Dirichlet series
| L(s) = 1 | − 5·2-s − 26·3-s − 367·4-s − 181·5-s + 130·6-s − 3.08e3·7-s + 2.48e3·8-s − 7.89e3·9-s + 905·10-s − 9.82e3·11-s + 9.54e3·12-s + 1.97e4·13-s + 1.54e4·14-s + 4.70e3·15-s + 5.94e4·16-s − 2.27e4·17-s + 3.94e4·18-s − 1.77e4·19-s + 6.64e4·20-s + 8.02e4·21-s + 4.91e4·22-s − 4.91e4·23-s − 6.47e4·24-s − 2.21e5·25-s − 9.88e4·26-s + 2.64e5·27-s + 1.13e6·28-s + ⋯ |
| L(s) = 1 | − 0.441·2-s − 0.555·3-s − 2.86·4-s − 0.647·5-s + 0.245·6-s − 3.40·7-s + 1.71·8-s − 3.61·9-s + 0.286·10-s − 2.22·11-s + 1.59·12-s + 2.49·13-s + 1.50·14-s + 0.360·15-s + 3.62·16-s − 1.12·17-s + 1.59·18-s − 0.594·19-s + 1.85·20-s + 1.89·21-s + 0.983·22-s − 0.841·23-s − 0.955·24-s − 2.83·25-s − 1.10·26-s + 2.58·27-s + 9.75·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{9} \cdot 13^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{9} \cdot 13^{9}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(7^{9} \cdot 13^{9}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.21222\times 10^{13}\) |
| Root analytic conductor: | \(5.33170\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 7^{9} \cdot 13^{9} ,\ ( \ : [7/2]^{9} ),\ -1 )\) |
Particular Values
| \(L(4)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 + p^{3} T )^{9} \) |
| 13 | \( ( 1 - p^{3} T )^{9} \) | |
| good | 2 | \( 1 + 5 T + 49 p^{3} T^{2} + 653 p T^{3} + 19629 p^{2} T^{4} + 9375 p^{5} T^{5} + 376273 p^{5} T^{6} + 2149785 p^{5} T^{7} + 25356305 p^{6} T^{8} + 10635679 p^{10} T^{9} + 25356305 p^{13} T^{10} + 2149785 p^{19} T^{11} + 376273 p^{26} T^{12} + 9375 p^{33} T^{13} + 19629 p^{37} T^{14} + 653 p^{43} T^{15} + 49 p^{52} T^{16} + 5 p^{56} T^{17} + p^{63} T^{18} \) |
| 3 | \( 1 + 26 T + 2858 p T^{2} + 163570 T^{3} + 12778661 p T^{4} + 57179948 p^{2} T^{5} + 1495690811 p^{4} T^{6} + 16740682934 p^{4} T^{7} + 1295317382471 p^{5} T^{8} + 4359704104324 p^{6} T^{9} + 1295317382471 p^{12} T^{10} + 16740682934 p^{18} T^{11} + 1495690811 p^{25} T^{12} + 57179948 p^{30} T^{13} + 12778661 p^{36} T^{14} + 163570 p^{42} T^{15} + 2858 p^{50} T^{16} + 26 p^{56} T^{17} + p^{63} T^{18} \) | |
| 5 | \( 1 + 181 T + 50842 p T^{2} + 62887829 T^{3} + 39888390822 T^{4} + 1891745308313 p T^{5} + 179851684980331 p^{2} T^{6} + 7921758757101194 p^{3} T^{7} + 653750404035339472 p^{4} T^{8} + 26452497285441749994 p^{5} T^{9} + 653750404035339472 p^{11} T^{10} + 7921758757101194 p^{17} T^{11} + 179851684980331 p^{23} T^{12} + 1891745308313 p^{29} T^{13} + 39888390822 p^{35} T^{14} + 62887829 p^{42} T^{15} + 50842 p^{50} T^{16} + 181 p^{56} T^{17} + p^{63} T^{18} \) | |
