Dirichlet series
| L(s) = 1 | + 32·2-s − 20·3-s − 5.12e3·4-s + 1.03e3·5-s − 640·6-s + 1.59e5·7-s − 1.58e5·8-s − 6.68e5·9-s + 3.30e4·10-s − 7.71e5·11-s + 1.02e5·12-s + 3.43e6·13-s + 5.10e6·14-s − 2.06e4·15-s + 1.24e7·16-s + 2.90e7·17-s − 2.13e7·18-s + 2.13e7·19-s − 5.29e6·20-s − 3.19e6·21-s − 2.46e7·22-s − 7.07e7·23-s + 3.16e6·24-s − 1.51e8·25-s + 1.09e8·26-s + 1.33e8·27-s − 8.17e8·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0475·3-s − 5/2·4-s + 0.147·5-s − 0.0336·6-s + 3.58·7-s − 1.70·8-s − 3.77·9-s + 0.104·10-s − 1.44·11-s + 0.118·12-s + 2.56·13-s + 2.53·14-s − 0.00703·15-s + 2.97·16-s + 4.95·17-s − 2.66·18-s + 1.98·19-s − 0.369·20-s − 0.170·21-s − 1.02·22-s − 2.29·23-s + 0.0810·24-s − 3.09·25-s + 1.81·26-s + 1.78·27-s − 8.97·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{11}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(23^{11}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.24945\times 10^{13}\) |
| Root analytic conductor: | \(4.20379\) |
| Motivic weight: | \(11\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((22,\ 23^{11} ,\ ( \ : [11/2]^{11} ),\ 1 )\) |
Particular Values
| \(L(6)\) | \(\approx\) | \(47.18546768\) |
| \(L(\frac12)\) | \(\approx\) | \(47.18546768\) |
| \(L(\frac{13}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 23 | \( ( 1 + p^{5} T )^{11} \) |
| good | 2 | \( 1 - p^{5} T + 3 p^{11} T^{2} - 202339 T^{3} + 5096619 p^{2} T^{4} - 98614353 p^{3} T^{5} + 410078509 p^{7} T^{6} - 139914498183 p^{4} T^{7} + 1639203231795 p^{6} T^{8} - 670633329835 p^{13} T^{9} + 189137681093 p^{20} T^{10} - 378534412964811 p^{15} T^{11} + 189137681093 p^{31} T^{12} - 670633329835 p^{35} T^{13} + 1639203231795 p^{39} T^{14} - 139914498183 p^{48} T^{15} + 410078509 p^{62} T^{16} - 98614353 p^{69} T^{17} + 5096619 p^{79} T^{18} - 202339 p^{88} T^{19} + 3 p^{110} T^{20} - p^{115} T^{21} + p^{121} T^{22} \) |
| 3 | \( 1 + 20 T + 668537 T^{2} - 106606144 T^{3} + 217473901057 T^{4} - 5886180855220 p^{2} T^{5} + 6681487416784825 p^{2} T^{6} - 119501273934056456 p^{4} T^{7} + 703168736465770379 p^{9} T^{8} - 28042194627658304140 p^{10} T^{9} + \)\(40\!\cdots\!25\)\( p^{10} T^{10} - \)\(66\!\cdots\!24\)\( p^{12} T^{11} + \)\(40\!\cdots\!25\)\( p^{21} T^{12} - 28042194627658304140 p^{32} T^{13} + 703168736465770379 p^{42} T^{14} - 119501273934056456 p^{48} T^{15} + 6681487416784825 p^{57} T^{16} - 5886180855220 p^{68} T^{17} + 217473901057 p^{77} T^{18} - 106606144 p^{88} T^{19} + 668537 p^{99} T^{20} + 20 p^{110} T^{21} + p^{121} T^{22} \) | |
