Dirichlet series
| L(s) = 1 | + 486·3-s − 1.53e3·4-s − 6.66e3·5-s + 3.05e4·7-s − 8.18e4·9-s − 1.11e5·11-s − 7.46e5·12-s − 3.23e6·15-s + 7.86e5·16-s + 1.43e6·17-s − 4.52e5·19-s + 1.02e7·20-s + 1.48e7·21-s + 8.53e5·23-s + 1.37e6·25-s − 7.80e7·27-s − 4.69e7·28-s + 6.01e7·29-s + 8.72e7·31-s − 5.40e7·33-s − 2.03e8·35-s + 1.25e8·36-s − 7.66e6·37-s + 1.06e9·43-s + 1.70e8·44-s + 5.45e8·45-s − 9.85e8·47-s + ⋯ |
| L(s) = 1 | + 2·3-s − 3/2·4-s − 2.13·5-s + 1.81·7-s − 1.38·9-s − 0.690·11-s − 3·12-s − 4.26·15-s + 3/4·16-s + 1.01·17-s − 0.182·19-s + 3.19·20-s + 3.63·21-s + 0.132·23-s + 0.140·25-s − 5.43·27-s − 2.72·28-s + 2.93·29-s + 3.04·31-s − 1.38·33-s − 3.88·35-s + 2.07·36-s − 0.110·37-s + 7.25·43-s + 1.03·44-s + 2.95·45-s − 4.29·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+5)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.45330\times 10^{11}\) |
| Root analytic conductor: | \(2.98244\) |
| Motivic weight: | \(10\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 7^{12} ,\ ( \ : [5]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{11}{2})\) | \(\approx\) | \(4.630139224\) |
| \(L(\frac12)\) | \(\approx\) | \(4.630139224\) |
| \(L(6)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{9} T^{2} + p^{18} T^{4} )^{3} \) |
| 7 | \( 1 - 624 p^{2} T + 4884774 p^{2} T^{2} + 2330121904 p^{4} T^{3} - 183708964935 p^{7} T^{4} + 401320689984 p^{11} T^{5} + 3127552449900 p^{15} T^{6} + 401320689984 p^{21} T^{7} - 183708964935 p^{27} T^{8} + 2330121904 p^{34} T^{9} + 4884774 p^{42} T^{10} - 624 p^{52} T^{11} + p^{60} T^{12} \) | |
| good | 3 | \( 1 - 2 p^{5} T + 106003 p T^{2} - 159518 p^{6} T^{3} + 4966369615 p^{2} T^{4} - 49565703500 p^{5} T^{5} + 14137137849332 p^{5} T^{6} - 100287728700400 p^{8} T^{7} + 21795573444836693 p^{8} T^{8} - 238722403842090178 p^{10} T^{9} + 10896320467516484561 p^{11} T^{10} + \)\(33\!\cdots\!66\)\( p^{13} T^{11} - \)\(14\!\cdots\!06\)\( p^{14} T^{12} + \)\(33\!\cdots\!66\)\( p^{23} T^{13} + 10896320467516484561 p^{31} T^{14} - 238722403842090178 p^{40} T^{15} + 21795573444836693 p^{48} T^{16} - 100287728700400 p^{58} T^{17} + 14137137849332 p^{65} T^{18} - 49565703500 p^{75} T^{19} + 4966369615 p^{82} T^{20} - 159518 p^{96} T^{21} + 106003 p^{101} T^{22} - 2 p^{115} T^{23} + p^{120} T^{24} \) |
