Dirichlet series
| L(s) = 1 | + 3.47e11·2-s + 9.27e19·3-s − 1.59e25·4-s + 9.58e28·5-s + 3.22e31·6-s + 4.19e34·7-s − 2.44e37·8-s − 6.07e39·9-s + 3.33e40·10-s + 2.58e43·11-s − 1.48e45·12-s + 1.75e46·13-s + 1.45e46·14-s + 8.89e48·15-s + 1.39e50·16-s − 1.84e50·17-s − 2.11e51·18-s + 7.00e52·19-s − 1.53e54·20-s + 3.89e54·21-s + 8.99e54·22-s + 8.24e56·23-s − 2.26e57·24-s − 3.38e58·25-s + 6.08e57·26-s − 9.59e59·27-s − 6.70e59·28-s + ⋯ |
| L(s) = 1 | + 0.111·2-s + 1.46·3-s − 1.65·4-s + 0.942·5-s + 0.164·6-s + 0.355·7-s − 0.811·8-s − 1.52·9-s + 0.105·10-s + 1.56·11-s − 2.42·12-s + 1.03·13-s + 0.0397·14-s + 1.38·15-s + 1.49·16-s − 0.159·17-s − 0.170·18-s + 0.598·19-s − 1.55·20-s + 0.522·21-s + 0.175·22-s + 2.53·23-s − 1.19·24-s − 3.27·25-s + 0.115·26-s − 3.80·27-s − 0.588·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(84-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+83/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.00816\times 10^{11}\) |
| Root analytic conductor: | \(6.60508\) |
| Motivic weight: | \(83\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 1,\ (\ :[83/2]^{7}),\ 1)\) |
Particular Values
| \(L(42)\) | \(\approx\) | \(3.757428831\) |
| \(L(\frac12)\) | \(\approx\) | \(3.757428831\) |
| \(L(\frac{85}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| good | 2 | \( 1 - 43431345177 p^{3} T + \)\(98\!\cdots\!87\)\( p^{14} T^{2} + \)\(63\!\cdots\!25\)\( p^{21} T^{3} + \)\(76\!\cdots\!11\)\( p^{37} T^{4} + \)\(33\!\cdots\!23\)\( p^{56} T^{5} + \)\(10\!\cdots\!59\)\( p^{82} T^{6} + \)\(33\!\cdots\!25\)\( p^{106} T^{7} + \)\(10\!\cdots\!59\)\( p^{165} T^{8} + \)\(33\!\cdots\!23\)\( p^{222} T^{9} + \)\(76\!\cdots\!11\)\( p^{286} T^{10} + \)\(63\!\cdots\!25\)\( p^{353} T^{11} + \)\(98\!\cdots\!87\)\( p^{429} T^{12} - 43431345177 p^{501} T^{13} + p^{581} T^{14} \) |
| 3 | \( 1 - 10307665859149819108 p^{2} T + \)\(67\!\cdots\!71\)\( p^{7} T^{2} - \)\(74\!\cdots\!00\)\( p^{17} T^{3} + \)\(14\!\cdots\!39\)\( p^{29} T^{4} - \)\(17\!\cdots\!32\)\( p^{43} T^{5} + \)\(37\!\cdots\!47\)\( p^{59} T^{6} - \)\(48\!\cdots\!00\)\( p^{77} T^{7} + \)\(37\!\cdots\!47\)\( p^{142} T^{8} - \)\(17\!\cdots\!32\)\( p^{209} T^{9} + \)\(14\!\cdots\!39\)\( p^{278} T^{10} - \)\(74\!\cdots\!00\)\( p^{349} T^{11} + \)\(67\!\cdots\!71\)\( p^{422} T^{12} - 10307665859149819108 p^{500} T^{13} + p^{581} T^{14} \) | |
