Dirichlet series
| L(s) = 1 | − 966·5-s − 7.69e3·7-s + 4.82e4·9-s + 4.76e4·11-s + 1.03e5·13-s − 4.02e5·17-s + 5.19e5·19-s + 6.83e5·23-s + 5.75e6·25-s + 2.54e6·27-s + 4.25e6·29-s + 1.04e6·31-s + 7.43e6·35-s − 4.88e6·37-s + 3.41e7·41-s − 5.52e7·43-s − 4.65e7·45-s + 2.68e7·47-s + 3.56e7·49-s − 2.17e7·53-s − 4.60e7·55-s − 3.12e7·59-s − 1.55e8·61-s − 3.71e8·63-s − 9.99e7·65-s − 2.12e8·67-s − 6.48e8·71-s + ⋯ |
| L(s) = 1 | − 0.691·5-s − 1.21·7-s + 2.45·9-s + 0.981·11-s + 1.00·13-s − 1.16·17-s + 0.915·19-s + 0.509·23-s + 2.94·25-s + 0.922·27-s + 1.11·29-s + 0.202·31-s + 0.837·35-s − 0.428·37-s + 1.88·41-s − 2.46·43-s − 1.69·45-s + 0.802·47-s + 0.883·49-s − 0.378·53-s − 0.678·55-s − 0.336·59-s − 1.43·61-s − 2.96·63-s − 0.694·65-s − 1.28·67-s − 3.02·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.35726\times 10^{21}\) |
| Root analytic conductor: | \(7.59499\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 7^{12} ,\ ( \ : [9/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(29.33400707\) |
| \(L(\frac12)\) | \(\approx\) | \(29.33400707\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7696 T + 3367046 p T^{2} + 255188688 p^{3} T^{3} - 996752657 p^{6} T^{4} - 44996907232 p^{10} T^{5} - 10647164886452 p^{12} T^{6} - 44996907232 p^{19} T^{7} - 996752657 p^{24} T^{8} + 255188688 p^{30} T^{9} + 3367046 p^{37} T^{10} + 7696 p^{45} T^{11} + p^{54} T^{12} \) | |
| good | 3 | \( 1 - 48239 T^{2} - 283024 p^{2} T^{3} + 966134899 T^{4} + 14668873240 p^{2} T^{5} - 557820125932 p^{2} T^{6} - 15173482732688 p^{5} T^{7} - 774981802772525 p^{5} T^{8} + 31320353847649216 p^{7} T^{9} + 9176107348460622899 p^{6} T^{10} - 85187220101317205704 p^{8} T^{11} - \)\(23\!\cdots\!06\)\( p^{8} T^{12} - 85187220101317205704 p^{17} T^{13} + 9176107348460622899 p^{24} T^{14} + 31320353847649216 p^{34} T^{15} - 774981802772525 p^{41} T^{16} - 15173482732688 p^{50} T^{17} - 557820125932 p^{56} T^{18} + 14668873240 p^{65} T^{19} + 966134899 p^{72} T^{20} - 283024 p^{83} T^{21} - 48239 p^{90} T^{22} + p^{108} T^{24} \) |
