\!64\)\( T^{13} + \)\(22\!\cdots\!66\)\( T^{14} - \)\(56\!\cdots\!56\)\( T^{15} + \)\(72\!\cdots\!54\)\( T^{16} - \)\(38\!\cdots\!52\)\( p T^{17} + \)\(72\!\cdots\!54\)\( p^{7} T^{18} - \)\(56\!\cdots\!56\)\( p^{14} T^{19} + \)\(22\!\cdots\!66\)\( p^{21} T^{20} - \)\(16\!\cdots\!64\)\( p^{28} T^{21} + \)\(61\!\cdots\!46\)\( p^{35} T^{22} - \)\(40\!\cdots\!80\)\( p^{42} T^{23} + \)\(13\!\cdots\!87\)\( p^{49} T^{24} - \)\(81\!\cdots\!54\)\( p^{56} T^{25} + \)\(25\!\cdots\!81\)\( p^{63} T^{26} - \)\(30\!\cdots\!34\)\( p^{71} T^{27} + \)\(37\!\cdots\!63\)\( p^{77} T^{28} - \)\(15\!\cdots\!28\)\( p^{84} T^{29} + \)\(40\!\cdots\!57\)\( p^{91} T^{30} - 1233303588264367974 p^{98} T^{31} + 2899708225263 p^{105} T^{32} - 478562 p^{112} T^{33} + p^{119} T^{34} \)
47 \( 1 - 703121 T + 4326064361200 T^{2} - 60206915045335618 p T^{3} + \)\(92\!\cdots\!70\)\( T^{4} - \)\(57\!\cdots\!83\)\( T^{5} + \)\(13\!\cdots\!35\)\( T^{6} - \)\(78\!\cdots\!88\)\( T^{7} + \)\(14\!\cdots\!25\)\( T^{8} - \)\(79\!\cdots\!59\)\( T^{9} + \)\(12\!\cdots\!84\)\( T^{10} - \)\(63\!\cdots\!88\)\( T^{11} + \)\(87\!\cdots\!64\)\( T^{12} - \)\(42\!\cdots\!81\)\( T^{13} + \)\(54\!\cdots\!55\)\( T^{14} - \)\(25\!\cdots\!50\)\( T^{15} + \)\(30\!\cdots\!66\)\( T^{16} - \)\(13\!\cdots\!16\)\( T^{17} + \)\(30\!\cdots\!66\)\( p^{7} T^{18} - \)\(25\!\cdots\!50\)\( p^{14} T^{19} + \)\(54\!\cdots\!55\)\( p^{21} T^{20} - \)\(42\!\cdots\!81\)\( p^{28} T^{21} + \)\(87\!\cdots\!64\)\( p^{35} T^{22} - \)\(63\!\cdots\!88\)\( p^{42} T^{23} + \)\(12\!\cdots\!84\)\( p^{49} T^{24} - \)\(79\!\cdots\!59\)\( p^{56} T^{25} + \)\(14\!\cdots\!25\)\( p^{63} T^{26} - \)\(78\!\cdots\!88\)\( p^{70} T^{27} + \)\(13\!\cdots\!35\)\( p^{77} T^{28} - \)\(57\!\cdots\!83\)\( p^{84} T^{29} + \)\(92\!\cdots\!70\)\( p^{91} T^{30} - 60206915045335618 p^{99} T^{31} + 4326064361200 p^{105} T^{32} - 703121 p^{112} T^{33} + p^{119} T^{34} \)
53 \( 1 + 1005974 T + 7497348324632 T^{2} + 5241523154844577932 T^{3} + \)\(24\!\cdots\!77\)\( T^{4} + \)\(86\!\cdots\!64\)\( T^{5} + \)\(49\!\cdots\!37\)\( T^{6} - \)\(24\!\cdots\!92\)\( T^{7} + \)\(77\!\cdots\!96\)\( T^{8} - \)\(28\!\cdots\!94\)\( T^{9} + \)\(12\!\cdots\!09\)\( T^{10} - \)\(47\!\cdots\!04\)\( T^{11} + \)\(20\!\cdots\!38\)\( T^{12} - \)\(64\!\cdots\!08\)\( T^{13} + \)\(29\!\cdots\!36\)\( T^{14} - \)\(11\!\cdots\!96\)\( T^{15} + \)\(34\!