| 5 | \( 1 - \)\(19\!\cdots\!94\)\( p T + \)\(68\!\cdots\!99\)\( p^{4} T^{2} - \)\(11\!\cdots\!48\)\( p^{9} T^{3} + \)\(98\!\cdots\!73\)\( p^{17} T^{4} - \)\(17\!\cdots\!54\)\( p^{27} T^{5} + \)\(21\!\cdots\!11\)\( p^{38} T^{6} + \)\(21\!\cdots\!36\)\( p^{50} T^{7} + \)\(21\!\cdots\!11\)\( p^{121} T^{8} - \)\(17\!\cdots\!54\)\( p^{193} T^{9} + \)\(98\!\cdots\!73\)\( p^{266} T^{10} - \)\(11\!\cdots\!48\)\( p^{341} T^{11} + \)\(68\!\cdots\!99\)\( p^{419} T^{12} - \)\(19\!\cdots\!94\)\( p^{499} T^{13} + p^{581} T^{14} \) | |
| 7 | \( 1 - \)\(59\!\cdots\!08\)\( p T + \)\(73\!\cdots\!51\)\( p^{3} T^{2} - \)\(22\!\cdots\!00\)\( p^{6} T^{3} + \)\(31\!\cdots\!97\)\( p^{12} T^{4} + \)\(15\!\cdots\!76\)\( p^{19} T^{5} + \)\(87\!\cdots\!47\)\( p^{27} T^{6} + \)\(15\!\cdots\!00\)\( p^{37} T^{7} + \)\(87\!\cdots\!47\)\( p^{110} T^{8} + \)\(15\!\cdots\!76\)\( p^{185} T^{9} + \)\(31\!\cdots\!97\)\( p^{261} T^{10} - \)\(22\!\cdots\!00\)\( p^{338} T^{11} + \)\(73\!\cdots\!51\)\( p^{418} T^{12} - \)\(59\!\cdots\!08\)\( p^{499} T^{13} + p^{581} T^{14} \) | |
| 11 | \( 1 - \)\(23\!\cdots\!84\)\( p T + \)\(88\!\cdots\!01\)\( p^{2} T^{2} - \)\(11\!\cdots\!44\)\( p^{5} T^{3} + \)\(25\!\cdots\!71\)\( p^{9} T^{4} - \)\(25\!\cdots\!72\)\( p^{14} T^{5} + \)\(34\!\cdots\!73\)\( p^{20} T^{6} - \)\(25\!\cdots\!72\)\( p^{26} T^{7} + \)\(34\!\cdots\!73\)\( p^{103} T^{8} - \)\(25\!\cdots\!72\)\( p^{180} T^{9} + \)\(25\!\cdots\!71\)\( p^{258} T^{10} - \)\(11\!\cdots\!44\)\( p^{337} T^{11} + \)\(88\!\cdots\!01\)\( p^{417} T^{12} - \)\(23\!\cdots\!84\)\( p^{499} T^{13} + p^{581} T^{14} \) | |
| 13 | \( 1 - \)\(17\!\cdots\!02\)\( T + \)\(89\!\cdots\!19\)\( p T^{2} - \)\(10\!\cdots\!00\)\( p^{3} T^{3} + \)\(16\!\cdots\!53\)\( p^{6} T^{4} - \)\(92\!\cdots\!46\)\( p^{10} T^{5} + \)\(65\!\cdots\!67\)\( p^{15} T^{6} - \)\(24\!\cdots\!00\)\( p^{20} T^{7} + \)\(65\!\cdots\!67\)\( p^{98} T^{8} - \)\(92\!\cdots\!46\)\( p^{176} T^{9} + \)\(16\!\cdots\!53\)\( p^{255} T^{10} - \)\(10\!\cdots\!00\)\( p^{335} T^{11} + \)\(89\!\cdots\!19\)\( p^{416} T^{12} - \)\(17\!\cdots\!02\)\( p^{498} T^{13} + p^{581} T^{14} \) | |