| 5 | \( 1 + 966 T - 4825469 T^{2} - 4157447406 T^{3} + 10842942162747 T^{4} + 1508992125017328 p T^{5} - 15240650837093129016 T^{6} - \)\(15\!\cdots\!32\)\( p T^{7} + \)\(41\!\cdots\!69\)\( p^{2} T^{8} + \)\(27\!\cdots\!86\)\( p^{4} T^{9} + \)\(70\!\cdots\!17\)\( p^{4} T^{10} - \)\(65\!\cdots\!74\)\( p^{5} T^{11} - \)\(10\!\cdots\!94\)\( p^{6} T^{12} - \)\(65\!\cdots\!74\)\( p^{14} T^{13} + \)\(70\!\cdots\!17\)\( p^{22} T^{14} + \)\(27\!\cdots\!86\)\( p^{31} T^{15} + \)\(41\!\cdots\!69\)\( p^{38} T^{16} - \)\(15\!\cdots\!32\)\( p^{46} T^{17} - 15240650837093129016 p^{54} T^{18} + 1508992125017328 p^{64} T^{19} + 10842942162747 p^{72} T^{20} - 4157447406 p^{81} T^{21} - 4825469 p^{90} T^{22} + 966 p^{99} T^{23} + p^{108} T^{24} \) | |
| 11 | \( 1 - 47640 T - 7891135223 T^{2} + 247416402168696 T^{3} + 41038550157877865379 T^{4} - \)\(71\!\cdots\!76\)\( T^{5} - \)\(13\!\cdots\!72\)\( T^{6} + \)\(40\!\cdots\!40\)\( T^{7} + \)\(31\!\cdots\!53\)\( T^{8} + \)\(18\!\cdots\!92\)\( T^{9} - \)\(49\!\cdots\!15\)\( p T^{10} - \)\(36\!\cdots\!48\)\( T^{11} + \)\(10\!\cdots\!66\)\( T^{12} - \)\(36\!\cdots\!48\)\( p^{9} T^{13} - \)\(49\!\cdots\!15\)\( p^{19} T^{14} + \)\(18\!\cdots\!92\)\( p^{27} T^{15} + \)\(31\!\cdots\!53\)\( p^{36} T^{16} + \)\(40\!\cdots\!40\)\( p^{45} T^{17} - \)\(13\!\cdots\!72\)\( p^{54} T^{18} - \)\(71\!\cdots\!76\)\( p^{63} T^{19} + 41038550157877865379 p^{72} T^{20} + 247416402168696 p^{81} T^{21} - 7891135223 p^{90} T^{22} - 47640 p^{99} T^{23} + p^{108} T^{24} \) | |
| 13 | \( ( 1 - 51716 T + 9950020650 T^{2} + 345706563560988 T^{3} + \)\(14\!\cdots\!83\)\( T^{4} - \)\(40\!\cdots\!16\)\( T^{5} + \)\(23\!\cdots\!24\)\( T^{6} - \)\(40\!\cdots\!16\)\( p^{9} T^{7} + \)\(14\!\cdots\!83\)\( p^{18} T^{8} + 345706563560988 p^{27} T^{9} + 9950020650 p^{36} T^{10} - 51716 p^{45} T^{11} + p^{54} T^{12} )^{2} \) | |
| 17 | \( 1 + 402234 T - 177497209361 T^{2} - 91486944626415450 T^{3} + \)\(33\!\cdots\!23\)\( T^{4} + \)\(43\!\cdots\!52\)\( T^{5} + \)\(76\!\cdots\!92\)\( T^{6} + \)\(60\!\cdots\!96\)\( T^{7} + \)\(75\!\cdots\!81\)\( T^{8} - \)\(37\!\cdots\!26\)\( T^{9} + \)\(15\!\cdots\!09\)\( T^{10} - \)\(14\!\cdots\!30\)\( T^{11} - \)\(22\!\cdots\!78\)\( T^{12} - \)\(14\!\cdots\!30\)\( p^{9} T^{13} + \)\(15\!\cdots\!09\)\( p^{18} T^{14} - \)\(37\!\cdots\!26\)\( p^{27} T^{15} + \)\(75\!\cdots\!81\)\( p^{36} T^{16} + \)\(60\!\cdots\!96\)\( p^{45} T^{17} + \)\(76\!\cdots\!92\)\( p^{54} T^{18} + \)\(43\!\cdots\!52\)\( p^{63} T^{19} + \)\(33\!\cdots\!23\)\( p^{72} T^{20} - 91486944626415450 p^{81} T^{21} - 177497209361 p^{90} T^{22} + 402234 p^{99} T^{23} + p^{108} T^{24} \) | |
