Dirichlet series
| L(s) = 1 | + 6·2-s − 4·3-s + 18·4-s + 108·5-s − 24·6-s + 398·7-s − 140·8-s − 3.39e3·9-s + 648·10-s + 1.68e3·11-s − 72·12-s − 3.92e3·13-s + 2.38e3·14-s − 432·15-s + 4.65e3·16-s − 2.03e4·18-s + 1.76e3·19-s + 1.94e3·20-s − 1.59e3·21-s + 1.01e4·22-s + 560·24-s + 5.83e3·25-s − 2.35e4·26-s + 7.75e3·27-s + 7.16e3·28-s − 9.01e4·29-s − 2.59e3·30-s + ⋯ |
| L(s) = 1 | + 3/4·2-s − 0.148·3-s + 9/32·4-s + 0.863·5-s − 1/9·6-s + 1.16·7-s − 0.273·8-s − 4.65·9-s + 0.647·10-s + 1.26·11-s − 0.0416·12-s − 1.78·13-s + 0.870·14-s − 0.127·15-s + 1.13·16-s − 3.49·18-s + 0.257·19-s + 0.242·20-s − 0.171·21-s + 0.950·22-s + 0.0405·24-s + 0.373·25-s − 1.34·26-s + 0.394·27-s + 0.326·28-s − 3.69·29-s − 0.0959·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{12}\right)^{s/2} \, \Gamma_{\C}(s+3)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(512011.\) |
| Root analytic conductor: | \(1.72936\) |
| Motivic weight: | \(6\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 13^{12} ,\ ( \ : [3]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{7}{2})\) | \(\approx\) | \(1.998347175\) |
| \(L(\frac12)\) | \(\approx\) | \(1.998347175\) |
| \(L(4)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 13 | \( 1 + 302 p T - 19022 p^{2} T^{2} - 18833362 p^{3} T^{3} - 2067806973 p^{4} T^{4} + 22347919980 p^{6} T^{5} + 48295301844 p^{9} T^{6} + 22347919980 p^{12} T^{7} - 2067806973 p^{16} T^{8} - 18833362 p^{21} T^{9} - 19022 p^{26} T^{10} + 302 p^{31} T^{11} + p^{36} T^{12} \) |
| good | 2 | \( 1 - 3 p T + 9 p T^{2} + 35 p^{2} T^{3} - 6655 T^{4} + 19167 p T^{5} - 50207 p T^{6} + 45397 p^{4} T^{7} - 288343 p^{6} T^{8} + 63599 p^{10} T^{9} + 13379 p^{13} T^{10} - 50219 p^{18} T^{11} + 466231 p^{19} T^{12} - 50219 p^{24} T^{13} + 13379 p^{25} T^{14} + 63599 p^{28} T^{15} - 288343 p^{30} T^{16} + 45397 p^{34} T^{17} - 50207 p^{37} T^{18} + 19167 p^{43} T^{19} - 6655 p^{48} T^{20} + 35 p^{56} T^{21} + 9 p^{61} T^{22} - 3 p^{67} T^{23} + p^{72} T^{24} \) |
| 3 | \( ( 1 + 2 T + 568 p T^{2} + 2110 p T^{3} + 8792 p^{5} T^{4} + 860522 p^{2} T^{5} + 6979622 p^{5} T^{6} + 860522 p^{8} T^{7} + 8792 p^{17} T^{8} + 2110 p^{19} T^{9} + 568 p^{25} T^{10} + 2 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 5 | \( 1 - 108 T + 5832 T^{2} + 31196 T^{3} - 770587876 T^{4} + 11984521956 p T^{5} - 79083470728 p^{2} T^{6} - 2889445085908 p^{3} T^{7} + 522261950165048 p^{4} T^{8} - 5269005752529644 p^{5} T^{9} + 8653934724018568 p^{6} T^{10} + 2755483427517264764 p^{7} T^{11} - \)\(24\!\cdots\!14\)\( p^{8} T^{12} + 2755483427517264764 p^{13} T^{13} + 8653934724018568 p^{18} T^{14} - 5269005752529644 p^{23} T^{15} + 522261950165048 p^{28} T^{16} - 2889445085908 p^{33} T^{17} - 79083470728 p^{38} T^{18} + 11984521956 p^{43} T^{19} - 770587876 p^{48} T^{20} + 31196 p^{54} T^{21} + 5832 p^{60} T^{22} - 108 p^{66} T^{23} + p^{72} T^{24} \) | |