| 5 | \( 1 + 6666 T + 43061751 T^{2} + 188313826734 T^{3} + 772793510066451 T^{4} + 528287194653044856 p T^{5} + \)\(14\!\cdots\!52\)\( p T^{6} + \)\(54\!\cdots\!68\)\( p^{2} T^{7} - \)\(13\!\cdots\!87\)\( p^{2} T^{8} - \)\(12\!\cdots\!26\)\( p^{3} T^{9} - \)\(74\!\cdots\!23\)\( p^{3} T^{10} - \)\(62\!\cdots\!98\)\( p^{4} T^{11} - \)\(21\!\cdots\!74\)\( p^{4} T^{12} - \)\(62\!\cdots\!98\)\( p^{14} T^{13} - \)\(74\!\cdots\!23\)\( p^{23} T^{14} - \)\(12\!\cdots\!26\)\( p^{33} T^{15} - \)\(13\!\cdots\!87\)\( p^{42} T^{16} + \)\(54\!\cdots\!68\)\( p^{52} T^{17} + \)\(14\!\cdots\!52\)\( p^{61} T^{18} + 528287194653044856 p^{71} T^{19} + 772793510066451 p^{80} T^{20} + 188313826734 p^{90} T^{21} + 43061751 p^{100} T^{22} + 6666 p^{110} T^{23} + p^{120} T^{24} \) | |
| 11 | \( 1 + 10110 p T - 59974635903 T^{2} - 19729258213110126 T^{3} + \)\(18\!\cdots\!35\)\( T^{4} + \)\(77\!\cdots\!04\)\( T^{5} + \)\(12\!\cdots\!76\)\( T^{6} - \)\(30\!\cdots\!48\)\( T^{7} - \)\(32\!\cdots\!19\)\( T^{8} - \)\(63\!\cdots\!38\)\( T^{9} - \)\(39\!\cdots\!97\)\( T^{10} + \)\(10\!\cdots\!66\)\( T^{11} + \)\(34\!\cdots\!94\)\( T^{12} + \)\(10\!\cdots\!66\)\( p^{10} T^{13} - \)\(39\!\cdots\!97\)\( p^{20} T^{14} - \)\(63\!\cdots\!38\)\( p^{30} T^{15} - \)\(32\!\cdots\!19\)\( p^{40} T^{16} - \)\(30\!\cdots\!48\)\( p^{50} T^{17} + \)\(12\!\cdots\!76\)\( p^{60} T^{18} + \)\(77\!\cdots\!04\)\( p^{70} T^{19} + \)\(18\!\cdots\!35\)\( p^{80} T^{20} - 19729258213110126 p^{90} T^{21} - 59974635903 p^{100} T^{22} + 10110 p^{111} T^{23} + p^{120} T^{24} \) | |
| 13 | \( 1 - 331245895908 T^{2} + \)\(53\!\cdots\!78\)\( T^{4} - \)\(51\!\cdots\!16\)\( T^{6} + \)\(35\!\cdots\!83\)\( T^{8} - \)\(25\!\cdots\!32\)\( T^{10} + \)\(41\!\cdots\!80\)\( T^{12} - \)\(25\!\cdots\!32\)\( p^{20} T^{14} + \)\(35\!\cdots\!83\)\( p^{40} T^{16} - \)\(51\!\cdots\!16\)\( p^{60} T^{18} + \)\(53\!\cdots\!78\)\( p^{80} T^{20} - 331245895908 p^{100} T^{22} + p^{120} T^{24} \) | |
| 17 | \( 1 - 1439502 T + 7412712475743 T^{2} - 9676318729972408650 T^{3} + \)\(24\!\cdots\!03\)\( T^{4} - \)\(36\!\cdots\!12\)\( T^{5} + \)\(67\!\cdots\!88\)\( T^{6} - \)\(11\!\cdots\!28\)\( T^{7} + \)\(19\!\cdots\!61\)\( T^{8} - \)\(28\!\cdots\!62\)\( T^{9} + \)\(47\!\cdots\!93\)\( T^{10} - \)\(57\!\cdots\!18\)\( T^{11} + \)\(97\!\cdots\!22\)\( T^{12} - \)\(57\!\cdots\!18\)\( p^{10} T^{13} + \)\(47\!\cdots\!93\)\( p^{20} T^{14} - \)\(28\!\cdots\!62\)\( p^{30} T^{15} + \)\(19\!\cdots\!61\)\( p^{40} T^{16} - \)\(11\!\cdots\!28\)\( p^{50} T^{17} + \)\(67\!\cdots\!88\)\( p^{60} T^{18} - \)\(36\!\cdots\!12\)\( p^{70} T^{19} + \)\(24\!\cdots\!03\)\( p^{80} T^{20} - 9676318729972408650 p^{90} T^{21} + 7412712475743 p^{100} T^{22} - 1439502 p^{110} T^{23} + p^{120} T^{24} \) | |