\cdots\!98\)\( T^{16} - \)\(15\!\cdots\!96\)\( T^{17} + \)\(34\!\cdots\!98\)\( p^{7} T^{18} - \)\(11\!\cdots\!96\)\( p^{14} T^{19} + \)\(29\!\cdots\!36\)\( p^{21} T^{20} - \)\(64\!\cdots\!08\)\( p^{28} T^{21} + \)\(20\!\cdots\!38\)\( p^{35} T^{22} - \)\(47\!\cdots\!04\)\( p^{42} T^{23} + \)\(12\!\cdots\!09\)\( p^{49} T^{24} - \)\(28\!\cdots\!94\)\( p^{56} T^{25} + \)\(77\!\cdots\!96\)\( p^{63} T^{26} - \)\(24\!\cdots\!92\)\( p^{70} T^{27} + \)\(49\!\cdots\!37\)\( p^{77} T^{28} + \)\(86\!\cdots\!64\)\( p^{84} T^{29} + \)\(24\!\cdots\!77\)\( p^{91} T^{30} + 5241523154844577932 p^{98} T^{31} + 7497348324632 p^{105} T^{32} + 1005974 p^{112} T^{33} + p^{119} T^{34} \)
61 \( 1 - 11510749 T + 100453494364256 T^{2} - \)\(64\!\cdots\!68\)\( T^{3} + \)\(34\!\cdots\!96\)\( T^{4} - \)\(16\!\cdots\!23\)\( T^{5} + \)\(66\!\cdots\!55\)\( T^{6} - \)\(24\!\cdots\!60\)\( T^{7} + \)\(82\!\cdots\!79\)\( T^{8} - \)\(25\!\cdots\!07\)\( T^{9} + \)\(73\!\cdots\!34\)\( T^{10} - \)\(19\!\cdots\!26\)\( T^{11} + \)\(48\!\cdots\!68\)\( T^{12} - \)\(11\!\cdots\!37\)\( T^{13} + \)\(24\!\cdots\!57\)\( T^{14} - \)\(51\!\cdots\!18\)\( T^{15} + \)\(99\!\cdots\!26\)\( T^{16} - \)\(18\!\cdots\!28\)\( T^{17} + \)\(99\!\cdots\!26\)\( p^{7} T^{18} - \)\(51\!\cdots\!18\)\( p^{14} T^{19} + \)\(24\!\cdots\!57\)\( p^{21} T^{20} - \)\(11\!\cdots\!37\)\( p^{28} T^{21} + \)\(48\!\cdots\!68\)\( p^{35} T^{22} - \)\(19\!\cdots\!26\)\( p^{42} T^{23} + \)\(73\!\cdots\!34\)\( p^{49} T^{24} - \)\(25\!\cdots\!07\)\( p^{56} T^{25} + \)\(82\!\cdots\!79\)\( p^{63} T^{26} - \)\(24\!\cdots\!60\)\( p^{70} T^{27} + \)\(66\!\cdots\!55\)\( p^{77} T^{28} - \)\(16\!\cdots\!23\)\( p^{84} T^{29} + \)\(34\!\cdots\!96\)\( p^{91} T^{30} - \)\(64\!\cdots\!68\)\( p^{98} T^{31} + 100453494364256 p^{105} T^{32} - 11510749 p^{112} T^{33} + p^{119} T^{34} \)
67 \( 1 - 14007144 T + 129191027948674 T^{2} - \)\(88\!\cdots\!28\)\( T^{3} + \)\(49\!\cdots\!25\)\( T^{4} - \)\(24\!\cdots\!20\)\( T^{5} + \)\(10\!\cdots\!91\)\( T^{6} - \)\(39\!\cdots\!36\)\( T^{7} + \)\(13\!\cdots\!30\)\( T^{8} - \)\(44\!\cdots\!92\)\( T^{9} + \)\(13\!\cdots\!67\)\( T^{10} - \)\(36\!\cdots\!60\)\( T^{11} + \)\(95\!\cdots\!30\)\( T^{12} - \)\(23\!\cdots\!08\)\( T^{13} + \)\(55\!\cdots\!36\)\( T^{14} - \)\(12\!\cdots\!00\)\( T^{15} + \)\(29\!\cdots\!06\)\( T^{16} - \)\(71\!\cdots\!40\)\( T^{17} + \)\(29\!