| 19 | \( 1 - 519960 T - 460162602399 T^{2} - 290097011577521512 T^{3} + \)\(27\!\cdots\!99\)\( T^{4} + \)\(19\!\cdots\!68\)\( T^{5} + \)\(37\!\cdots\!80\)\( T^{6} - \)\(47\!\cdots\!44\)\( T^{7} - \)\(43\!\cdots\!31\)\( T^{8} - \)\(38\!\cdots\!72\)\( p T^{9} + \)\(18\!\cdots\!75\)\( T^{10} + \)\(49\!\cdots\!28\)\( T^{11} + \)\(13\!\cdots\!42\)\( T^{12} + \)\(49\!\cdots\!28\)\( p^{9} T^{13} + \)\(18\!\cdots\!75\)\( p^{18} T^{14} - \)\(38\!\cdots\!72\)\( p^{28} T^{15} - \)\(43\!\cdots\!31\)\( p^{36} T^{16} - \)\(47\!\cdots\!44\)\( p^{45} T^{17} + \)\(37\!\cdots\!80\)\( p^{54} T^{18} + \)\(19\!\cdots\!68\)\( p^{63} T^{19} + \)\(27\!\cdots\!99\)\( p^{72} T^{20} - 290097011577521512 p^{81} T^{21} - 460162602399 p^{90} T^{22} - 519960 p^{99} T^{23} + p^{108} T^{24} \) | |
| 23 | \( 1 - 683124 T - 4386944499755 T^{2} + 9491695528806084 p T^{3} + \)\(11\!\cdots\!03\)\( T^{4} + \)\(26\!\cdots\!92\)\( T^{5} - \)\(11\!\cdots\!12\)\( T^{6} - \)\(48\!\cdots\!20\)\( T^{7} - \)\(13\!\cdots\!11\)\( T^{8} - \)\(12\!\cdots\!00\)\( T^{9} + \)\(66\!\cdots\!19\)\( T^{10} + \)\(11\!\cdots\!48\)\( T^{11} - \)\(15\!\cdots\!30\)\( T^{12} + \)\(11\!\cdots\!48\)\( p^{9} T^{13} + \)\(66\!\cdots\!19\)\( p^{18} T^{14} - \)\(12\!\cdots\!00\)\( p^{27} T^{15} - \)\(13\!\cdots\!11\)\( p^{36} T^{16} - \)\(48\!\cdots\!20\)\( p^{45} T^{17} - \)\(11\!\cdots\!12\)\( p^{54} T^{18} + \)\(26\!\cdots\!92\)\( p^{63} T^{19} + \)\(11\!\cdots\!03\)\( p^{72} T^{20} + 9491695528806084 p^{82} T^{21} - 4386944499755 p^{90} T^{22} - 683124 p^{99} T^{23} + p^{108} T^{24} \) | |
| 29 | \( ( 1 - 2126340 T + 21437677879562 T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(44\!\cdots\!63\)\( T^{4} - \)\(12\!\cdots\!52\)\( T^{5} + \)\(91\!\cdots\!32\)\( T^{6} - \)\(12\!\cdots\!52\)\( p^{9} T^{7} + \)\(44\!\cdots\!63\)\( p^{18} T^{8} - \)\(10\!\cdots\!00\)\( p^{27} T^{9} + 21437677879562 p^{36} T^{10} - 2126340 p^{45} T^{11} + p^{54} T^{12} )^{2} \) | |