| 19 | \( 1 + 452814 T + 25791852626985 T^{2} + 11647963549639742742 T^{3} + \)\(34\!\cdots\!31\)\( T^{4} + \)\(18\!\cdots\!20\)\( T^{5} + \)\(34\!\cdots\!60\)\( T^{6} + \)\(23\!\cdots\!32\)\( T^{7} + \)\(29\!\cdots\!05\)\( T^{8} + \)\(23\!\cdots\!78\)\( T^{9} + \)\(21\!\cdots\!11\)\( T^{10} + \)\(18\!\cdots\!74\)\( T^{11} + \)\(13\!\cdots\!10\)\( T^{12} + \)\(18\!\cdots\!74\)\( p^{10} T^{13} + \)\(21\!\cdots\!11\)\( p^{20} T^{14} + \)\(23\!\cdots\!78\)\( p^{30} T^{15} + \)\(29\!\cdots\!05\)\( p^{40} T^{16} + \)\(23\!\cdots\!32\)\( p^{50} T^{17} + \)\(34\!\cdots\!60\)\( p^{60} T^{18} + \)\(18\!\cdots\!20\)\( p^{70} T^{19} + \)\(34\!\cdots\!31\)\( p^{80} T^{20} + 11647963549639742742 p^{90} T^{21} + 25791852626985 p^{100} T^{22} + 452814 p^{110} T^{23} + p^{120} T^{24} \) | |
| 23 | \( 1 - 853074 T - 126099256146927 T^{2} - \)\(19\!\cdots\!50\)\( T^{3} + \)\(71\!\cdots\!87\)\( T^{4} + \)\(27\!\cdots\!92\)\( T^{5} - \)\(11\!\cdots\!32\)\( p T^{6} - \)\(12\!\cdots\!96\)\( T^{7} + \)\(89\!\cdots\!13\)\( T^{8} + \)\(30\!\cdots\!54\)\( T^{9} - \)\(35\!\cdots\!93\)\( T^{10} - \)\(28\!\cdots\!42\)\( T^{11} + \)\(14\!\cdots\!10\)\( T^{12} - \)\(28\!\cdots\!42\)\( p^{10} T^{13} - \)\(35\!\cdots\!93\)\( p^{20} T^{14} + \)\(30\!\cdots\!54\)\( p^{30} T^{15} + \)\(89\!\cdots\!13\)\( p^{40} T^{16} - \)\(12\!\cdots\!96\)\( p^{50} T^{17} - \)\(11\!\cdots\!32\)\( p^{61} T^{18} + \)\(27\!\cdots\!92\)\( p^{70} T^{19} + \)\(71\!\cdots\!87\)\( p^{80} T^{20} - \)\(19\!\cdots\!50\)\( p^{90} T^{21} - 126099256146927 p^{100} T^{22} - 853074 p^{110} T^{23} + p^{120} T^{24} \) | |
| 29 | \( ( 1 - 30078624 T + 1932667776747222 T^{2} - \)\(41\!\cdots\!36\)\( T^{3} + \)\(17\!\cdots\!51\)\( T^{4} - \)\(29\!\cdots\!84\)\( T^{5} + \)\(90\!\cdots\!48\)\( T^{6} - \)\(29\!\cdots\!84\)\( p^{10} T^{7} + \)\(17\!\cdots\!51\)\( p^{20} T^{8} - \)\(41\!\cdots\!36\)\( p^{30} T^{9} + 1932667776747222 p^{40} T^{10} - 30078624 p^{50} T^{11} + p^{60} T^{12} )^{2} \) | |