| 7 | \( 1 - 398 T + 79202 T^{2} - 28188670 T^{3} - 10722345216 T^{4} + 3939021777462 T^{5} - 45885560492542 p T^{6} - 176763204490308034 T^{7} + \)\(55\!\cdots\!08\)\( T^{8} - \)\(12\!\cdots\!14\)\( T^{9} + \)\(16\!\cdots\!54\)\( T^{10} - \)\(90\!\cdots\!14\)\( T^{11} - \)\(37\!\cdots\!10\)\( T^{12} - \)\(90\!\cdots\!14\)\( p^{6} T^{13} + \)\(16\!\cdots\!54\)\( p^{12} T^{14} - \)\(12\!\cdots\!14\)\( p^{18} T^{15} + \)\(55\!\cdots\!08\)\( p^{24} T^{16} - 176763204490308034 p^{30} T^{17} - 45885560492542 p^{37} T^{18} + 3939021777462 p^{42} T^{19} - 10722345216 p^{48} T^{20} - 28188670 p^{54} T^{21} + 79202 p^{60} T^{22} - 398 p^{66} T^{23} + p^{72} T^{24} \) | |
| 11 | \( 1 - 1686 T + 1421298 T^{2} - 859064714 p T^{3} + 21784578331358 T^{4} - 16083795355398150 T^{5} + 40803428417812876874 T^{6} - \)\(11\!\cdots\!94\)\( T^{7} + \)\(99\!\cdots\!75\)\( T^{8} - \)\(10\!\cdots\!04\)\( T^{9} + \)\(34\!\cdots\!28\)\( T^{10} - \)\(39\!\cdots\!32\)\( T^{11} + \)\(22\!\cdots\!72\)\( T^{12} - \)\(39\!\cdots\!32\)\( p^{6} T^{13} + \)\(34\!\cdots\!28\)\( p^{12} T^{14} - \)\(10\!\cdots\!04\)\( p^{18} T^{15} + \)\(99\!\cdots\!75\)\( p^{24} T^{16} - \)\(11\!\cdots\!94\)\( p^{30} T^{17} + 40803428417812876874 p^{36} T^{18} - 16083795355398150 p^{42} T^{19} + 21784578331358 p^{48} T^{20} - 859064714 p^{55} T^{21} + 1421298 p^{60} T^{22} - 1686 p^{66} T^{23} + p^{72} T^{24} \) | |
| 17 | \( 1 - 95568036 T^{2} + 4520630311261512 T^{4} - \)\(18\!\cdots\!40\)\( T^{6} + \)\(66\!\cdots\!80\)\( T^{8} - \)\(19\!\cdots\!68\)\( T^{10} + \)\(48\!\cdots\!06\)\( T^{12} - \)\(19\!\cdots\!68\)\( p^{12} T^{14} + \)\(66\!\cdots\!80\)\( p^{24} T^{16} - \)\(18\!\cdots\!40\)\( p^{36} T^{18} + 4520630311261512 p^{48} T^{20} - 95568036 p^{60} T^{22} + p^{72} T^{24} \) | |
| 19 | \( 1 - 1766 T + 1559378 T^{2} - 696664382926 T^{3} + 60749858441442 p T^{4} + 28961508664554671754 T^{5} + \)\(18\!\cdots\!30\)\( T^{6} - \)\(58\!\cdots\!02\)\( T^{7} - \)\(11\!\cdots\!37\)\( T^{8} - \)\(42\!\cdots\!84\)\( T^{9} + \)\(33\!\cdots\!00\)\( T^{10} - \)\(27\!\cdots\!60\)\( T^{11} + \)\(24\!\cdots\!32\)\( T^{12} - \)\(27\!\cdots\!60\)\( p^{6} T^{13} + \)\(33\!\cdots\!00\)\( p^{12} T^{14} - \)\(42\!\cdots\!84\)\( p^{18} T^{15} - \)\(11\!\cdots\!37\)\( p^{24} T^{16} - \)\(58\!\cdots\!02\)\( p^{30} T^{17} + \)\(18\!\cdots\!30\)\( p^{36} T^{18} + 28961508664554671754 p^{42} T^{19} + 60749858441442 p^{49} T^{20} - 696664382926 p^{54} T^{21} + 1559378 p^{60} T^{22} - 1766 p^{66} T^{23} + p^{72} T^{24} \) | |