| 31 | \( 1 - 1040564 T - 78378769383923 T^{2} + 86136167514674739420 T^{3} + \)\(38\!\cdots\!39\)\( T^{4} - \)\(16\!\cdots\!80\)\( p T^{5} - \)\(82\!\cdots\!68\)\( T^{6} + \)\(16\!\cdots\!36\)\( T^{7} - \)\(18\!\cdots\!55\)\( T^{8} - \)\(42\!\cdots\!16\)\( T^{9} + \)\(34\!\cdots\!89\)\( p T^{10} + \)\(40\!\cdots\!36\)\( T^{11} - \)\(37\!\cdots\!26\)\( T^{12} + \)\(40\!\cdots\!36\)\( p^{9} T^{13} + \)\(34\!\cdots\!89\)\( p^{19} T^{14} - \)\(42\!\cdots\!16\)\( p^{27} T^{15} - \)\(18\!\cdots\!55\)\( p^{36} T^{16} + \)\(16\!\cdots\!36\)\( p^{45} T^{17} - \)\(82\!\cdots\!68\)\( p^{54} T^{18} - \)\(16\!\cdots\!80\)\( p^{64} T^{19} + \)\(38\!\cdots\!39\)\( p^{72} T^{20} + 86136167514674739420 p^{81} T^{21} - 78378769383923 p^{90} T^{22} - 1040564 p^{99} T^{23} + p^{108} T^{24} \) | |
| 37 | \( 1 + 4886646 T - 458218243124781 T^{2} + \)\(17\!\cdots\!82\)\( T^{3} + \)\(13\!\cdots\!87\)\( T^{4} - \)\(10\!\cdots\!44\)\( T^{5} - \)\(18\!\cdots\!96\)\( T^{6} + \)\(28\!\cdots\!84\)\( T^{7} + \)\(11\!\cdots\!93\)\( T^{8} - \)\(37\!\cdots\!02\)\( T^{9} + \)\(13\!\cdots\!37\)\( T^{10} + \)\(22\!\cdots\!18\)\( T^{11} - \)\(32\!\cdots\!78\)\( T^{12} + \)\(22\!\cdots\!18\)\( p^{9} T^{13} + \)\(13\!\cdots\!37\)\( p^{18} T^{14} - \)\(37\!\cdots\!02\)\( p^{27} T^{15} + \)\(11\!\cdots\!93\)\( p^{36} T^{16} + \)\(28\!\cdots\!84\)\( p^{45} T^{17} - \)\(18\!\cdots\!96\)\( p^{54} T^{18} - \)\(10\!\cdots\!44\)\( p^{63} T^{19} + \)\(13\!\cdots\!87\)\( p^{72} T^{20} + \)\(17\!\cdots\!82\)\( p^{81} T^{21} - 458218243124781 p^{90} T^{22} + 4886646 p^{99} T^{23} + p^{108} T^{24} \) | |
| 41 | \( ( 1 - 17091564 T + 1100496527703858 T^{2} - \)\(18\!\cdots\!20\)\( T^{3} + \)\(52\!\cdots\!83\)\( T^{4} - \)\(85\!\cdots\!24\)\( T^{5} + \)\(17\!\cdots\!12\)\( T^{6} - \)\(85\!\cdots\!24\)\( p^{9} T^{7} + \)\(52\!\cdots\!83\)\( p^{18} T^{8} - \)\(18\!\cdots\!20\)\( p^{27} T^{9} + 1100496527703858 p^{36} T^{10} - 17091564 p^{45} T^{11} + p^{54} T^{12} )^{2} \) | |
| 43 | \( ( 1 + 27622536 T + 1806440435059506 T^{2} + \)\(30\!\cdots\!88\)\( T^{3} + \)\(10\!\cdots\!87\)\( T^{4} + \)\(11\!\cdots\!48\)\( T^{5} + \)\(40\!\cdots\!88\)\( T^{6} + \)\(11\!\cdots\!48\)\( p^{9} T^{7} + \)\(10\!\cdots\!87\)\( p^{18} T^{8} + \)\(30\!\cdots\!88\)\( p^{27} T^{9} + 1806440435059506 p^{36} T^{10} + 27622536 p^{45} T^{11} + p^{54} T^{12} )^{2} \) | |