| 31 | \( 1 - 87231186 T + 5617910629301721 T^{2} - \)\(26\!\cdots\!54\)\( T^{3} + \)\(10\!\cdots\!63\)\( T^{4} - \)\(36\!\cdots\!84\)\( T^{5} + \)\(11\!\cdots\!24\)\( T^{6} - \)\(37\!\cdots\!48\)\( T^{7} + \)\(11\!\cdots\!97\)\( T^{8} - \)\(39\!\cdots\!58\)\( T^{9} + \)\(12\!\cdots\!99\)\( T^{10} - \)\(40\!\cdots\!94\)\( T^{11} + \)\(12\!\cdots\!42\)\( T^{12} - \)\(40\!\cdots\!94\)\( p^{10} T^{13} + \)\(12\!\cdots\!99\)\( p^{20} T^{14} - \)\(39\!\cdots\!58\)\( p^{30} T^{15} + \)\(11\!\cdots\!97\)\( p^{40} T^{16} - \)\(37\!\cdots\!48\)\( p^{50} T^{17} + \)\(11\!\cdots\!24\)\( p^{60} T^{18} - \)\(36\!\cdots\!84\)\( p^{70} T^{19} + \)\(10\!\cdots\!63\)\( p^{80} T^{20} - \)\(26\!\cdots\!54\)\( p^{90} T^{21} + 5617910629301721 p^{100} T^{22} - 87231186 p^{110} T^{23} + p^{120} T^{24} \) | |
| 37 | \( 1 + 7666506 T - 22550860947604401 T^{2} - \)\(28\!\cdots\!22\)\( T^{3} + \)\(27\!\cdots\!31\)\( T^{4} + \)\(39\!\cdots\!64\)\( T^{5} - \)\(24\!\cdots\!08\)\( T^{6} - \)\(29\!\cdots\!32\)\( T^{7} + \)\(16\!\cdots\!85\)\( T^{8} + \)\(13\!\cdots\!34\)\( T^{9} - \)\(99\!\cdots\!79\)\( T^{10} - \)\(27\!\cdots\!86\)\( T^{11} + \)\(51\!\cdots\!10\)\( T^{12} - \)\(27\!\cdots\!86\)\( p^{10} T^{13} - \)\(99\!\cdots\!79\)\( p^{20} T^{14} + \)\(13\!\cdots\!34\)\( p^{30} T^{15} + \)\(16\!\cdots\!85\)\( p^{40} T^{16} - \)\(29\!\cdots\!32\)\( p^{50} T^{17} - \)\(24\!\cdots\!08\)\( p^{60} T^{18} + \)\(39\!\cdots\!64\)\( p^{70} T^{19} + \)\(27\!\cdots\!31\)\( p^{80} T^{20} - \)\(28\!\cdots\!22\)\( p^{90} T^{21} - 22550860947604401 p^{100} T^{22} + 7666506 p^{110} T^{23} + p^{120} T^{24} \) | |
| 41 | \( 1 - 95274393138015876 T^{2} + \)\(45\!\cdots\!86\)\( T^{4} - \)\(14\!\cdots\!88\)\( T^{6} + \)\(35\!\cdots\!59\)\( T^{8} - \)\(65\!\cdots\!44\)\( T^{10} + \)\(97\!\cdots\!48\)\( T^{12} - \)\(65\!\cdots\!44\)\( p^{20} T^{14} + \)\(35\!\cdots\!59\)\( p^{40} T^{16} - \)\(14\!\cdots\!88\)\( p^{60} T^{18} + \)\(45\!\cdots\!86\)\( p^{80} T^{20} - 95274393138015876 p^{100} T^{22} + p^{120} T^{24} \) | |
| 43 | \( ( 1 - 533401668 T + 215437601578901706 T^{2} - \)\(61\!\cdots\!44\)\( T^{3} + \)\(14\!\cdots\!31\)\( T^{4} - \)\(27\!\cdots\!08\)\( T^{5} + \)\(43\!\cdots\!96\)\( T^{6} - \)\(27\!\cdots\!08\)\( p^{10} T^{7} + \)\(14\!\cdots\!31\)\( p^{20} T^{8} - \)\(61\!\cdots\!44\)\( p^{30} T^{9} + 215437601578901706 p^{40} T^{10} - 533401668 p^{50} T^{11} + p^{60} T^{12} )^{2} \) | |