\cdots\!06\)\( p^{7} T^{18} - \)\(12\!\cdots\!00\)\( p^{14} T^{19} + \)\(55\!\cdots\!36\)\( p^{21} T^{20} - \)\(23\!\cdots\!08\)\( p^{28} T^{21} + \)\(95\!\cdots\!30\)\( p^{35} T^{22} - \)\(36\!\cdots\!60\)\( p^{42} T^{23} + \)\(13\!\cdots\!67\)\( p^{49} T^{24} - \)\(44\!\cdots\!92\)\( p^{56} T^{25} + \)\(13\!\cdots\!30\)\( p^{63} T^{26} - \)\(39\!\cdots\!36\)\( p^{70} T^{27} + \)\(10\!\cdots\!91\)\( p^{77} T^{28} - \)\(24\!\cdots\!20\)\( p^{84} T^{29} + \)\(49\!\cdots\!25\)\( p^{91} T^{30} - \)\(88\!\cdots\!28\)\( p^{98} T^{31} + 129191027948674 p^{105} T^{32} - 14007144 p^{112} T^{33} + p^{119} T^{34} \)
71 \( 1 - 5229074 T + 69785484651195 T^{2} - \)\(25\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!79\)\( T^{4} - \)\(58\!\cdots\!86\)\( T^{5} + \)\(43\!\cdots\!91\)\( T^{6} - \)\(88\!\cdots\!36\)\( T^{7} + \)\(65\!\cdots\!43\)\( T^{8} - \)\(87\!\cdots\!86\)\( T^{9} + \)\(77\!\cdots\!73\)\( T^{10} - \)\(45\!\cdots\!84\)\( T^{11} + \)\(77\!\cdots\!40\)\( T^{12} + \)\(20\!\cdots\!00\)\( T^{13} + \)\(67\!\cdots\!28\)\( T^{14} + \)\(74\!\cdots\!80\)\( T^{15} + \)\(58\!\cdots\!64\)\( T^{16} + \)\(89\!\cdots\!56\)\( T^{17} + \)\(58\!\cdots\!64\)\( p^{7} T^{18} + \)\(74\!\cdots\!80\)\( p^{14} T^{19} + \)\(67\!\cdots\!28\)\( p^{21} T^{20} + \)\(20\!\cdots\!00\)\( p^{28} T^{21} + \)\(77\!\cdots\!40\)\( p^{35} T^{22} - \)\(45\!\cdots\!84\)\( p^{42} T^{23} + \)\(77\!\cdots\!73\)\( p^{49} T^{24} - \)\(87\!\cdots\!86\)\( p^{56} T^{25} + \)\(65\!\cdots\!43\)\( p^{63} T^{26} - \)\(88\!\cdots\!36\)\( p^{70} T^{27} + \)\(43\!\cdots\!91\)\( p^{77} T^{28} - \)\(58\!\cdots\!86\)\( p^{84} T^{29} + \)\(21\!\cdots\!79\)\( p^{91} T^{30} - \)\(25\!\cdots\!96\)\( p^{98} T^{31} + 69785484651195 p^{105} T^{32} - 5229074 p^{112} T^{33} + p^{119} T^{34} \)
73 \( 1 - 5452211 T + 123594067803620 T^{2} - \)\(61\!\cdots\!48\)\( T^{3} + \)\(74\!\cdots\!46\)\( T^{4} - \)\(34\!\cdots\!65\)\( T^{5} + \)\(29\!\cdots\!85\)\( T^{6} - \)\(12\!\cdots\!24\)\( T^{7} + \)\(87\!\cdots\!01\)\( T^{8} - \)\(34\!\cdots\!01\)\( T^{9} + \)\(20\!\cdots\!04\)\( T^{10} - \)\(10\!\cdots\!70\)\( p T^{11} + \)\(37\!\cdots\!92\)\( T^{12} - \)\(12\!\cdots\!91\)\( T^{13} + \)\(58\!\cdots\!29\)\( T^{14} - \)\(18\!\cdots\!50\)\( T^{15} + \)\(77\!\cdots\!30\)\( T^{16} - \)\(22\!\cdots\!32\)\( T^{17} + \)\(77\!\cdots\!30\)\( p^{7} T^{18} - \)\(18\!