| 17 | \( 1 + \)\(10\!\cdots\!42\)\( p T + \)\(22\!\cdots\!67\)\( p^{2} T^{2} + \)\(45\!\cdots\!00\)\( p^{3} T^{3} + \)\(21\!\cdots\!17\)\( p^{4} T^{4} + \)\(21\!\cdots\!26\)\( p^{7} T^{5} + \)\(96\!\cdots\!27\)\( p^{11} T^{6} + \)\(58\!\cdots\!00\)\( p^{15} T^{7} + \)\(96\!\cdots\!27\)\( p^{94} T^{8} + \)\(21\!\cdots\!26\)\( p^{173} T^{9} + \)\(21\!\cdots\!17\)\( p^{253} T^{10} + \)\(45\!\cdots\!00\)\( p^{335} T^{11} + \)\(22\!\cdots\!67\)\( p^{417} T^{12} + \)\(10\!\cdots\!42\)\( p^{499} T^{13} + p^{581} T^{14} \) | |
| 19 | \( 1 - \)\(70\!\cdots\!60\)\( T + \)\(32\!\cdots\!27\)\( p T^{2} - \)\(42\!\cdots\!60\)\( p^{3} T^{3} + \)\(75\!\cdots\!99\)\( p^{5} T^{4} - \)\(38\!\cdots\!00\)\( p^{8} T^{5} + \)\(16\!\cdots\!65\)\( p^{12} T^{6} - \)\(36\!\cdots\!00\)\( p^{16} T^{7} + \)\(16\!\cdots\!65\)\( p^{95} T^{8} - \)\(38\!\cdots\!00\)\( p^{174} T^{9} + \)\(75\!\cdots\!99\)\( p^{254} T^{10} - \)\(42\!\cdots\!60\)\( p^{335} T^{11} + \)\(32\!\cdots\!27\)\( p^{416} T^{12} - \)\(70\!\cdots\!60\)\( p^{498} T^{13} + p^{581} T^{14} \) | |
| 23 | \( 1 - \)\(82\!\cdots\!32\)\( T + \)\(22\!\cdots\!79\)\( p T^{2} - \)\(43\!\cdots\!00\)\( p^{2} T^{3} + \)\(31\!\cdots\!37\)\( p^{4} T^{4} - \)\(87\!\cdots\!52\)\( p^{7} T^{5} + \)\(21\!\cdots\!61\)\( p^{10} T^{6} - \)\(57\!\cdots\!00\)\( p^{13} T^{7} + \)\(21\!\cdots\!61\)\( p^{93} T^{8} - \)\(87\!\cdots\!52\)\( p^{173} T^{9} + \)\(31\!\cdots\!37\)\( p^{253} T^{10} - \)\(43\!\cdots\!00\)\( p^{334} T^{11} + \)\(22\!\cdots\!79\)\( p^{416} T^{12} - \)\(82\!\cdots\!32\)\( p^{498} T^{13} + p^{581} T^{14} \) | |
| 29 | \( 1 - \)\(64\!\cdots\!90\)\( T + \)\(10\!\cdots\!23\)\( T^{2} - \)\(23\!\cdots\!40\)\( p T^{3} + \)\(64\!\cdots\!01\)\( p^{2} T^{4} - \)\(44\!\cdots\!50\)\( p^{4} T^{5} + \)\(31\!\cdots\!15\)\( p^{6} T^{6} - \)\(17\!\cdots\!00\)\( p^{8} T^{7} + \)\(31\!\cdots\!15\)\( p^{89} T^{8} - \)\(44\!\cdots\!50\)\( p^{170} T^{9} + \)\(64\!\cdots\!01\)\( p^{251} T^{10} - \)\(23\!\cdots\!40\)\( p^{333} T^{11} + \)\(10\!\cdots\!23\)\( p^{415} T^{12} - \)\(64\!\cdots\!90\)\( p^{498} T^{13} + p^{581} T^{14} \) | |
| 31 | \( 1 + \)\(37\!\cdots\!56\)\( T + \)\(24\!\cdots\!81\)\( T^{2} + \)\(62\!\cdots\!16\)\( p T^{3} + \)\(29\!\cdots\!21\)\( p^{2} T^{4} + \)\(32\!\cdots\!88\)\( p^{4} T^{5} + \)\(25\!\cdots\!73\)\( p^{6} T^{6} + \)\(28\!\cdots\!68\)\( p^{8} T^{7} + \)\(25\!\cdots\!73\)\( p^{89} T^{8} + \)\(32\!\cdots\!88\)\( p^{170} T^{9} + \)\(29\!\cdots\!21\)\( p^{251} T^{10} + \)\(62\!\cdots\!16\)\( p^{333} T^{11} + \)\(24\!\cdots\!81\)\( p^{415} T^{12} + \)\(37\!\cdots\!56\)\( p^{498} T^{13} + p^{581} T^{14} \) | |