| 5 | \( 1 - 1034 T + 152308011 T^{2} - 404352475308 T^{3} + 13807929098377051 T^{4} - 9934829444945386826 p T^{5} + \)\(43\!\cdots\!57\)\( p^{2} T^{6} - \)\(28\!\cdots\!92\)\( p^{3} T^{7} + \)\(11\!\cdots\!58\)\( p^{4} T^{8} - \)\(78\!\cdots\!28\)\( p^{5} T^{9} + \)\(25\!\cdots\!34\)\( p^{6} T^{10} - \)\(18\!\cdots\!64\)\( p^{7} T^{11} + \)\(25\!\cdots\!34\)\( p^{17} T^{12} - \)\(78\!\cdots\!28\)\( p^{27} T^{13} + \)\(11\!\cdots\!58\)\( p^{37} T^{14} - \)\(28\!\cdots\!92\)\( p^{47} T^{15} + \)\(43\!\cdots\!57\)\( p^{57} T^{16} - 9934829444945386826 p^{67} T^{17} + 13807929098377051 p^{77} T^{18} - 404352475308 p^{88} T^{19} + 152308011 p^{99} T^{20} - 1034 p^{110} T^{21} + p^{121} T^{22} \) | |
| 7 | \( 1 - 159584 T + 21430630505 T^{2} - 1936951291247600 T^{3} + 22221406987807308461 p T^{4} - \)\(29\!\cdots\!72\)\( p^{3} T^{5} + \)\(18\!\cdots\!25\)\( p^{3} T^{6} - \)\(14\!\cdots\!04\)\( p^{4} T^{7} + \)\(10\!\cdots\!86\)\( p^{5} T^{8} - \)\(10\!\cdots\!84\)\( p^{7} T^{9} + \)\(50\!\cdots\!42\)\( p^{7} T^{10} - \)\(32\!\cdots\!92\)\( p^{8} T^{11} + \)\(50\!\cdots\!42\)\( p^{18} T^{12} - \)\(10\!\cdots\!84\)\( p^{29} T^{13} + \)\(10\!\cdots\!86\)\( p^{38} T^{14} - \)\(14\!\cdots\!04\)\( p^{48} T^{15} + \)\(18\!\cdots\!25\)\( p^{58} T^{16} - \)\(29\!\cdots\!72\)\( p^{69} T^{17} + 22221406987807308461 p^{78} T^{18} - 1936951291247600 p^{88} T^{19} + 21430630505 p^{99} T^{20} - 159584 p^{110} T^{21} + p^{121} T^{22} \) | |
| 11 | \( 1 + 771396 T + 1610354568081 T^{2} + 838694347423798376 T^{3} + \)\(10\!\cdots\!75\)\( T^{4} + \)\(33\!\cdots\!20\)\( T^{5} + \)\(37\!\cdots\!63\)\( T^{6} + \)\(37\!\cdots\!64\)\( T^{7} + \)\(94\!\cdots\!06\)\( T^{8} - \)\(14\!\cdots\!52\)\( T^{9} + \)\(22\!\cdots\!54\)\( T^{10} - \)\(70\!\cdots\!32\)\( T^{11} + \)\(22\!\cdots\!54\)\( p^{11} T^{12} - \)\(14\!\cdots\!52\)\( p^{22} T^{13} + \)\(94\!\cdots\!06\)\( p^{33} T^{14} + \)\(37\!\cdots\!64\)\( p^{44} T^{15} + \)\(37\!\cdots\!63\)\( p^{55} T^{16} + \)\(33\!\cdots\!20\)\( p^{66} T^{17} + \)\(10\!\cdots\!75\)\( p^{77} T^{18} + 838694347423798376 p^{88} T^{19} + 1610354568081 p^{99} T^{20} + 771396 p^{110} T^{21} + p^{121} T^{22} \) | |
| 13 | \( 1 - 3433434 T + 14687481254431 T^{2} - 39036989376663117612 T^{3} + \)\(10\!\cdots\!33\)\( T^{4} - \)\(17\!\cdots\!22\)\( p T^{5} + \)\(47\!\cdots\!11\)\( T^{6} - \)\(84\!\cdots\!20\)\( T^{7} + \)\(14\!\cdots\!33\)\( T^{8} - \)\(23\!\cdots\!10\)\( T^{9} + \)\(35\!\cdots\!07\)\( T^{10} - \)\(47\!\cdots\!96\)\( T^{11} + \)\(35\!\cdots\!07\)\( p^{11} T^{12} - \)\(23\!\cdots\!10\)\( p^{22} T^{13} + \)\(14\!\cdots\!33\)\( p^{33} T^{14} - \)\(84\!\cdots\!20\)\( p^{44} T^{15} + \)\(47\!\cdots\!11\)\( p^{55} T^{16} - \)\(17\!\cdots\!22\)\( p^{67} T^{17} + \)\(10\!\cdots\!33\)\( p^{77} T^{18} - 39036989376663117612 p^{88} T^{19} + 14687481254431 p^{99} T^{20} - 3433434 p^{110} T^{21} + p^{121} T^{22} \) | |