| 23 | \( 1 - 814212660 T^{2} + 317916327712063914 T^{4} - \)\(86\!\cdots\!44\)\( T^{6} + \)\(19\!\cdots\!11\)\( T^{8} - \)\(36\!\cdots\!04\)\( T^{10} + \)\(58\!\cdots\!32\)\( T^{12} - \)\(36\!\cdots\!04\)\( p^{12} T^{14} + \)\(19\!\cdots\!11\)\( p^{24} T^{16} - \)\(86\!\cdots\!44\)\( p^{36} T^{18} + 317916327712063914 p^{48} T^{20} - 814212660 p^{60} T^{22} + p^{72} T^{24} \) | |
| 29 | \( ( 1 + 45054 T + 3480794474 T^{2} + 115344513309902 T^{3} + 5022872858956651955 T^{4} + \)\(12\!\cdots\!56\)\( T^{5} + \)\(39\!\cdots\!64\)\( T^{6} + \)\(12\!\cdots\!56\)\( p^{6} T^{7} + 5022872858956651955 p^{12} T^{8} + 115344513309902 p^{18} T^{9} + 3480794474 p^{24} T^{10} + 45054 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 31 | \( 1 - 61014 T + 1861354098 T^{2} - 53209300169710 T^{3} + 2357813398795510374 T^{4} - \)\(94\!\cdots\!82\)\( T^{5} + \)\(27\!\cdots\!46\)\( T^{6} - \)\(72\!\cdots\!38\)\( T^{7} + \)\(25\!\cdots\!35\)\( T^{8} - \)\(10\!\cdots\!80\)\( T^{9} + \)\(34\!\cdots\!28\)\( T^{10} - \)\(96\!\cdots\!40\)\( T^{11} + \)\(27\!\cdots\!24\)\( T^{12} - \)\(96\!\cdots\!40\)\( p^{6} T^{13} + \)\(34\!\cdots\!28\)\( p^{12} T^{14} - \)\(10\!\cdots\!80\)\( p^{18} T^{15} + \)\(25\!\cdots\!35\)\( p^{24} T^{16} - \)\(72\!\cdots\!38\)\( p^{30} T^{17} + \)\(27\!\cdots\!46\)\( p^{36} T^{18} - \)\(94\!\cdots\!82\)\( p^{42} T^{19} + 2357813398795510374 p^{48} T^{20} - 53209300169710 p^{54} T^{21} + 1861354098 p^{60} T^{22} - 61014 p^{66} T^{23} + p^{72} T^{24} \) | |
| 37 | \( 1 + 40212 T + 808502472 T^{2} - 94108369185124 T^{3} - 2212353752837064900 T^{4} + \)\(60\!\cdots\!48\)\( T^{5} + \)\(30\!\cdots\!64\)\( T^{6} + \)\(15\!\cdots\!64\)\( T^{7} - \)\(30\!\cdots\!60\)\( T^{8} - \)\(71\!\cdots\!72\)\( T^{9} + \)\(73\!\cdots\!72\)\( T^{10} + \)\(20\!\cdots\!08\)\( p T^{11} + \)\(59\!\cdots\!98\)\( p^{2} T^{12} + \)\(20\!\cdots\!08\)\( p^{7} T^{13} + \)\(73\!\cdots\!72\)\( p^{12} T^{14} - \)\(71\!\cdots\!72\)\( p^{18} T^{15} - \)\(30\!\cdots\!60\)\( p^{24} T^{16} + \)\(15\!\cdots\!64\)\( p^{30} T^{17} + \)\(30\!\cdots\!64\)\( p^{36} T^{18} + \)\(60\!\cdots\!48\)\( p^{42} T^{19} - 2212353752837064900 p^{48} T^{20} - 94108369185124 p^{54} T^{21} + 808502472 p^{60} T^{22} + 40212 p^{66} T^{23} + p^{72} T^{24} \) | |