| 37 | \( 1 - \)\(11\!\cdots\!46\)\( T + \)\(15\!\cdots\!19\)\( p T^{2} - \)\(23\!\cdots\!00\)\( p^{2} T^{3} + \)\(27\!\cdots\!09\)\( p^{3} T^{4} - \)\(22\!\cdots\!06\)\( p^{5} T^{5} + \)\(23\!\cdots\!07\)\( p^{7} T^{6} - \)\(24\!\cdots\!00\)\( p^{9} T^{7} + \)\(23\!\cdots\!07\)\( p^{90} T^{8} - \)\(22\!\cdots\!06\)\( p^{171} T^{9} + \)\(27\!\cdots\!09\)\( p^{252} T^{10} - \)\(23\!\cdots\!00\)\( p^{334} T^{11} + \)\(15\!\cdots\!19\)\( p^{416} T^{12} - \)\(11\!\cdots\!46\)\( p^{498} T^{13} + p^{581} T^{14} \) | |
| 41 | \( 1 - \)\(42\!\cdots\!94\)\( p T + \)\(30\!\cdots\!31\)\( p^{2} T^{2} - \)\(94\!\cdots\!04\)\( p^{3} T^{3} + \)\(37\!\cdots\!81\)\( p^{4} T^{4} - \)\(91\!\cdots\!02\)\( p^{5} T^{5} + \)\(25\!\cdots\!63\)\( p^{6} T^{6} - \)\(50\!\cdots\!12\)\( p^{7} T^{7} + \)\(25\!\cdots\!63\)\( p^{89} T^{8} - \)\(91\!\cdots\!02\)\( p^{171} T^{9} + \)\(37\!\cdots\!81\)\( p^{253} T^{10} - \)\(94\!\cdots\!04\)\( p^{335} T^{11} + \)\(30\!\cdots\!31\)\( p^{417} T^{12} - \)\(42\!\cdots\!94\)\( p^{499} T^{13} + p^{581} T^{14} \) | |
| 43 | \( 1 - \)\(15\!\cdots\!92\)\( T + \)\(28\!\cdots\!57\)\( T^{2} - \)\(28\!\cdots\!00\)\( T^{3} + \)\(73\!\cdots\!79\)\( p T^{4} - \)\(13\!\cdots\!76\)\( p^{2} T^{5} + \)\(24\!\cdots\!47\)\( p^{3} T^{6} - \)\(34\!\cdots\!00\)\( p^{4} T^{7} + \)\(24\!\cdots\!47\)\( p^{86} T^{8} - \)\(13\!\cdots\!76\)\( p^{168} T^{9} + \)\(73\!\cdots\!79\)\( p^{250} T^{10} - \)\(28\!\cdots\!00\)\( p^{332} T^{11} + \)\(28\!\cdots\!57\)\( p^{415} T^{12} - \)\(15\!\cdots\!92\)\( p^{498} T^{13} + p^{581} T^{14} \) | |
| 47 | \( 1 - \)\(11\!\cdots\!76\)\( T + \)\(81\!\cdots\!73\)\( T^{2} - \)\(92\!\cdots\!00\)\( p T^{3} + \)\(84\!\cdots\!93\)\( p^{2} T^{4} - \)\(63\!\cdots\!44\)\( p^{3} T^{5} + \)\(41\!\cdots\!21\)\( p^{4} T^{6} - \)\(23\!\cdots\!00\)\( p^{5} T^{7} + \)\(41\!\cdots\!21\)\( p^{87} T^{8} - \)\(63\!\cdots\!44\)\( p^{169} T^{9} + \)\(84\!\cdots\!93\)\( p^{251} T^{10} - \)\(92\!\cdots\!00\)\( p^{333} T^{11} + \)\(81\!\cdots\!73\)\( p^{415} T^{12} - \)\(11\!\cdots\!76\)\( p^{498} T^{13} + p^{581} T^{14} \) | |