| 17 | \( 1 - 29035398 T + 36222148777795 p T^{2} - \)\(92\!\cdots\!28\)\( T^{3} + \)\(11\!\cdots\!51\)\( T^{4} - \)\(12\!\cdots\!82\)\( T^{5} + \)\(12\!\cdots\!65\)\( T^{6} - \)\(10\!\cdots\!12\)\( T^{7} + \)\(84\!\cdots\!90\)\( T^{8} - \)\(35\!\cdots\!36\)\( p T^{9} + \)\(39\!\cdots\!82\)\( T^{10} - \)\(24\!\cdots\!76\)\( T^{11} + \)\(39\!\cdots\!82\)\( p^{11} T^{12} - \)\(35\!\cdots\!36\)\( p^{23} T^{13} + \)\(84\!\cdots\!90\)\( p^{33} T^{14} - \)\(10\!\cdots\!12\)\( p^{44} T^{15} + \)\(12\!\cdots\!65\)\( p^{55} T^{16} - \)\(12\!\cdots\!82\)\( p^{66} T^{17} + \)\(11\!\cdots\!51\)\( p^{77} T^{18} - \)\(92\!\cdots\!28\)\( p^{88} T^{19} + 36222148777795 p^{100} T^{20} - 29035398 p^{110} T^{21} + p^{121} T^{22} \) | |
| 19 | \( 1 - 21398428 T + 903499633180685 T^{2} - \)\(15\!\cdots\!32\)\( T^{3} + \)\(38\!\cdots\!71\)\( T^{4} - \)\(56\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!99\)\( T^{6} - \)\(13\!\cdots\!92\)\( T^{7} + \)\(21\!\cdots\!14\)\( T^{8} - \)\(24\!\cdots\!00\)\( T^{9} + \)\(32\!\cdots\!98\)\( T^{10} - \)\(32\!\cdots\!84\)\( T^{11} + \)\(32\!\cdots\!98\)\( p^{11} T^{12} - \)\(24\!\cdots\!00\)\( p^{22} T^{13} + \)\(21\!\cdots\!14\)\( p^{33} T^{14} - \)\(13\!\cdots\!92\)\( p^{44} T^{15} + \)\(10\!\cdots\!99\)\( p^{55} T^{16} - \)\(56\!\cdots\!16\)\( p^{66} T^{17} + \)\(38\!\cdots\!71\)\( p^{77} T^{18} - \)\(15\!\cdots\!32\)\( p^{88} T^{19} + 903499633180685 p^{99} T^{20} - 21398428 p^{110} T^{21} + p^{121} T^{22} \) | |
| 29 | \( 1 - 226699042 T + 126656219731067895 T^{2} - \)\(22\!\cdots\!80\)\( T^{3} + \)\(70\!\cdots\!09\)\( T^{4} - \)\(10\!\cdots\!98\)\( T^{5} + \)\(23\!\cdots\!15\)\( T^{6} - \)\(29\!\cdots\!80\)\( T^{7} + \)\(53\!\cdots\!09\)\( T^{8} - \)\(57\!\cdots\!98\)\( T^{9} + \)\(87\!\cdots\!99\)\( T^{10} - \)\(81\!\cdots\!24\)\( T^{11} + \)\(87\!\cdots\!99\)\( p^{11} T^{12} - \)\(57\!\cdots\!98\)\( p^{22} T^{13} + \)\(53\!\cdots\!09\)\( p^{33} T^{14} - \)\(29\!\cdots\!80\)\( p^{44} T^{15} + \)\(23\!\cdots\!15\)\( p^{55} T^{16} - \)\(10\!\cdots\!98\)\( p^{66} T^{17} + \)\(70\!\cdots\!09\)\( p^{77} T^{18} - \)\(22\!\cdots\!80\)\( p^{88} T^{19} + 126656219731067895 p^{99} T^{20} - 226699042 p^{110} T^{21} + p^{121} T^{22} \) | |