| 41 | \( 1 + 190416 T + 18129126528 T^{2} + 1488204135776624 T^{3} + 64138724847656792258 T^{4} - \)\(31\!\cdots\!60\)\( T^{5} - \)\(64\!\cdots\!96\)\( T^{6} - \)\(70\!\cdots\!64\)\( T^{7} - \)\(50\!\cdots\!05\)\( T^{8} - \)\(13\!\cdots\!76\)\( T^{9} + \)\(57\!\cdots\!56\)\( T^{10} + \)\(14\!\cdots\!24\)\( T^{11} + \)\(14\!\cdots\!88\)\( T^{12} + \)\(14\!\cdots\!24\)\( p^{6} T^{13} + \)\(57\!\cdots\!56\)\( p^{12} T^{14} - \)\(13\!\cdots\!76\)\( p^{18} T^{15} - \)\(50\!\cdots\!05\)\( p^{24} T^{16} - \)\(70\!\cdots\!64\)\( p^{30} T^{17} - \)\(64\!\cdots\!96\)\( p^{36} T^{18} - \)\(31\!\cdots\!60\)\( p^{42} T^{19} + 64138724847656792258 p^{48} T^{20} + 1488204135776624 p^{54} T^{21} + 18129126528 p^{60} T^{22} + 190416 p^{66} T^{23} + p^{72} T^{24} \) | |
| 43 | \( 1 - 36018404640 T^{2} + \)\(71\!\cdots\!12\)\( T^{4} - \)\(98\!\cdots\!24\)\( T^{6} + \)\(10\!\cdots\!56\)\( T^{8} - \)\(87\!\cdots\!36\)\( T^{10} + \)\(60\!\cdots\!62\)\( T^{12} - \)\(87\!\cdots\!36\)\( p^{12} T^{14} + \)\(10\!\cdots\!56\)\( p^{24} T^{16} - \)\(98\!\cdots\!24\)\( p^{36} T^{18} + \)\(71\!\cdots\!12\)\( p^{48} T^{20} - 36018404640 p^{60} T^{22} + p^{72} T^{24} \) | |
| 47 | \( 1 - 562446 T + 158172751458 T^{2} - 709421491030610 p T^{3} + \)\(58\!\cdots\!68\)\( T^{4} - \)\(81\!\cdots\!86\)\( T^{5} + \)\(91\!\cdots\!62\)\( T^{6} - \)\(79\!\cdots\!34\)\( T^{7} + \)\(33\!\cdots\!20\)\( T^{8} + \)\(42\!\cdots\!90\)\( T^{9} - \)\(12\!\cdots\!38\)\( T^{10} + \)\(19\!\cdots\!94\)\( T^{11} - \)\(23\!\cdots\!42\)\( T^{12} + \)\(19\!\cdots\!94\)\( p^{6} T^{13} - \)\(12\!\cdots\!38\)\( p^{12} T^{14} + \)\(42\!\cdots\!90\)\( p^{18} T^{15} + \)\(33\!\cdots\!20\)\( p^{24} T^{16} - \)\(79\!\cdots\!34\)\( p^{30} T^{17} + \)\(91\!\cdots\!62\)\( p^{36} T^{18} - \)\(81\!\cdots\!86\)\( p^{42} T^{19} + \)\(58\!\cdots\!68\)\( p^{48} T^{20} - 709421491030610 p^{55} T^{21} + 158172751458 p^{60} T^{22} - 562446 p^{66} T^{23} + p^{72} T^{24} \) | |
| 53 | \( ( 1 - 254568 T + 108652497230 T^{2} - 20683934542944136 T^{3} + \)\(52\!\cdots\!83\)\( T^{4} - \)\(79\!\cdots\!72\)\( T^{5} + \)\(14\!\cdots\!32\)\( T^{6} - \)\(79\!\cdots\!72\)\( p^{6} T^{7} + \)\(52\!\cdots\!83\)\( p^{12} T^{8} - 20683934542944136 p^{18} T^{9} + 108652497230 p^{24} T^{10} - 254568 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 59 | \( 1 + 994458 T + 494473356882 T^{2} + 189903786626655986 T^{3} + \)\(68\!\cdots\!90\)\( T^{4} + \)\(22\!\cdots\!94\)\( T^{5} + \)\(67\!\cdots\!70\)\( T^{6} + \)\(18\!\cdots\!14\)\( T^{7} + \)\(48\!\cdots\!75\)\( T^{8} + \)\(11\!\cdots\!88\)\( T^{9} + \)\(27\!\cdots\!76\)\( T^{10} + \)\(61\!