| 47 | \( 1 + 985909398 T + 690878122207732449 T^{2} + \)\(36\!\cdots\!38\)\( T^{3} + \)\(34\!\cdots\!17\)\( p T^{4} + \)\(65\!\cdots\!64\)\( T^{5} + \)\(23\!\cdots\!24\)\( T^{6} + \)\(78\!\cdots\!68\)\( T^{7} + \)\(24\!\cdots\!33\)\( T^{8} + \)\(69\!\cdots\!42\)\( T^{9} + \)\(18\!\cdots\!95\)\( T^{10} + \)\(46\!\cdots\!82\)\( T^{11} + \)\(11\!\cdots\!10\)\( T^{12} + \)\(46\!\cdots\!82\)\( p^{10} T^{13} + \)\(18\!\cdots\!95\)\( p^{20} T^{14} + \)\(69\!\cdots\!42\)\( p^{30} T^{15} + \)\(24\!\cdots\!33\)\( p^{40} T^{16} + \)\(78\!\cdots\!68\)\( p^{50} T^{17} + \)\(23\!\cdots\!24\)\( p^{60} T^{18} + \)\(65\!\cdots\!64\)\( p^{70} T^{19} + \)\(34\!\cdots\!17\)\( p^{81} T^{20} + \)\(36\!\cdots\!38\)\( p^{90} T^{21} + 690878122207732449 p^{100} T^{22} + 985909398 p^{110} T^{23} + p^{120} T^{24} \) | |
| 53 | \( 1 + 600022554 T - 312626877283981761 T^{2} - \)\(98\!\cdots\!98\)\( T^{3} + \)\(12\!\cdots\!15\)\( T^{4} + \)\(46\!\cdots\!96\)\( T^{5} - \)\(20\!\cdots\!08\)\( T^{6} + \)\(74\!\cdots\!48\)\( T^{7} + \)\(21\!\cdots\!25\)\( T^{8} - \)\(14\!\cdots\!94\)\( T^{9} + \)\(31\!\cdots\!61\)\( T^{10} + \)\(18\!\cdots\!62\)\( T^{11} - \)\(66\!\cdots\!98\)\( T^{12} + \)\(18\!\cdots\!62\)\( p^{10} T^{13} + \)\(31\!\cdots\!61\)\( p^{20} T^{14} - \)\(14\!\cdots\!94\)\( p^{30} T^{15} + \)\(21\!\cdots\!25\)\( p^{40} T^{16} + \)\(74\!\cdots\!48\)\( p^{50} T^{17} - \)\(20\!\cdots\!08\)\( p^{60} T^{18} + \)\(46\!\cdots\!96\)\( p^{70} T^{19} + \)\(12\!\cdots\!15\)\( p^{80} T^{20} - \)\(98\!\cdots\!98\)\( p^{90} T^{21} - 312626877283981761 p^{100} T^{22} + 600022554 p^{110} T^{23} + p^{120} T^{24} \) | |
| 59 | \( 1 + 2101762050 T + 3554886883547928465 T^{2} + \)\(43\!\cdots\!50\)\( T^{3} + \)\(45\!\cdots\!95\)\( T^{4} + \)\(38\!\cdots\!20\)\( T^{5} + \)\(27\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!25\)\( T^{8} - \)\(47\!\cdots\!50\)\( T^{9} - \)\(79\!\cdots\!05\)\( T^{10} - \)\(81\!\cdots\!50\)\( T^{11} - \)\(64\!\cdots\!18\)\( T^{12} - \)\(81\!\cdots\!50\)\( p^{10} T^{13} - \)\(79\!\cdots\!05\)\( p^{20} T^{14} - \)\(47\!\cdots\!50\)\( p^{30} T^{15} + \)\(27\!\cdots\!25\)\( p^{40} T^{16} + \)\(13\!\cdots\!00\)\( p^{50} T^{17} + \)\(27\!\cdots\!20\)\( p^{60} T^{18} + \)\(38\!\cdots\!20\)\( p^{70} T^{19} + \)\(45\!\cdots\!95\)\( p^{80} T^{20} + \)\(43\!\cdots\!50\)\( p^{90} T^{21} + 3554886883547928465 p^{100} T^{22} + 2101762050 p^{110} T^{23} + p^{120} T^{24} \) | |