| 11 | \( 1 + 9826 T + 149036972 T^{2} + 1161582473264 T^{3} + 10129159353676881 T^{4} + 63957247060353262648 T^{5} + \)\(37\!\cdots\!15\)\( p T^{6} + \)\(21\!\cdots\!52\)\( T^{7} + \)\(11\!\cdots\!03\)\( T^{8} + \)\(50\!\cdots\!80\)\( T^{9} + \)\(11\!\cdots\!03\)\( p^{7} T^{10} + \)\(21\!\cdots\!52\)\( p^{14} T^{11} + \)\(37\!\cdots\!15\)\( p^{22} T^{12} + 63957247060353262648 p^{28} T^{13} + 10129159353676881 p^{35} T^{14} + 1161582473264 p^{42} T^{15} + 149036972 p^{49} T^{16} + 9826 p^{56} T^{17} + p^{63} T^{18} \) | |
| 17 | \( 1 + 22766 T + 2861926361 T^{2} + 65328444064744 T^{3} + 3830373581918849064 T^{4} + \)\(85\!\cdots\!92\)\( T^{5} + \)\(31\!\cdots\!72\)\( T^{6} + \)\(65\!\cdots\!72\)\( T^{7} + \)\(18\!\cdots\!26\)\( T^{8} + \)\(33\!\cdots\!84\)\( T^{9} + \)\(18\!\cdots\!26\)\( p^{7} T^{10} + \)\(65\!\cdots\!72\)\( p^{14} T^{11} + \)\(31\!\cdots\!72\)\( p^{21} T^{12} + \)\(85\!\cdots\!92\)\( p^{28} T^{13} + 3830373581918849064 p^{35} T^{14} + 65328444064744 p^{42} T^{15} + 2861926361 p^{49} T^{16} + 22766 p^{56} T^{17} + p^{63} T^{18} \) | |
| 19 | \( 1 + 17769 T + 3592327200 T^{2} + 5663481166663 p T^{3} + 7030191564607880880 T^{4} + \)\(25\!\cdots\!85\)\( T^{5} + \)\(10\!\cdots\!93\)\( T^{6} + \)\(34\!\cdots\!14\)\( T^{7} + \)\(13\!\cdots\!92\)\( T^{8} + \)\(34\!\cdots\!98\)\( T^{9} + \)\(13\!\cdots\!92\)\( p^{7} T^{10} + \)\(34\!\cdots\!14\)\( p^{14} T^{11} + \)\(10\!\cdots\!93\)\( p^{21} T^{12} + \)\(25\!\cdots\!85\)\( p^{28} T^{13} + 7030191564607880880 p^{35} T^{14} + 5663481166663 p^{43} T^{15} + 3592327200 p^{49} T^{16} + 17769 p^{56} T^{17} + p^{63} T^{18} \) | |
| 23 | \( 1 + 49103 T + 23714433803 T^{2} + 1089573613925612 T^{3} + \)\(27\!\cdots\!14\)\( T^{4} + \)\(11\!\cdots\!38\)\( T^{5} + \)\(19\!\cdots\!46\)\( T^{6} + \)\(70\!\cdots\!68\)\( T^{7} + \)\(93\!\cdots\!59\)\( T^{8} + \)\(29\!\cdots\!33\)\( T^{9} + \)\(93\!\cdots\!59\)\( p^{7} T^{10} + \)\(70\!\cdots\!68\)\( p^{14} T^{11} + \)\(19\!\cdots\!46\)\( p^{21} T^{12} + \)\(11\!\cdots\!38\)\( p^{28} T^{13} + \)\(27\!\cdots\!14\)\( p^{35} T^{14} + 1089573613925612 p^{42} T^{15} + 23714433803 p^{49} T^{16} + 49103 p^{56} T^{17} + p^{63} T^{18} \) | |
| 29 | \( 1 + 487455 T + 215553650538 T^{2} + 62198341218327107 T^{3} + \)\(16\!\cdots\!38\)\( T^{4} + \)\(33\!\cdots\!79\)\( T^{5} + \)\(62\!\cdots\!63\)\( T^{6} + \)\(10\!\cdots\!06\)\( T^{7} + \)\(15\!\cdots\!04\)\( T^{8} + \)\(21\!\cdots\!74\)\( T^{9} + \)\(15\!\cdots\!04\)\( p^{7} T^{10} + \)\(10\!\cdots\!06\)\( p^{14} T^{11} + \)\(62\!\cdots\!63\)\( p^{21} T^{12} + \)\(33\!\cdots\!79\)\( p^{28} T^{13} + \)\(16\!\cdots\!38\)\( p^{35} T^{14} + 62198341218327107 p^{42} T^{15} + 215553650538 p^{49} T^{16} + 487455 p^{56} T^{17} + p^{63} T^{18} \) | |