| 47 | \( 1 - 26858748 T - 1664507917630371 T^{2} - \)\(23\!\cdots\!52\)\( T^{3} + \)\(19\!\cdots\!83\)\( T^{4} + \)\(10\!\cdots\!68\)\( T^{5} + \)\(44\!\cdots\!12\)\( T^{6} - \)\(11\!\cdots\!92\)\( T^{7} - \)\(28\!\cdots\!59\)\( T^{8} + \)\(24\!\cdots\!72\)\( T^{9} + \)\(15\!\cdots\!67\)\( T^{10} + \)\(51\!\cdots\!32\)\( T^{11} - \)\(18\!\cdots\!26\)\( T^{12} + \)\(51\!\cdots\!32\)\( p^{9} T^{13} + \)\(15\!\cdots\!67\)\( p^{18} T^{14} + \)\(24\!\cdots\!72\)\( p^{27} T^{15} - \)\(28\!\cdots\!59\)\( p^{36} T^{16} - \)\(11\!\cdots\!92\)\( p^{45} T^{17} + \)\(44\!\cdots\!12\)\( p^{54} T^{18} + \)\(10\!\cdots\!68\)\( p^{63} T^{19} + \)\(19\!\cdots\!83\)\( p^{72} T^{20} - \)\(23\!\cdots\!52\)\( p^{81} T^{21} - 1664507917630371 p^{90} T^{22} - 26858748 p^{99} T^{23} + p^{108} T^{24} \) | |
| 53 | \( 1 + 21717558 T - 7942129959887661 T^{2} - \)\(52\!\cdots\!78\)\( T^{3} + \)\(18\!\cdots\!59\)\( T^{4} + \)\(28\!\cdots\!68\)\( T^{5} + \)\(65\!\cdots\!44\)\( T^{6} - \)\(46\!\cdots\!72\)\( T^{7} - \)\(35\!\cdots\!67\)\( T^{8} - \)\(10\!\cdots\!66\)\( T^{9} + \)\(11\!\cdots\!13\)\( T^{10} + \)\(32\!\cdots\!94\)\( T^{11} + \)\(20\!\cdots\!78\)\( T^{12} + \)\(32\!\cdots\!94\)\( p^{9} T^{13} + \)\(11\!\cdots\!13\)\( p^{18} T^{14} - \)\(10\!\cdots\!66\)\( p^{27} T^{15} - \)\(35\!\cdots\!67\)\( p^{36} T^{16} - \)\(46\!\cdots\!72\)\( p^{45} T^{17} + \)\(65\!\cdots\!44\)\( p^{54} T^{18} + \)\(28\!\cdots\!68\)\( p^{63} T^{19} + \)\(18\!\cdots\!59\)\( p^{72} T^{20} - \)\(52\!\cdots\!78\)\( p^{81} T^{21} - 7942129959887661 p^{90} T^{22} + 21717558 p^{99} T^{23} + p^{108} T^{24} \) | |
| 59 | \( 1 + 31298904 T - 30681028071663191 T^{2} - \)\(16\!\cdots\!24\)\( T^{3} + \)\(44\!\cdots\!63\)\( T^{4} + \)\(27\!\cdots\!36\)\( T^{5} - \)\(71\!\cdots\!32\)\( p T^{6} - \)\(18\!\cdots\!68\)\( T^{7} + \)\(41\!\cdots\!37\)\( T^{8} + \)\(15\!\cdots\!00\)\( T^{9} - \)\(49\!\cdots\!45\)\( T^{10} + \)\(23\!\cdots\!20\)\( T^{11} + \)\(50\!\cdots\!38\)\( T^{12} + \)\(23\!\cdots\!20\)\( p^{9} T^{13} - \)\(49\!\cdots\!45\)\( p^{18} T^{14} + \)\(15\!\cdots\!00\)\( p^{27} T^{15} + \)\(41\!\cdots\!37\)\( p^{36} T^{16} - \)\(18\!\cdots\!68\)\( p^{45} T^{17} - \)\(71\!\cdots\!32\)\( p^{55} T^{18} + \)\(27\!\cdots\!36\)\( p^{63} T^{19} + \)\(44\!\cdots\!63\)\( p^{72} T^{20} - \)\(16\!\cdots\!24\)\( p^{81} T^{21} - 30681028071663191 p^{90} T^{22} + 31298904 p^{99} T^{23} + p^{108} T^{24} \) | |