\cdots\!50\)\( p^{14} T^{19} + \)\(58\!\cdots\!29\)\( p^{21} T^{20} - \)\(12\!\cdots\!91\)\( p^{28} T^{21} + \)\(37\!\cdots\!92\)\( p^{35} T^{22} - \)\(10\!\cdots\!70\)\( p^{43} T^{23} + \)\(20\!\cdots\!04\)\( p^{49} T^{24} - \)\(34\!\cdots\!01\)\( p^{56} T^{25} + \)\(87\!\cdots\!01\)\( p^{63} T^{26} - \)\(12\!\cdots\!24\)\( p^{70} T^{27} + \)\(29\!\cdots\!85\)\( p^{77} T^{28} - \)\(34\!\cdots\!65\)\( p^{84} T^{29} + \)\(74\!\cdots\!46\)\( p^{91} T^{30} - \)\(61\!\cdots\!48\)\( p^{98} T^{31} + 123594067803620 p^{105} T^{32} - 5452211 p^{112} T^{33} + p^{119} T^{34} \)
79 \( 1 - 15275654 T + 259184787665727 T^{2} - \)\(27\!\cdots\!86\)\( T^{3} + \)\(29\!\cdots\!05\)\( T^{4} - \)\(24\!\cdots\!84\)\( T^{5} + \)\(20\!\cdots\!59\)\( T^{6} - \)\(14\!\cdots\!06\)\( T^{7} + \)\(10\!\cdots\!69\)\( T^{8} - \)\(64\!\cdots\!46\)\( T^{9} + \)\(39\!\cdots\!03\)\( T^{10} - \)\(22\!\cdots\!92\)\( T^{11} + \)\(12\!\cdots\!58\)\( T^{12} - \)\(65\!\cdots\!52\)\( T^{13} + \)\(33\!\cdots\!82\)\( T^{14} - \)\(15\!\cdots\!36\)\( T^{15} + \)\(74\!\cdots\!86\)\( T^{16} - \)\(32\!\cdots\!08\)\( T^{17} + \)\(74\!\cdots\!86\)\( p^{7} T^{18} - \)\(15\!\cdots\!36\)\( p^{14} T^{19} + \)\(33\!\cdots\!82\)\( p^{21} T^{20} - \)\(65\!\cdots\!52\)\( p^{28} T^{21} + \)\(12\!\cdots\!58\)\( p^{35} T^{22} - \)\(22\!\cdots\!92\)\( p^{42} T^{23} + \)\(39\!\cdots\!03\)\( p^{49} T^{24} - \)\(64\!\cdots\!46\)\( p^{56} T^{25} + \)\(10\!\cdots\!69\)\( p^{63} T^{26} - \)\(14\!\cdots\!06\)\( p^{70} T^{27} + \)\(20\!\cdots\!59\)\( p^{77} T^{28} - \)\(24\!\cdots\!84\)\( p^{84} T^{29} + \)\(29\!\cdots\!05\)\( p^{91} T^{30} - \)\(27\!\cdots\!86\)\( p^{98} T^{31} + 259184787665727 p^{105} T^{32} - 15275654 p^{112} T^{33} + p^{119} T^{34} \)
83 \( 1 - 7826609 T + 222788927048745 T^{2} - \)\(17\!\cdots\!03\)\( T^{3} + \)\(26\!\cdots\!14\)\( T^{4} - \)\(20\!\cdots\!43\)\( T^{5} + \)\(21\!\cdots\!18\)\( T^{6} - \)\(16\!\cdots\!18\)\( T^{7} + \)\(13\!\cdots\!02\)\( T^{8} - \)\(94\!\cdots\!07\)\( T^{9} + \)\(67\!\cdots\!88\)\( T^{10} - \)\(44\!\cdots\!93\)\( T^{11} + \)\(28\!\cdots\!93\)\( T^{12} - \)\(17\!\cdots\!83\)\( T^{13} + \)\(10\!\cdots\!79\)\( T^{14} - \)\(59\!\cdots\!78\)\( T^{15} + \)\(33\!\cdots\!68\)\( T^{16} - \)\(17\!\cdots\!52\)\( T^{17} + \)\(33\!\cdots\!68\)\( p^{7} T^{18} - \)\(59\!\cdots\!78\)\( p^{14} T^{19} + \)\(10\!\cdots\!79\)\( p^{21} T^{20} - \)\(17\!