| 31 | \( 1 + 63843 T + 136044121359 T^{2} + 5262314268356752 T^{3} + \)\(91\!\cdots\!14\)\( T^{4} + \)\(15\!\cdots\!90\)\( T^{5} + \)\(13\!\cdots\!62\)\( p T^{6} + \)\(58\!\cdots\!68\)\( T^{7} + \)\(13\!\cdots\!19\)\( T^{8} - \)\(59\!\cdots\!35\)\( T^{9} + \)\(13\!\cdots\!19\)\( p^{7} T^{10} + \)\(58\!\cdots\!68\)\( p^{14} T^{11} + \)\(13\!\cdots\!62\)\( p^{22} T^{12} + \)\(15\!\cdots\!90\)\( p^{28} T^{13} + \)\(91\!\cdots\!14\)\( p^{35} T^{14} + 5262314268356752 p^{42} T^{15} + 136044121359 p^{49} T^{16} + 63843 p^{56} T^{17} + p^{63} T^{18} \) | |
| 37 | \( 1 + 796926 T + 569309959560 T^{2} + 249623265438378006 T^{3} + \)\(10\!\cdots\!01\)\( T^{4} + \)\(33\!\cdots\!44\)\( T^{5} + \)\(12\!\cdots\!91\)\( T^{6} + \)\(43\!\cdots\!58\)\( T^{7} + \)\(16\!\cdots\!39\)\( T^{8} + \)\(52\!\cdots\!84\)\( T^{9} + \)\(16\!\cdots\!39\)\( p^{7} T^{10} + \)\(43\!\cdots\!58\)\( p^{14} T^{11} + \)\(12\!\cdots\!91\)\( p^{21} T^{12} + \)\(33\!\cdots\!44\)\( p^{28} T^{13} + \)\(10\!\cdots\!01\)\( p^{35} T^{14} + 249623265438378006 p^{42} T^{15} + 569309959560 p^{49} T^{16} + 796926 p^{56} T^{17} + p^{63} T^{18} \) | |
| 41 | \( 1 + 1567546 T + 2421654205004 T^{2} + 2379656663703135038 T^{3} + \)\(21\!\cdots\!25\)\( T^{4} + \)\(15\!\cdots\!56\)\( T^{5} + \)\(10\!\cdots\!83\)\( T^{6} + \)\(61\!\cdots\!26\)\( T^{7} + \)\(32\!\cdots\!35\)\( T^{8} + \)\(14\!\cdots\!40\)\( T^{9} + \)\(32\!\cdots\!35\)\( p^{7} T^{10} + \)\(61\!\cdots\!26\)\( p^{14} T^{11} + \)\(10\!\cdots\!83\)\( p^{21} T^{12} + \)\(15\!\cdots\!56\)\( p^{28} T^{13} + \)\(21\!\cdots\!25\)\( p^{35} T^{14} + 2379656663703135038 p^{42} T^{15} + 2421654205004 p^{49} T^{16} + 1567546 p^{56} T^{17} + p^{63} T^{18} \) | |
| 43 | \( 1 + 277899 T + 1186311179664 T^{2} + 237010036296326543 T^{3} + \)\(58\!\cdots\!00\)\( T^{4} + \)\(10\!\cdots\!07\)\( T^{5} + \)\(16\!\cdots\!41\)\( T^{6} + \)\(40\!\cdots\!54\)\( T^{7} + \)\(34\!\cdots\!76\)\( T^{8} + \)\(12\!\cdots\!70\)\( T^{9} + \)\(34\!\cdots\!76\)\( p^{7} T^{10} + \)\(40\!\cdots\!54\)\( p^{14} T^{11} + \)\(16\!\cdots\!41\)\( p^{21} T^{12} + \)\(10\!\cdots\!07\)\( p^{28} T^{13} + \)\(58\!\cdots\!00\)\( p^{35} T^{14} + 237010036296326543 p^{42} T^{15} + 1186311179664 p^{49} T^{16} + 277899 p^{56} T^{17} + p^{63} T^{18} \) | |
| 47 | \( 1 - 1077367 T + 1869334436291 T^{2} - 1171229971997269364 T^{3} + \)\(15\!\cdots\!22\)\( T^{4} - \)\(72\!\cdots\!62\)\( T^{5} + \)\(10\!\cdots\!78\)\( T^{6} - \)\(45\!\cdots\!88\)\( T^{7} + \)\(66\!\cdots\!07\)\( T^{8} - \)\(26\!\cdots\!97\)\( T^{9} + \)\(66\!\cdots\!07\)\( p^{7} T^{10} - \)\(45\!\cdots\!88\)\( p^{14} T^{11} + \)\(10\!\cdots\!78\)\( p^{21} T^{12} - \)\(72\!\cdots\!62\)\( p^{28} T^{13} + \)\(15\!\cdots\!22\)\( p^{35} T^{14} - 1171229971997269364 p^{42} T^{15} + 1869334436291 p^{49} T^{16} - 1077367 p^{56} T^{17} + p^{63} T^{18} \) | |