| 53 | \( 1 - \)\(27\!\cdots\!22\)\( T + \)\(59\!\cdots\!27\)\( T^{2} - \)\(29\!\cdots\!00\)\( p T^{3} + \)\(64\!\cdots\!93\)\( p^{2} T^{4} - \)\(28\!\cdots\!82\)\( p^{3} T^{5} + \)\(44\!\cdots\!79\)\( p^{4} T^{6} - \)\(16\!\cdots\!00\)\( p^{5} T^{7} + \)\(44\!\cdots\!79\)\( p^{87} T^{8} - \)\(28\!\cdots\!82\)\( p^{169} T^{9} + \)\(64\!\cdots\!93\)\( p^{251} T^{10} - \)\(29\!\cdots\!00\)\( p^{333} T^{11} + \)\(59\!\cdots\!27\)\( p^{415} T^{12} - \)\(27\!\cdots\!22\)\( p^{498} T^{13} + p^{581} T^{14} \) | |
| 59 | \( 1 + \)\(23\!\cdots\!20\)\( T + \)\(40\!\cdots\!67\)\( p T^{2} + \)\(15\!\cdots\!80\)\( p^{2} T^{3} + \)\(21\!\cdots\!01\)\( p^{4} T^{4} + \)\(38\!\cdots\!00\)\( p^{4} T^{5} + \)\(19\!\cdots\!35\)\( p^{5} T^{6} + \)\(85\!\cdots\!00\)\( p^{6} T^{7} + \)\(19\!\cdots\!35\)\( p^{88} T^{8} + \)\(38\!\cdots\!00\)\( p^{170} T^{9} + \)\(21\!\cdots\!01\)\( p^{253} T^{10} + \)\(15\!\cdots\!80\)\( p^{334} T^{11} + \)\(40\!\cdots\!67\)\( p^{416} T^{12} + \)\(23\!\cdots\!20\)\( p^{498} T^{13} + p^{581} T^{14} \) | |
| 61 | \( 1 - \)\(17\!\cdots\!74\)\( T + \)\(89\!\cdots\!71\)\( T^{2} - \)\(24\!\cdots\!04\)\( p T^{3} + \)\(98\!\cdots\!41\)\( p^{2} T^{4} - \)\(23\!\cdots\!42\)\( p^{3} T^{5} + \)\(63\!\cdots\!03\)\( p^{4} T^{6} - \)\(12\!\cdots\!92\)\( p^{5} T^{7} + \)\(63\!\cdots\!03\)\( p^{87} T^{8} - \)\(23\!\cdots\!42\)\( p^{169} T^{9} + \)\(98\!\cdots\!41\)\( p^{251} T^{10} - \)\(24\!\cdots\!04\)\( p^{333} T^{11} + \)\(89\!\cdots\!71\)\( p^{415} T^{12} - \)\(17\!\cdots\!74\)\( p^{498} T^{13} + p^{581} T^{14} \) | |
| 67 | \( 1 - \)\(42\!\cdots\!36\)\( T + \)\(28\!\cdots\!39\)\( p T^{2} - \)\(14\!\cdots\!00\)\( p^{2} T^{3} + \)\(58\!\cdots\!39\)\( p^{3} T^{4} - \)\(23\!\cdots\!12\)\( p^{4} T^{5} + \)\(71\!\cdots\!63\)\( p^{5} T^{6} - \)\(24\!\cdots\!00\)\( p^{6} T^{7} + \)\(71\!\cdots\!63\)\( p^{88} T^{8} - \)\(23\!\cdots\!12\)\( p^{170} T^{9} + \)\(58\!\cdots\!39\)\( p^{252} T^{10} - \)\(14\!\cdots\!00\)\( p^{334} T^{11} + \)\(28\!\cdots\!39\)\( p^{416} T^{12} - \)\(42\!\cdots\!36\)\( p^{498} T^{13} + p^{581} T^{14} \) | |