\cdots\!83\)\( p^{28} T^{21} + \)\(28\!\cdots\!93\)\( p^{35} T^{22} - \)\(44\!\cdots\!93\)\( p^{42} T^{23} + \)\(67\!\cdots\!88\)\( p^{49} T^{24} - \)\(94\!\cdots\!07\)\( p^{56} T^{25} + \)\(13\!\cdots\!02\)\( p^{63} T^{26} - \)\(16\!\cdots\!18\)\( p^{70} T^{27} + \)\(21\!\cdots\!18\)\( p^{77} T^{28} - \)\(20\!\cdots\!43\)\( p^{84} T^{29} + \)\(26\!\cdots\!14\)\( p^{91} T^{30} - \)\(17\!\cdots\!03\)\( p^{98} T^{31} + 222788927048745 p^{105} T^{32} - 7826609 p^{112} T^{33} + p^{119} T^{34} \)
89 \( 1 + 6436185 T + 566699800687054 T^{2} + \)\(35\!\cdots\!08\)\( T^{3} + \)\(15\!\cdots\!64\)\( T^{4} + \)\(94\!\cdots\!51\)\( T^{5} + \)\(27\!\cdots\!61\)\( T^{6} + \)\(16\!\cdots\!40\)\( T^{7} + \)\(38\!\cdots\!97\)\( p T^{8} + \)\(20\!\cdots\!59\)\( T^{9} + \)\(33\!\cdots\!38\)\( T^{10} + \)\(18\!\cdots\!66\)\( T^{11} + \)\(26\!\cdots\!46\)\( T^{12} + \)\(14\!\cdots\!13\)\( T^{13} + \)\(16\!\cdots\!09\)\( T^{14} + \)\(84\!\cdots\!54\)\( T^{15} + \)\(88\!\cdots\!90\)\( T^{16} + \)\(41\!\cdots\!08\)\( T^{17} + \)\(88\!\cdots\!90\)\( p^{7} T^{18} + \)\(84\!\cdots\!54\)\( p^{14} T^{19} + \)\(16\!\cdots\!09\)\( p^{21} T^{20} + \)\(14\!\cdots\!13\)\( p^{28} T^{21} + \)\(26\!\cdots\!46\)\( p^{35} T^{22} + \)\(18\!\cdots\!66\)\( p^{42} T^{23} + \)\(33\!\cdots\!38\)\( p^{49} T^{24} + \)\(20\!\cdots\!59\)\( p^{56} T^{25} + \)\(38\!\cdots\!97\)\( p^{64} T^{26} + \)\(16\!\cdots\!40\)\( p^{70} T^{27} + \)\(27\!\cdots\!61\)\( p^{77} T^{28} + \)\(94\!\cdots\!51\)\( p^{84} T^{29} + \)\(15\!\cdots\!64\)\( p^{91} T^{30} + \)\(35\!\cdots\!08\)\( p^{98} T^{31} + 566699800687054 p^{105} T^{32} + 6436185 p^{112} T^{33} + p^{119} T^{34} \)
97 \( 1 - 26377540 T + 1168711970753462 T^{2} - \)\(23\!\cdots\!54\)\( T^{3} + \)\(61\!\cdots\!13\)\( T^{4} - \)\(10\!\cdots\!12\)\( T^{5} + \)\(19\!\cdots\!89\)\( T^{6} - \)\(28\!\cdots\!02\)\( T^{7} + \)\(45\!\cdots\!94\)\( T^{8} - \)\(57\!\cdots\!16\)\( T^{9} + \)\(81\!\cdots\!21\)\( T^{10} - \)\(90\!\cdots\!28\)\( T^{11} + \)\(11\!\cdots\!26\)\( T^{12} - \)\(11\!\cdots\!96\)\( T^{13} + \)\(13\!\cdots\!88\)\( T^{14} - \)\(12\!\cdots\!44\)\( T^{15} + \)\(12\!\cdots\!70\)\( T^{16} - \)\(10\!\cdots\!76\)\( T^{17} + \)\(12\!\cdots\!70\)\( p^{7} T^{18} - \)\(12\!\cdots\!44\)\( p^{14} T^{19} + \)\(13\!\cdots\!88\)\( p^{21} T^{20} - \)\(11\!\cdots\!96\)\( p^{28} T^{21} + \)\(11\!