| 61 | \( 1 + 2201391150 T + 5461134675530221647 T^{2} + \)\(84\!\cdots\!50\)\( T^{3} + \)\(13\!\cdots\!39\)\( T^{4} + \)\(17\!\cdots\!72\)\( T^{5} + \)\(21\!\cdots\!28\)\( T^{6} + \)\(24\!\cdots\!28\)\( T^{7} + \)\(25\!\cdots\!53\)\( T^{8} + \)\(25\!\cdots\!26\)\( T^{9} + \)\(24\!\cdots\!77\)\( T^{10} + \)\(22\!\cdots\!66\)\( T^{11} + \)\(19\!\cdots\!10\)\( T^{12} + \)\(22\!\cdots\!66\)\( p^{10} T^{13} + \)\(24\!\cdots\!77\)\( p^{20} T^{14} + \)\(25\!\cdots\!26\)\( p^{30} T^{15} + \)\(25\!\cdots\!53\)\( p^{40} T^{16} + \)\(24\!\cdots\!28\)\( p^{50} T^{17} + \)\(21\!\cdots\!28\)\( p^{60} T^{18} + \)\(17\!\cdots\!72\)\( p^{70} T^{19} + \)\(13\!\cdots\!39\)\( p^{80} T^{20} + \)\(84\!\cdots\!50\)\( p^{90} T^{21} + 5461134675530221647 p^{100} T^{22} + 2201391150 p^{110} T^{23} + p^{120} T^{24} \) | |
| 67 | \( 1 + 1590058326 T - 1957295100843987303 T^{2} - \)\(93\!\cdots\!74\)\( T^{3} - \)\(69\!\cdots\!65\)\( T^{4} + \)\(14\!\cdots\!68\)\( T^{5} + \)\(26\!\cdots\!12\)\( T^{6} + \)\(81\!\cdots\!64\)\( T^{7} - \)\(33\!\cdots\!99\)\( T^{8} + \)\(24\!\cdots\!98\)\( T^{9} + \)\(56\!\cdots\!51\)\( T^{10} - \)\(36\!\cdots\!94\)\( p T^{11} - \)\(13\!\cdots\!78\)\( T^{12} - \)\(36\!\cdots\!94\)\( p^{11} T^{13} + \)\(56\!\cdots\!51\)\( p^{20} T^{14} + \)\(24\!\cdots\!98\)\( p^{30} T^{15} - \)\(33\!\cdots\!99\)\( p^{40} T^{16} + \)\(81\!\cdots\!64\)\( p^{50} T^{17} + \)\(26\!\cdots\!12\)\( p^{60} T^{18} + \)\(14\!\cdots\!68\)\( p^{70} T^{19} - \)\(69\!\cdots\!65\)\( p^{80} T^{20} - \)\(93\!\cdots\!74\)\( p^{90} T^{21} - 1957295100843987303 p^{100} T^{22} + 1590058326 p^{110} T^{23} + p^{120} T^{24} \) | |
| 71 | \( ( 1 + 3869780580 T + 20573561494948089786 T^{2} + \)\(54\!\cdots\!12\)\( T^{3} + \)\(16\!\cdots\!95\)\( T^{4} + \)\(33\!\cdots\!36\)\( T^{5} + \)\(73\!\cdots\!96\)\( T^{6} + \)\(33\!\cdots\!36\)\( p^{10} T^{7} + \)\(16\!\cdots\!95\)\( p^{20} T^{8} + \)\(54\!\cdots\!12\)\( p^{30} T^{9} + 20573561494948089786 p^{40} T^{10} + 3869780580 p^{50} T^{11} + p^{60} T^{12} )^{2} \) | |