\cdots\!96\)\( T^{11} + \)\(13\!\cdots\!24\)\( T^{12} + \)\(61\!\cdots\!96\)\( p^{6} T^{13} + \)\(27\!\cdots\!76\)\( p^{12} T^{14} + \)\(11\!\cdots\!88\)\( p^{18} T^{15} + \)\(48\!\cdots\!75\)\( p^{24} T^{16} + \)\(18\!\cdots\!14\)\( p^{30} T^{17} + \)\(67\!\cdots\!70\)\( p^{36} T^{18} + \)\(22\!\cdots\!94\)\( p^{42} T^{19} + \)\(68\!\cdots\!90\)\( p^{48} T^{20} + 189903786626655986 p^{54} T^{21} + 494473356882 p^{60} T^{22} + 994458 p^{66} T^{23} + p^{72} T^{24} \) | |
| 61 | \( ( 1 - 506848 T + 282902598538 T^{2} - 105182501059034848 T^{3} + \)\(35\!\cdots\!39\)\( T^{4} - \)\(97\!\cdots\!56\)\( T^{5} + \)\(24\!\cdots\!88\)\( T^{6} - \)\(97\!\cdots\!56\)\( p^{6} T^{7} + \)\(35\!\cdots\!39\)\( p^{12} T^{8} - 105182501059034848 p^{18} T^{9} + 282902598538 p^{24} T^{10} - 506848 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 67 | \( 1 + 1442386 T + 1040238686498 T^{2} + 574663178423383274 T^{3} + \)\(28\!\cdots\!54\)\( T^{4} + \)\(12\!\cdots\!82\)\( T^{5} + \)\(49\!\cdots\!98\)\( T^{6} + \)\(18\!\cdots\!90\)\( T^{7} + \)\(66\!\cdots\!55\)\( T^{8} + \)\(34\!\cdots\!08\)\( p T^{9} + \)\(77\!\cdots\!36\)\( T^{10} + \)\(25\!\cdots\!32\)\( T^{11} + \)\(78\!\cdots\!36\)\( T^{12} + \)\(25\!\cdots\!32\)\( p^{6} T^{13} + \)\(77\!\cdots\!36\)\( p^{12} T^{14} + \)\(34\!\cdots\!08\)\( p^{19} T^{15} + \)\(66\!\cdots\!55\)\( p^{24} T^{16} + \)\(18\!\cdots\!90\)\( p^{30} T^{17} + \)\(49\!\cdots\!98\)\( p^{36} T^{18} + \)\(12\!\cdots\!82\)\( p^{42} T^{19} + \)\(28\!\cdots\!54\)\( p^{48} T^{20} + 574663178423383274 p^{54} T^{21} + 1040238686498 p^{60} T^{22} + 1442386 p^{66} T^{23} + p^{72} T^{24} \) | |
| 71 | \( 1 + 655866 T + 215080104978 T^{2} + 141159966904204586 T^{3} + \)\(93\!\cdots\!48\)\( T^{4} + \)\(25\!\cdots\!74\)\( T^{5} + \)\(62\!\cdots\!38\)\( T^{6} + \)\(33\!\cdots\!78\)\( T^{7} + \)\(73\!\cdots\!60\)\( T^{8} - \)\(16\!\cdots\!30\)\( T^{9} - \)\(63\!\cdots\!38\)\( T^{10} - \)\(36\!\cdots\!50\)\( T^{11} - \)\(20\!\cdots\!42\)\( T^{12} - \)\(36\!\cdots\!50\)\( p^{6} T^{13} - \)\(63\!\cdots\!38\)\( p^{12} T^{14} - \)\(16\!\cdots\!30\)\( p^{18} T^{15} + \)\(73\!\cdots\!60\)\( p^{24} T^{16} + \)\(33\!\cdots\!78\)\( p^{30} T^{17} + \)\(62\!\cdots\!38\)\( p^{36} T^{18} + \)\(25\!\cdots\!74\)\( p^{42} T^{19} + \)\(93\!\cdots\!48\)\( p^{48} T^{20} + 141159966904204586 p^{54} T^{21} + 215080104978 p^{60} T^{22} + 655866 p^{66} T^{23} + p^{72} T^{24} \) | |