| 61 | \( 1 + 155592542 T - 15672401127142805 T^{2} - \)\(44\!\cdots\!46\)\( T^{3} - \)\(45\!\cdots\!01\)\( T^{4} + \)\(46\!\cdots\!76\)\( T^{5} + \)\(49\!\cdots\!56\)\( T^{6} + \)\(23\!\cdots\!60\)\( T^{7} - \)\(40\!\cdots\!27\)\( T^{8} - \)\(88\!\cdots\!66\)\( T^{9} - \)\(22\!\cdots\!35\)\( T^{10} + \)\(60\!\cdots\!30\)\( T^{11} + \)\(83\!\cdots\!70\)\( T^{12} + \)\(60\!\cdots\!30\)\( p^{9} T^{13} - \)\(22\!\cdots\!35\)\( p^{18} T^{14} - \)\(88\!\cdots\!66\)\( p^{27} T^{15} - \)\(40\!\cdots\!27\)\( p^{36} T^{16} + \)\(23\!\cdots\!60\)\( p^{45} T^{17} + \)\(49\!\cdots\!56\)\( p^{54} T^{18} + \)\(46\!\cdots\!76\)\( p^{63} T^{19} - \)\(45\!\cdots\!01\)\( p^{72} T^{20} - \)\(44\!\cdots\!46\)\( p^{81} T^{21} - 15672401127142805 p^{90} T^{22} + 155592542 p^{99} T^{23} + p^{108} T^{24} \) | |
| 67 | \( 1 + 212196992 T - 71489125899378319 T^{2} - \)\(24\!\cdots\!64\)\( T^{3} + \)\(18\!\cdots\!35\)\( T^{4} + \)\(13\!\cdots\!52\)\( T^{5} + \)\(32\!\cdots\!00\)\( T^{6} - \)\(45\!\cdots\!84\)\( T^{7} - \)\(45\!\cdots\!83\)\( T^{8} + \)\(10\!\cdots\!52\)\( T^{9} + \)\(20\!\cdots\!91\)\( T^{10} - \)\(10\!\cdots\!44\)\( T^{11} - \)\(63\!\cdots\!90\)\( T^{12} - \)\(10\!\cdots\!44\)\( p^{9} T^{13} + \)\(20\!\cdots\!91\)\( p^{18} T^{14} + \)\(10\!\cdots\!52\)\( p^{27} T^{15} - \)\(45\!\cdots\!83\)\( p^{36} T^{16} - \)\(45\!\cdots\!84\)\( p^{45} T^{17} + \)\(32\!\cdots\!00\)\( p^{54} T^{18} + \)\(13\!\cdots\!52\)\( p^{63} T^{19} + \)\(18\!\cdots\!35\)\( p^{72} T^{20} - \)\(24\!\cdots\!64\)\( p^{81} T^{21} - 71489125899378319 p^{90} T^{22} + 212196992 p^{99} T^{23} + p^{108} T^{24} \) | |
| 71 | \( ( 1 + 324259824 T + 3620604412976326 p T^{2} + \)\(56\!\cdots\!76\)\( T^{3} + \)\(25\!\cdots\!15\)\( T^{4} + \)\(42\!\cdots\!48\)\( T^{5} + \)\(14\!\cdots\!00\)\( T^{6} + \)\(42\!\cdots\!48\)\( p^{9} T^{7} + \)\(25\!\cdots\!15\)\( p^{18} T^{8} + \)\(56\!\cdots\!76\)\( p^{27} T^{9} + 3620604412976326 p^{37} T^{10} + 324259824 p^{45} T^{11} + p^{54} T^{12} )^{2} \) | |
| 73 | \( 1 + 549596530 T - 49656956025488233 T^{2} - \)\(83\!\cdots\!30\)\( T^{3} - \)\(47\!\cdots\!81\)\( T^{4} + \)\(61\!\cdots\!64\)\( T^{5} + \)\(72\!\cdots\!60\)\( T^{6} - \)\(18\!\cdots\!52\)\( T^{7} - \)\(17\!\cdots\!35\)\( T^{8} + \)\(47\!\cdots\!26\)\( T^{9} - \)\(42\!\cdots\!47\)\( T^{10} + \)\(14\!\cdots\!70\)\( T^{11} + \)\(40\!\cdots\!86\)\( T^{12} + \)\(14\!