| 73 | \( 1 - 2008593834 T + 14491956889889080455 T^{2} - \)\(26\!\cdots\!02\)\( T^{3} + \)\(93\!\cdots\!19\)\( T^{4} - \)\(19\!\cdots\!60\)\( T^{5} + \)\(43\!\cdots\!48\)\( T^{6} - \)\(11\!\cdots\!52\)\( T^{7} + \)\(22\!\cdots\!29\)\( T^{8} - \)\(81\!\cdots\!74\)\( p T^{9} + \)\(13\!\cdots\!41\)\( T^{10} - \)\(39\!\cdots\!58\)\( p T^{11} + \)\(69\!\cdots\!02\)\( T^{12} - \)\(39\!\cdots\!58\)\( p^{11} T^{13} + \)\(13\!\cdots\!41\)\( p^{20} T^{14} - \)\(81\!\cdots\!74\)\( p^{31} T^{15} + \)\(22\!\cdots\!29\)\( p^{40} T^{16} - \)\(11\!\cdots\!52\)\( p^{50} T^{17} + \)\(43\!\cdots\!48\)\( p^{60} T^{18} - \)\(19\!\cdots\!60\)\( p^{70} T^{19} + \)\(93\!\cdots\!19\)\( p^{80} T^{20} - \)\(26\!\cdots\!02\)\( p^{90} T^{21} + 14491956889889080455 p^{100} T^{22} - 2008593834 p^{110} T^{23} + p^{120} T^{24} \) | |
| 79 | \( 1 - 2310562242 T - 25991968824378503487 T^{2} + \)\(11\!\cdots\!26\)\( T^{3} + \)\(26\!\cdots\!19\)\( T^{4} - \)\(23\!\cdots\!60\)\( T^{5} + \)\(49\!\cdots\!64\)\( T^{6} + \)\(30\!\cdots\!40\)\( T^{7} - \)\(56\!\cdots\!71\)\( T^{8} - \)\(25\!\cdots\!66\)\( T^{9} + \)\(99\!\cdots\!83\)\( T^{10} + \)\(91\!\cdots\!22\)\( T^{11} - \)\(11\!\cdots\!22\)\( T^{12} + \)\(91\!\cdots\!22\)\( p^{10} T^{13} + \)\(99\!\cdots\!83\)\( p^{20} T^{14} - \)\(25\!\cdots\!66\)\( p^{30} T^{15} - \)\(56\!\cdots\!71\)\( p^{40} T^{16} + \)\(30\!\cdots\!40\)\( p^{50} T^{17} + \)\(49\!\cdots\!64\)\( p^{60} T^{18} - \)\(23\!\cdots\!60\)\( p^{70} T^{19} + \)\(26\!\cdots\!19\)\( p^{80} T^{20} + \)\(11\!\cdots\!26\)\( p^{90} T^{21} - 25991968824378503487 p^{100} T^{22} - 2310562242 p^{110} T^{23} + p^{120} T^{24} \) | |
| 83 | \( 1 - 62489070939388550220 T^{2} + \)\(26\!\cdots\!38\)\( T^{4} - \)\(79\!\cdots\!20\)\( T^{6} + \)\(19\!\cdots\!63\)\( T^{8} - \)\(38\!\cdots\!60\)\( T^{10} + \)\(66\!\cdots\!52\)\( T^{12} - \)\(38\!\cdots\!60\)\( p^{20} T^{14} + \)\(19\!\cdots\!63\)\( p^{40} T^{16} - \)\(79\!\cdots\!20\)\( p^{60} T^{18} + \)\(26\!\cdots\!38\)\( p^{80} T^{20} - 62489070939388550220 p^{100} T^{22} + p^{120} T^{24} \) | |
| 89 | \( 1 - 2541648690 T + 51996691064873684319 T^{2} - \)\(12\!\cdots\!10\)\( T^{3} + \)\(30\!\cdots\!75\)\( T^{4} + \)\(82\!\cdots\!64\)\( T^{5} - \)\(53\!\cdots\!60\)\( T^{6} + \)\(61\!\cdots\!04\)\( T^{7} - \)\(16\!\cdots\!23\)\( T^{8} + \)\(75\!\cdots\!42\)\( T^{9} + \)\(47\!\cdots\!85\)\( T^{10} - \)\(39\!\cdots\!18\)\( T^{11} + \)\(36\!\cdots\!26\)\( T^{12} - \)\(39\!\cdots\!18\)\( p^{10} T^{13} + \)\(47\!\cdots\!85\)\( p^{20} T^{14} + \)\(75\!\cdots\!42\)\( p^{30} T^{15} - \)\(16\!\cdots\!23\)\( p^{40} T^{16} + \)\(61\!\cdots\!04\)\( p^{50} T^{17} - \)\(53\!\cdots\!60\)\( p^{60} T^{18} + \)\(82\!\cdots\!64\)\( p^{70} T^{19} + \)\(30\!\cdots\!75\)\( p^{80} T^{20} - \)\(12\!\cdots\!10\)\( p^{90} T^{21} + 51996691064873684319 p^{100} T^{22} - 2541648690 p^{110} T^{23} + p^{120} T^{24} \) | |