| 73 | \( 1 - 2588228 T + 3349462089992 T^{2} - 2980435039024420492 T^{3} + \)\(20\!\cdots\!26\)\( T^{4} - \)\(11\!\cdots\!48\)\( T^{5} + \)\(49\!\cdots\!84\)\( T^{6} - \)\(16\!\cdots\!44\)\( T^{7} + \)\(27\!\cdots\!95\)\( T^{8} + \)\(10\!\cdots\!08\)\( T^{9} - \)\(12\!\cdots\!48\)\( T^{10} + \)\(77\!\cdots\!00\)\( T^{11} - \)\(34\!\cdots\!20\)\( T^{12} + \)\(77\!\cdots\!00\)\( p^{6} T^{13} - \)\(12\!\cdots\!48\)\( p^{12} T^{14} + \)\(10\!\cdots\!08\)\( p^{18} T^{15} + \)\(27\!\cdots\!95\)\( p^{24} T^{16} - \)\(16\!\cdots\!44\)\( p^{30} T^{17} + \)\(49\!\cdots\!84\)\( p^{36} T^{18} - \)\(11\!\cdots\!48\)\( p^{42} T^{19} + \)\(20\!\cdots\!26\)\( p^{48} T^{20} - 2980435039024420492 p^{54} T^{21} + 3349462089992 p^{60} T^{22} - 2588228 p^{66} T^{23} + p^{72} T^{24} \) | |
| 79 | \( ( 1 + 37658 T + 781884604234 T^{2} - 44764114305384646 T^{3} + \)\(35\!\cdots\!55\)\( T^{4} - \)\(18\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!76\)\( T^{6} - \)\(18\!\cdots\!44\)\( p^{6} T^{7} + \)\(35\!\cdots\!55\)\( p^{12} T^{8} - 44764114305384646 p^{18} T^{9} + 781884604234 p^{24} T^{10} + 37658 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 83 | \( 1 - 894966 T + 400482070578 T^{2} - 28674110295070414 T^{3} - \)\(31\!\cdots\!42\)\( T^{4} + \)\(47\!\cdots\!86\)\( T^{5} - \)\(29\!\cdots\!02\)\( T^{6} + \)\(46\!\cdots\!02\)\( T^{7} - \)\(19\!\cdots\!85\)\( T^{8} - \)\(15\!\cdots\!48\)\( T^{9} + \)\(81\!\cdots\!84\)\( T^{10} - \)\(39\!\cdots\!56\)\( T^{11} + \)\(26\!\cdots\!72\)\( T^{12} - \)\(39\!\cdots\!56\)\( p^{6} T^{13} + \)\(81\!\cdots\!84\)\( p^{12} T^{14} - \)\(15\!\cdots\!48\)\( p^{18} T^{15} - \)\(19\!\cdots\!85\)\( p^{24} T^{16} + \)\(46\!\cdots\!02\)\( p^{30} T^{17} - \)\(29\!\cdots\!02\)\( p^{36} T^{18} + \)\(47\!\cdots\!86\)\( p^{42} T^{19} - \)\(31\!\cdots\!42\)\( p^{48} T^{20} - 28674110295070414 p^{54} T^{21} + 400482070578 p^{60} T^{22} - 894966 p^{66} T^{23} + p^{72} T^{24} \) | |
| 89 | \( 1 + 977376 T + 477631922688 T^{2} + 751481099068671296 T^{3} + \)\(91\!\cdots\!82\)\( T^{4} + \)\(32\!\cdots\!88\)\( T^{5} + \)\(15\!\cdots\!80\)\( T^{6} + \)\(21\!\cdots\!88\)\( T^{7} - \)\(48\!\cdots\!97\)\( T^{8} - \)\(14\!\cdots\!96\)\( T^{9} - \)\(50\!\cdots\!00\)\( T^{10} - \)\(76\!\cdots\!96\)\( T^{11} - \)\(10\!\cdots\!76\)\( T^{12} - \)\(76\!\cdots\!96\)\( p^{6} T^{13} - \)\(50\!\cdots\!00\)\( p^{12} T^{14} - \)\(14\!\cdots\!96\)\( p^{18} T^{15} - \)\(48\!\cdots\!97\)\( p^{24} T^{16} + \)\(21\!\cdots\!88\)\( p^{30} T^{17} + \)\(15\!\cdots\!80\)\( p^{36} T^{18} + \)\(32\!\cdots\!88\)\( p^{42} T^{19} + \)\(91\!\cdots\!82\)\( p^{48} T^{20} + 751481099068671296 p^{54} T^{21} + 477631922688 p^{60} T^{22} + 977376 p^{66} T^{23} + p^{72} T^{24} \) | |