\cdots\!70\)\( p^{9} T^{13} - \)\(42\!\cdots\!47\)\( p^{18} T^{14} + \)\(47\!\cdots\!26\)\( p^{27} T^{15} - \)\(17\!\cdots\!35\)\( p^{36} T^{16} - \)\(18\!\cdots\!52\)\( p^{45} T^{17} + \)\(72\!\cdots\!60\)\( p^{54} T^{18} + \)\(61\!\cdots\!64\)\( p^{63} T^{19} - \)\(47\!\cdots\!81\)\( p^{72} T^{20} - \)\(83\!\cdots\!30\)\( p^{81} T^{21} - 49656956025488233 p^{90} T^{22} + 549596530 p^{99} T^{23} + p^{108} T^{24} \) | |
| 79 | \( 1 + 227611204 T - 509452402242280307 T^{2} - \)\(48\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!19\)\( T^{4} + \)\(21\!\cdots\!92\)\( T^{5} - \)\(32\!\cdots\!04\)\( T^{6} + \)\(11\!\cdots\!04\)\( T^{7} + \)\(51\!\cdots\!13\)\( T^{8} - \)\(24\!\cdots\!04\)\( T^{9} - \)\(67\!\cdots\!25\)\( T^{10} + \)\(14\!\cdots\!80\)\( T^{11} + \)\(82\!\cdots\!34\)\( T^{12} + \)\(14\!\cdots\!80\)\( p^{9} T^{13} - \)\(67\!\cdots\!25\)\( p^{18} T^{14} - \)\(24\!\cdots\!04\)\( p^{27} T^{15} + \)\(51\!\cdots\!13\)\( p^{36} T^{16} + \)\(11\!\cdots\!04\)\( p^{45} T^{17} - \)\(32\!\cdots\!04\)\( p^{54} T^{18} + \)\(21\!\cdots\!92\)\( p^{63} T^{19} + \)\(15\!\cdots\!19\)\( p^{72} T^{20} - \)\(48\!\cdots\!00\)\( p^{81} T^{21} - 509452402242280307 p^{90} T^{22} + 227611204 p^{99} T^{23} + p^{108} T^{24} \) | |
| 83 | \( ( 1 + 494669112 T + 10054652749638550 p T^{2} + \)\(30\!\cdots\!32\)\( T^{3} + \)\(30\!\cdots\!71\)\( T^{4} + \)\(86\!\cdots\!08\)\( T^{5} + \)\(69\!\cdots\!16\)\( T^{6} + \)\(86\!\cdots\!08\)\( p^{9} T^{7} + \)\(30\!\cdots\!71\)\( p^{18} T^{8} + \)\(30\!\cdots\!32\)\( p^{27} T^{9} + 10054652749638550 p^{37} T^{10} + 494669112 p^{45} T^{11} + p^{54} T^{12} )^{2} \) | |
| 89 | \( 1 + 1746541986 T + 533591123842112487 T^{2} - \)\(70\!\cdots\!18\)\( T^{3} - \)\(33\!\cdots\!49\)\( T^{4} + \)\(26\!\cdots\!88\)\( T^{5} + \)\(14\!\cdots\!92\)\( T^{6} - \)\(93\!\cdots\!12\)\( T^{7} - \)\(51\!\cdots\!27\)\( T^{8} + \)\(33\!\cdots\!82\)\( T^{9} + \)\(23\!\cdots\!77\)\( T^{10} - \)\(93\!\cdots\!34\)\( T^{11} - \)\(14\!\cdots\!78\)\( T^{12} - \)\(93\!\cdots\!34\)\( p^{9} T^{13} + \)\(23\!\cdots\!77\)\( p^{18} T^{14} + \)\(33\!\cdots\!82\)\( p^{27} T^{15} - \)\(51\!\cdots\!27\)\( p^{36} T^{16} - \)\(93\!\cdots\!12\)\( p^{45} T^{17} + \)\(14\!\cdots\!92\)\( p^{54} T^{18} + \)\(26\!\cdots\!88\)\( p^{63} T^{19} - \)\(33\!\cdots\!49\)\( p^{72} T^{20} - \)\(70\!\cdots\!18\)\( p^{81} T^{21} + 533591123842112487 p^{90} T^{22} + 1746541986 p^{99} T^{23} + p^{108} T^{24} \) | |