| 53 | \( 1 + 7322659 T + 30874701005774 T^{2} + 91815856119771974095 T^{3} + \)\(21\!\cdots\!66\)\( T^{4} + \)\(41\!\cdots\!67\)\( T^{5} + \)\(69\!\cdots\!35\)\( T^{6} + \)\(10\!\cdots\!22\)\( T^{7} + \)\(13\!\cdots\!80\)\( T^{8} + \)\(15\!\cdots\!58\)\( T^{9} + \)\(13\!\cdots\!80\)\( p^{7} T^{10} + \)\(10\!\cdots\!22\)\( p^{14} T^{11} + \)\(69\!\cdots\!35\)\( p^{21} T^{12} + \)\(41\!\cdots\!67\)\( p^{28} T^{13} + \)\(21\!\cdots\!66\)\( p^{35} T^{14} + 91815856119771974095 p^{42} T^{15} + 30874701005774 p^{49} T^{16} + 7322659 p^{56} T^{17} + p^{63} T^{18} \) | |
| 59 | \( 1 + 169804 T + 11511206462567 T^{2} - 1933845725927155648 T^{3} + \)\(67\!\cdots\!92\)\( T^{4} - \)\(32\!\cdots\!04\)\( T^{5} + \)\(27\!\cdots\!92\)\( T^{6} - \)\(17\!\cdots\!12\)\( T^{7} + \)\(83\!\cdots\!46\)\( T^{8} - \)\(56\!\cdots\!92\)\( T^{9} + \)\(83\!\cdots\!46\)\( p^{7} T^{10} - \)\(17\!\cdots\!12\)\( p^{14} T^{11} + \)\(27\!\cdots\!92\)\( p^{21} T^{12} - \)\(32\!\cdots\!04\)\( p^{28} T^{13} + \)\(67\!\cdots\!92\)\( p^{35} T^{14} - 1933845725927155648 p^{42} T^{15} + 11511206462567 p^{49} T^{16} + 169804 p^{56} T^{17} + p^{63} T^{18} \) | |
| 61 | \( 1 + 6352284 T + 26405100148602 T^{2} + 83513736444473526462 T^{3} + \)\(22\!\cdots\!23\)\( T^{4} + \)\(54\!\cdots\!52\)\( T^{5} + \)\(12\!\cdots\!69\)\( T^{6} + \)\(24\!\cdots\!18\)\( T^{7} + \)\(47\!\cdots\!21\)\( T^{8} + \)\(87\!\cdots\!64\)\( T^{9} + \)\(47\!\cdots\!21\)\( p^{7} T^{10} + \)\(24\!\cdots\!18\)\( p^{14} T^{11} + \)\(12\!\cdots\!69\)\( p^{21} T^{12} + \)\(54\!\cdots\!52\)\( p^{28} T^{13} + \)\(22\!\cdots\!23\)\( p^{35} T^{14} + 83513736444473526462 p^{42} T^{15} + 26405100148602 p^{49} T^{16} + 6352284 p^{56} T^{17} + p^{63} T^{18} \) | |
| 67 | \( 1 - 921120 T + 25297841142456 T^{2} - 16668527009857462098 T^{3} + \)\(50\!\cdots\!51\)\( p T^{4} - \)\(17\!\cdots\!60\)\( T^{5} + \)\(33\!\cdots\!45\)\( T^{6} - \)\(12\!\cdots\!30\)\( T^{7} + \)\(24\!\cdots\!67\)\( T^{8} - \)\(77\!\cdots\!44\)\( T^{9} + \)\(24\!\cdots\!67\)\( p^{7} T^{10} - \)\(12\!\cdots\!30\)\( p^{14} T^{11} + \)\(33\!\cdots\!45\)\( p^{21} T^{12} - \)\(17\!\cdots\!60\)\( p^{28} T^{13} + \)\(50\!\cdots\!51\)\( p^{36} T^{14} - 16668527009857462098 p^{42} T^{15} + 25297841142456 p^{49} T^{16} - 921120 p^{56} T^{17} + p^{63} T^{18} \) | |