| 97 | \( 1 - 983388 T + 483525979272 T^{2} - 532915345861301044 T^{3} + \)\(79\!\cdots\!30\)\( T^{4} - \)\(71\!\cdots\!48\)\( T^{5} + \)\(46\!\cdots\!32\)\( T^{6} - \)\(68\!\cdots\!88\)\( T^{7} + \)\(13\!\cdots\!95\)\( T^{8} - \)\(89\!\cdots\!64\)\( T^{9} + \)\(46\!\cdots\!56\)\( T^{10} - \)\(52\!\cdots\!88\)\( T^{11} + \)\(52\!\cdots\!08\)\( T^{12} - \)\(52\!\cdots\!88\)\( p^{6} T^{13} + \)\(46\!\cdots\!56\)\( p^{12} T^{14} - \)\(89\!\cdots\!64\)\( p^{18} T^{15} + \)\(13\!\cdots\!95\)\( p^{24} T^{16} - \)\(68\!\cdots\!88\)\( p^{30} T^{17} + \)\(46\!\cdots\!32\)\( p^{36} T^{18} - \)\(71\!\cdots\!48\)\( p^{42} T^{19} + \)\(79\!\cdots\!30\)\( p^{48} T^{20} - 532915345861301044 p^{54} T^{21} + 483525979272 p^{60} T^{22} - 983388 p^{66} T^{23} + p^{72} T^{24} \) | |
| 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
−6.42997037227850686615595357179, −6.38019103824701829643610889080, −6.18105194041926002579172058369, −6.10814302173413882934807115089, −5.74668540480848899089570389690, −5.63582779683439791703237382172, −5.61000219935485713789004375493, −5.49785883633525511569949997502, −5.28652861802477691551645407877, −4.98134388997065816822511361284, −4.92935246235027418454324145032, −4.68370083190392673519968836507, −4.14059153918663501922715523942, −3.80372861088556971791478337985, −3.77244250785574721895084480769, −3.56304407949843614556580986971, −3.23372779221072048128359094616, −2.69617062528233393597593469683, −2.54319385957153063342264040241, −2.48428543440956923572269466925, −2.28821731794185273070785333206, −1.54671117598442303582915104988, −1.30315968327935357353648199589, −0.53440404126271950367759641134, −0.24417087655973949509290138804, 0.24417087655973949509290138804, 0.53440404126271950367759641134, 1.30315968327935357353648199589, 1.54671117598442303582915104988, 2.28821731794185273070785333206, 2.48428543440956923572269466925, 2.54319385957153063342264040241, 2.69617062528233393597593469683, 3.23372779221072048128359094616, 3.56304407949843614556580986971, 3.77244250785574721895084480769, 3.80372861088556971791478337985, 4.14059153918663501922715523942, 4.68370083190392673519968836507, 4.92935246235027418454324145032, 4.98134388997065816822511361284, 5.28652861802477691551645407877, 5.49785883633525511569949997502, 5.61000219935485713789004375493, 5.63582779683439791703237382172, 5.74668540480848899089570389690, 6.10814302173413882934807115089, 6.18105194041926002579172058369, 6.38019103824701829643610889080, 6.42997037227850686615595357179
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.