| 97 | \( ( 1 - 1153357660 T + 3878466323319733154 T^{2} - \)\(36\!\cdots\!60\)\( T^{3} + \)\(65\!\cdots\!71\)\( T^{4} - \)\(50\!\cdots\!60\)\( T^{5} + \)\(64\!\cdots\!28\)\( T^{6} - \)\(50\!\cdots\!60\)\( p^{9} T^{7} + \)\(65\!\cdots\!71\)\( p^{18} T^{8} - \)\(36\!\cdots\!60\)\( p^{27} T^{9} + 3878466323319733154 p^{36} T^{10} - 1153357660 p^{45} T^{11} + p^{54} T^{12} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.08124278967714636033420157293, −2.95381694362796981856230714054, −2.94021778231694412862610663271, −2.83269474340738242269265836625, −2.60325130720949880223703549937, −2.56456485366096994412034482499, −2.36241445472418013773195724268, −2.21039413930761175887158657600, −1.99772537727437443533035094575, −1.94774580995131556477779890689, −1.84408105982263562290565294835, −1.59582881036863748092663060441, −1.50124852460389049820602399876, −1.35890562671999878322208197809, −1.33388317212845330445124460724, −1.22629373250227278372016389326, −1.12296909183574492901564741753, −1.00467424531747654008786597143, −0.921098231541845519234461356425, −0.819819904338279281488838168695, −0.53160257826246389734184932658, −0.39345484259855760346928806995, −0.37150723415457877223369647462, −0.25862016996642981486764035382, −0.17698183566914278888074389332, 0.17698183566914278888074389332, 0.25862016996642981486764035382, 0.37150723415457877223369647462, 0.39345484259855760346928806995, 0.53160257826246389734184932658, 0.819819904338279281488838168695, 0.921098231541845519234461356425, 1.00467424531747654008786597143, 1.12296909183574492901564741753, 1.22629373250227278372016389326, 1.33388317212845330445124460724, 1.35890562671999878322208197809, 1.50124852460389049820602399876, 1.59582881036863748092663060441, 1.84408105982263562290565294835, 1.94774580995131556477779890689, 1.99772537727437443533035094575, 2.21039413930761175887158657600, 2.36241445472418013773195724268, 2.56456485366096994412034482499, 2.60325130720949880223703549937, 2.83269474340738242269265836625, 2.94021778231694412862610663271, 2.95381694362796981856230714054, 3.08124278967714636033420157293
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.