| 71 | \( 1 - 3786654 T + 55155750187671 T^{2} - \)\(18\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!84\)\( T^{4} - \)\(42\!\cdots\!92\)\( T^{5} + \)\(25\!\cdots\!20\)\( T^{6} - \)\(65\!\cdots\!40\)\( T^{7} + \)\(30\!\cdots\!54\)\( T^{8} - \)\(70\!\cdots\!44\)\( T^{9} + \)\(30\!\cdots\!54\)\( p^{7} T^{10} - \)\(65\!\cdots\!40\)\( p^{14} T^{11} + \)\(25\!\cdots\!20\)\( p^{21} T^{12} - \)\(42\!\cdots\!92\)\( p^{28} T^{13} + \)\(14\!\cdots\!84\)\( p^{35} T^{14} - \)\(18\!\cdots\!20\)\( p^{42} T^{15} + 55155750187671 p^{49} T^{16} - 3786654 p^{56} T^{17} + p^{63} T^{18} \) | |
| 73 | \( 1 - 5792889 T + 93995158533489 T^{2} - \)\(44\!\cdots\!84\)\( T^{3} + \)\(39\!\cdots\!86\)\( T^{4} - \)\(15\!\cdots\!82\)\( T^{5} + \)\(98\!\cdots\!50\)\( T^{6} - \)\(33\!\cdots\!16\)\( T^{7} + \)\(15\!\cdots\!51\)\( T^{8} - \)\(44\!\cdots\!95\)\( T^{9} + \)\(15\!\cdots\!51\)\( p^{7} T^{10} - \)\(33\!\cdots\!16\)\( p^{14} T^{11} + \)\(98\!\cdots\!50\)\( p^{21} T^{12} - \)\(15\!\cdots\!82\)\( p^{28} T^{13} + \)\(39\!\cdots\!86\)\( p^{35} T^{14} - \)\(44\!\cdots\!84\)\( p^{42} T^{15} + 93995158533489 p^{49} T^{16} - 5792889 p^{56} T^{17} + p^{63} T^{18} \) | |
| 79 | \( 1 - 3464037 T + 107081646021147 T^{2} - \)\(36\!\cdots\!44\)\( T^{3} + \)\(57\!\cdots\!14\)\( T^{4} - \)\(18\!\cdots\!86\)\( T^{5} + \)\(20\!\cdots\!82\)\( T^{6} - \)\(62\!\cdots\!84\)\( T^{7} + \)\(53\!\cdots\!71\)\( T^{8} - \)\(14\!\cdots\!19\)\( T^{9} + \)\(53\!\cdots\!71\)\( p^{7} T^{10} - \)\(62\!\cdots\!84\)\( p^{14} T^{11} + \)\(20\!\cdots\!82\)\( p^{21} T^{12} - \)\(18\!\cdots\!86\)\( p^{28} T^{13} + \)\(57\!\cdots\!14\)\( p^{35} T^{14} - \)\(36\!\cdots\!44\)\( p^{42} T^{15} + 107081646021147 p^{49} T^{16} - 3464037 p^{56} T^{17} + p^{63} T^{18} \) | |
| 83 | \( 1 - 6834945 T + 138640893549960 T^{2} - \)\(86\!\cdots\!09\)\( T^{3} + \)\(97\!\cdots\!24\)\( T^{4} - \)\(53\!\cdots\!49\)\( T^{5} + \)\(45\!\cdots\!41\)\( T^{6} - \)\(21\!\cdots\!70\)\( T^{7} + \)\(15\!\cdots\!84\)\( T^{8} - \)\(67\!\cdots\!30\)\( T^{9} + \)\(15\!\cdots\!84\)\( p^{7} T^{10} - \)\(21\!\cdots\!70\)\( p^{14} T^{11} + \)\(45\!\cdots\!41\)\( p^{21} T^{12} - \)\(53\!\cdots\!49\)\( p^{28} T^{13} + \)\(97\!\cdots\!24\)\( p^{35} T^{14} - \)\(86\!\cdots\!09\)\( p^{42} T^{15} + 138640893549960 p^{49} T^{16} - 6834945 p^{56} T^{17} + p^{63} T^{18} \) | |
| 89 | \( 1 + 20408371 T + 3989035507366 p T^{2} + \)\(46\!\cdots\!95\)\( T^{3} + \)\(54\!\cdots\!86\)\( T^{4} + \)\(54\!\cdots\!79\)\( T^{5} + \)\(50\!\cdots\!27\)\( T^{6} + \)\(41\!\cdots\!82\)\( T^{7} + \)\(31\!\cdots\!96\)\( T^{8} + \)\(21\!\cdots\!10\)\( T^{9} + \)\(31\!\cdots\!96\)\( p^{7} T^{10} + \)\(41\!\cdots\!82\)\( p^{14} T^{11} + \)\(50\!\cdots\!27\)\( p^{21} T^{12} + \)\(54\!\cdots\!79\)\( p^{28} T^{13} + \)\(54\!\cdots\!86\)\( p^{35} T^{14} + \)\(46\!\cdots\!95\)\( p^{42} T^{15} + 3989035507366 p^{50} T^{16} + 20408371 p^{56} T^{17} + p^{63} T^{18} \) | |