| 71 | \( 1 + \)\(35\!\cdots\!96\)\( p T + \)\(85\!\cdots\!61\)\( p^{2} T^{2} + \)\(14\!\cdots\!16\)\( p^{3} T^{3} + \)\(21\!\cdots\!41\)\( p^{4} T^{4} + \)\(26\!\cdots\!48\)\( p^{5} T^{5} + \)\(30\!\cdots\!33\)\( p^{6} T^{6} + \)\(30\!\cdots\!28\)\( p^{7} T^{7} + \)\(30\!\cdots\!33\)\( p^{89} T^{8} + \)\(26\!\cdots\!48\)\( p^{171} T^{9} + \)\(21\!\cdots\!41\)\( p^{253} T^{10} + \)\(14\!\cdots\!16\)\( p^{335} T^{11} + \)\(85\!\cdots\!61\)\( p^{417} T^{12} + \)\(35\!\cdots\!96\)\( p^{499} T^{13} + p^{581} T^{14} \) | |
| 73 | \( 1 + \)\(30\!\cdots\!18\)\( T + \)\(31\!\cdots\!79\)\( p T^{2} + \)\(84\!\cdots\!00\)\( p^{2} T^{3} + \)\(54\!\cdots\!01\)\( p^{3} T^{4} + \)\(97\!\cdots\!66\)\( p^{4} T^{5} + \)\(56\!\cdots\!23\)\( p^{5} T^{6} + \)\(82\!\cdots\!00\)\( p^{6} T^{7} + \)\(56\!\cdots\!23\)\( p^{88} T^{8} + \)\(97\!\cdots\!66\)\( p^{170} T^{9} + \)\(54\!\cdots\!01\)\( p^{252} T^{10} + \)\(84\!\cdots\!00\)\( p^{334} T^{11} + \)\(31\!\cdots\!79\)\( p^{416} T^{12} + \)\(30\!\cdots\!18\)\( p^{498} T^{13} + p^{581} T^{14} \) | |
| 79 | \( 1 - \)\(29\!\cdots\!60\)\( p T + \)\(66\!\cdots\!53\)\( p^{2} T^{2} - \)\(10\!\cdots\!40\)\( p^{3} T^{3} + \)\(13\!\cdots\!61\)\( p^{4} T^{4} - \)\(14\!\cdots\!00\)\( p^{5} T^{5} + \)\(12\!\cdots\!65\)\( p^{6} T^{6} - \)\(98\!\cdots\!00\)\( p^{7} T^{7} + \)\(12\!\cdots\!65\)\( p^{89} T^{8} - \)\(14\!\cdots\!00\)\( p^{171} T^{9} + \)\(13\!\cdots\!61\)\( p^{253} T^{10} - \)\(10\!\cdots\!40\)\( p^{335} T^{11} + \)\(66\!\cdots\!53\)\( p^{417} T^{12} - \)\(29\!\cdots\!60\)\( p^{499} T^{13} + p^{581} T^{14} \) | |
| 83 | \( 1 + \)\(46\!\cdots\!88\)\( T + \)\(75\!\cdots\!37\)\( T^{2} + \)\(23\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!57\)\( T^{4} + \)\(71\!\cdots\!16\)\( T^{5} + \)\(74\!\cdots\!09\)\( T^{6} + \)\(16\!\cdots\!00\)\( T^{7} + \)\(74\!\cdots\!09\)\( p^{83} T^{8} + \)\(71\!\cdots\!16\)\( p^{166} T^{9} + \)\(27\!\cdots\!57\)\( p^{249} T^{10} + \)\(23\!\cdots\!00\)\( p^{332} T^{11} + \)\(75\!\cdots\!37\)\( p^{415} T^{12} + \)\(46\!\cdots\!88\)\( p^{498} T^{13} + p^{581} T^{14} \) | |
| 89 | \( 1 - \)\(10\!\cdots\!70\)\( T + \)\(33\!\cdots\!83\)\( T^{2} - \)\(26\!\cdots\!80\)\( T^{3} + \)\(49\!\cdots\!81\)\( T^{4} - \)\(31\!\cdots\!50\)\( T^{5} + \)\(44\!\cdots\!15\)\( T^{6} - \)\(23\!\cdots\!00\)\( T^{7} + \)\(44\!\cdots\!15\)\( p^{83} T^{8} - \)\(31\!\cdots\!50\)\( p^{166} T^{9} + \)\(49\!\cdots\!81\)\( p^{249} T^{10} - \)\(26\!\cdots\!80\)\( p^{332} T^{11} + \)\(33\!\cdots\!83\)\( p^{415} T^{12} - \)\(10\!\cdots\!70\)\( p^{498} T^{13} + p^{581} T^{14} \) | |