\cdots\!26\)\( p^{35} T^{22} - \)\(90\!\cdots\!28\)\( p^{42} T^{23} + \)\(81\!\cdots\!21\)\( p^{49} T^{24} - \)\(57\!\cdots\!16\)\( p^{56} T^{25} + \)\(45\!\cdots\!94\)\( p^{63} T^{26} - \)\(28\!\cdots\!02\)\( p^{70} T^{27} + \)\(19\!\cdots\!89\)\( p^{77} T^{28} - \)\(10\!\cdots\!12\)\( p^{84} T^{29} + \)\(61\!\cdots\!13\)\( p^{91} T^{30} - \)\(23\!\cdots\!54\)\( p^{98} T^{31} + 1168711970753462 p^{105} T^{32} - 26377540 p^{112} T^{33} + p^{119} T^{34} \)
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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

−1.87925526490131357501477134275, −1.80195588374330216384607915709, −1.68464473817070593883085396031, −1.67486529059361995549623860224, −1.67387161851102881113617326463, −1.67144284774928147642343226479, −1.66709054492680548484409165125, −1.55866135331196834498945166944, −1.43865393110429910051739118575, −1.16199140779909799718345720033, −1.13160060623441683884505152388, −1.02026254511472121368843112605, −0.976296513947561666190648431919, −0.819779363042382687118424548634, −0.74490707756597320379114915980, −0.70334622483954518117393701212, −0.61977552223271549743630254131, −0.55822223480724481547935235938, −0.52393267577116903636853752988, −0.49463714486490465142525450717, −0.42055776672956387320373967960, −0.38786162398710739281472581645, −0.37132349566897025727966174151, −0.07762229341996068008928851978, −0.04779777867369096037625480846, 0.04779777867369096037625480846, 0.07762229341996068008928851978, 0.37132349566897025727966174151, 0.38786162398710739281472581645, 0.42055776672956387320373967960, 0.49463714486490465142525450717, 0.52393267577116903636853752988, 0.55822223480724481547935235938, 0.61977552223271549743630254131, 0.70334622483954518117393701212, 0.74490707756597320379114915980, 0.819779363042382687118424548634, 0.976296513947561666190648431919, 1.02026254511472121368843112605, 1.13160060623441683884505152388, 1.16199140779909799718345720033, 1.43865393110429910051739118575, 1.55866135331196834498945166944, 1.66709054492680548484409165125, 1.67144284774928147642343226479, 1.67387161851102881113617326463, 1.67486529059361995549623860224, 1.68464473817070593883085396031, 1.80195588374330216384607915709, 1.87925526490131357501477134275

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

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