| 97 | \( 1 - 41644125 T + 982790355598281 T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!66\)\( T^{4} - \)\(34\!\cdots\!78\)\( T^{5} + \)\(41\!\cdots\!62\)\( T^{6} - \)\(46\!\cdots\!00\)\( T^{7} + \)\(46\!\cdots\!83\)\( T^{8} - \)\(43\!\cdots\!39\)\( T^{9} + \)\(46\!\cdots\!83\)\( p^{7} T^{10} - \)\(46\!\cdots\!00\)\( p^{14} T^{11} + \)\(41\!\cdots\!62\)\( p^{21} T^{12} - \)\(34\!\cdots\!78\)\( p^{28} T^{13} + \)\(25\!\cdots\!66\)\( p^{35} T^{14} - \)\(17\!\cdots\!00\)\( p^{42} T^{15} + 982790355598281 p^{49} T^{16} - 41644125 p^{56} T^{17} + p^{63} T^{18} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.19175145188198726297973956519, −5.13978625374325176944667867056, −5.10347929398800207091980518823, −4.94192139842278689718592259196, −4.79673481247408652321443049339, −4.56170797026164947582303115591, −4.21673779000135756440304330459, −3.89672815502102575638541121161, −3.87149442873373892919446498246, −3.67999195039187100354863123877, −3.60085473309128521613366773934, −3.57038044990405343388975565955, −3.52663397569566864305344730612, −3.30535359078773361311988378547, −3.15703091552755063106793971437, −2.74620011669511967285979721546, −2.72551537988363500379532921733, −2.49758646553294322302156472477, −2.17219626448841765687356467838, −2.10231416428804268686953780943, −1.84187405367196858949913377150, −1.54950365524627196511265655647, −1.29747157780935328742476209569, −1.16007154630248780308661781620, −0.888522798364425778889058611502, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.888522798364425778889058611502, 1.16007154630248780308661781620, 1.29747157780935328742476209569, 1.54950365524627196511265655647, 1.84187405367196858949913377150, 2.10231416428804268686953780943, 2.17219626448841765687356467838, 2.49758646553294322302156472477, 2.72551537988363500379532921733, 2.74620011669511967285979721546, 3.15703091552755063106793971437, 3.30535359078773361311988378547, 3.52663397569566864305344730612, 3.57038044990405343388975565955, 3.60085473309128521613366773934, 3.67999195039187100354863123877, 3.87149442873373892919446498246, 3.89672815502102575638541121161, 4.21673779000135756440304330459, 4.56170797026164947582303115591, 4.79673481247408652321443049339, 4.94192139842278689718592259196, 5.10347929398800207091980518823, 5.13978625374325176944667867056, 5.19175145188198726297973956519
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.