| 97 | \( 1 + \)\(74\!\cdots\!74\)\( T + \)\(59\!\cdots\!23\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!37\)\( T^{4} + \)\(40\!\cdots\!38\)\( T^{5} + \)\(13\!\cdots\!51\)\( T^{6} + \)\(36\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!51\)\( p^{83} T^{8} + \)\(40\!\cdots\!38\)\( p^{166} T^{9} + \)\(11\!\cdots\!37\)\( p^{249} T^{10} + \)\(27\!\cdots\!00\)\( p^{332} T^{11} + \)\(59\!\cdots\!23\)\( p^{415} T^{12} + \)\(74\!\cdots\!74\)\( p^{498} T^{13} + p^{581} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.82608038975066911071322838467, −5.50285822364972256162219031727, −5.49885798871510605018542993899, −5.33549708135052045496851953267, −4.82198524221240653246091862956, −4.44693571510118527574394580946, −4.18036974471018778576715920083, −4.08109611737468558625664737265, −3.97008373389652681899794330110, −3.83794527649381369338478814487, −3.32389226197368291992344336393, −3.11414269670588140766638183753, −3.09357457012830528605586553896, −2.68510262677656696749332295313, −2.54919841922959855902120281457, −2.43354507653183999128339962278, −2.15851733495935603386191137335, −1.66076858784604658417093581971, −1.64981835493912290174345703995, −1.37436608868908131576040606483, −0.878639372700450754888891206178, −0.76286909682885747196394090707, −0.68650228161668592560560146658, −0.58892235545161152057114558656, −0.098883314690710113132541806307, 0.098883314690710113132541806307, 0.58892235545161152057114558656, 0.68650228161668592560560146658, 0.76286909682885747196394090707, 0.878639372700450754888891206178, 1.37436608868908131576040606483, 1.64981835493912290174345703995, 1.66076858784604658417093581971, 2.15851733495935603386191137335, 2.43354507653183999128339962278, 2.54919841922959855902120281457, 2.68510262677656696749332295313, 3.09357457012830528605586553896, 3.11414269670588140766638183753, 3.32389226197368291992344336393, 3.83794527649381369338478814487, 3.97008373389652681899794330110, 4.08109611737468558625664737265, 4.18036974471018778576715920083, 4.44693571510118527574394580946, 4.82198524221240653246091862956, 5.33549708135052045496851953267, 5.49885798871510605018542993899, 5.50285822364972256162219031727, 5.82608038975066911071322838467
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.