| 97 | \( 1 - \)\(25\!\cdots\!60\)\( T^{2} + \)\(47\!\cdots\!02\)\( T^{4} - \)\(65\!\cdots\!20\)\( T^{6} + \)\(72\!\cdots\!11\)\( T^{8} - \)\(68\!\cdots\!60\)\( T^{10} + \)\(53\!\cdots\!68\)\( T^{12} - \)\(68\!\cdots\!60\)\( p^{20} T^{14} + \)\(72\!\cdots\!11\)\( p^{40} T^{16} - \)\(65\!\cdots\!20\)\( p^{60} T^{18} + \)\(47\!\cdots\!02\)\( p^{80} T^{20} - \)\(25\!\cdots\!60\)\( p^{100} T^{22} + p^{120} T^{24} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.78739351024307073232016782887, −4.77134693748676828940404318951, −4.67518256529300557048185409482, −4.55425793032650860312453221801, −4.53962538649922967481664027584, −4.48663990796486900172775711112, −4.10096506155357922810779377079, −3.97592395717945952878778776025, −3.63326778175601009316617289638, −3.19674533845582700727570658662, −3.12056911905251418423064959254, −3.05847466182100120367350228274, −3.03427357099666248471262191373, −2.95622700609571270300971460494, −2.52962999200428636274574617108, −2.32327372896061432163838692224, −2.17137564714115660644907899396, −1.75853904760654267008471293088, −1.63629701347823013069773609143, −1.09962673794057555234088261618, −1.00840013893256538640344361197, −0.881309422326726906787495305426, −0.35688558393097108074056757498, −0.35050679396020455966466436706, −0.30194979093020839288105805716, 0.30194979093020839288105805716, 0.35050679396020455966466436706, 0.35688558393097108074056757498, 0.881309422326726906787495305426, 1.00840013893256538640344361197, 1.09962673794057555234088261618, 1.63629701347823013069773609143, 1.75853904760654267008471293088, 2.17137564714115660644907899396, 2.32327372896061432163838692224, 2.52962999200428636274574617108, 2.95622700609571270300971460494, 3.03427357099666248471262191373, 3.05847466182100120367350228274, 3.12056911905251418423064959254, 3.19674533845582700727570658662, 3.63326778175601009316617289638, 3.97592395717945952878778776025, 4.10096506155357922810779377079, 4.48663990796486900172775711112, 4.53962538649922967481664027584, 4.55425793032650860312453221801, 4.67518256529300557048185409482, 4.77134693748676828940404318951, 4.78739351024307073232016782887
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.