| 31 | \( 1 - 251932328 T + 78570872736309821 T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(46\!\cdots\!01\)\( T^{4} - \)\(85\!\cdots\!16\)\( T^{5} + \)\(20\!\cdots\!17\)\( T^{6} - \)\(37\!\cdots\!72\)\( T^{7} + \)\(75\!\cdots\!57\)\( T^{8} - \)\(39\!\cdots\!96\)\( p T^{9} + \)\(23\!\cdots\!73\)\( p^{2} T^{10} - \)\(32\!\cdots\!56\)\( T^{11} + \)\(23\!\cdots\!73\)\( p^{13} T^{12} - \)\(39\!\cdots\!96\)\( p^{23} T^{13} + \)\(75\!\cdots\!57\)\( p^{33} T^{14} - \)\(37\!\cdots\!72\)\( p^{44} T^{15} + \)\(20\!\cdots\!17\)\( p^{55} T^{16} - \)\(85\!\cdots\!16\)\( p^{66} T^{17} + \)\(46\!\cdots\!01\)\( p^{77} T^{18} - \)\(17\!\cdots\!00\)\( p^{88} T^{19} + 78570872736309821 p^{99} T^{20} - 251932328 p^{110} T^{21} + p^{121} T^{22} \) | |
| 37 | \( 1 - 573876170 T + 31562781693832303 p T^{2} - \)\(59\!\cdots\!40\)\( T^{3} + \)\(66\!\cdots\!75\)\( T^{4} - \)\(29\!\cdots\!74\)\( T^{5} + \)\(24\!\cdots\!57\)\( T^{6} - \)\(26\!\cdots\!52\)\( p T^{7} + \)\(67\!\cdots\!26\)\( T^{8} - \)\(23\!\cdots\!52\)\( T^{9} + \)\(14\!\cdots\!26\)\( T^{10} - \)\(45\!\cdots\!24\)\( T^{11} + \)\(14\!\cdots\!26\)\( p^{11} T^{12} - \)\(23\!\cdots\!52\)\( p^{22} T^{13} + \)\(67\!\cdots\!26\)\( p^{33} T^{14} - \)\(26\!\cdots\!52\)\( p^{45} T^{15} + \)\(24\!\cdots\!57\)\( p^{55} T^{16} - \)\(29\!\cdots\!74\)\( p^{66} T^{17} + \)\(66\!\cdots\!75\)\( p^{77} T^{18} - \)\(59\!\cdots\!40\)\( p^{88} T^{19} + 31562781693832303 p^{100} T^{20} - 573876170 p^{110} T^{21} + p^{121} T^{22} \) | |
| 41 | \( 1 + 1733596378 T + 4977186208038793939 T^{2} + \)\(69\!\cdots\!80\)\( T^{3} + \)\(27\!\cdots\!93\)\( p T^{4} + \)\(13\!\cdots\!50\)\( T^{5} + \)\(16\!\cdots\!59\)\( T^{6} + \)\(16\!\cdots\!80\)\( T^{7} + \)\(16\!\cdots\!29\)\( T^{8} + \)\(14\!\cdots\!02\)\( T^{9} + \)\(12\!\cdots\!59\)\( T^{10} + \)\(92\!\cdots\!80\)\( T^{11} + \)\(12\!\cdots\!59\)\( p^{11} T^{12} + \)\(14\!\cdots\!02\)\( p^{22} T^{13} + \)\(16\!\cdots\!29\)\( p^{33} T^{14} + \)\(16\!\cdots\!80\)\( p^{44} T^{15} + \)\(16\!\cdots\!59\)\( p^{55} T^{16} + \)\(13\!\cdots\!50\)\( p^{66} T^{17} + \)\(27\!\cdots\!93\)\( p^{78} T^{18} + \)\(69\!\cdots\!80\)\( p^{88} T^{19} + 4977186208038793939 p^{99} T^{20} + 1733596378 p^{110} T^{21} + p^{121} T^{22} \) | |
| 43 | \( 1 - 647370308 T + 3107396728530397461 T^{2} - \)\(14\!\cdots\!32\)\( T^{3} + \)\(76\!\cdots\!47\)\( T^{4} - \)\(39\!\cdots\!16\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} - \)\(53\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!14\)\( T^{8} - \)\(76\!\cdots\!76\)\( T^{9} + \)\(19\!\cdots\!38\)\( T^{10} - \)\(69\!\cdots\!72\)\( T^{11} + \)\(19\!\cdots\!38\)\( p^{11} T^{12} - \)\(76\!\cdots\!76\)\( p^{22} T^{13} + \)\(17\!\cdots\!14\)\( p^{33} T^{14} - \)\(53\!\cdots\!00\)\( p^{44} T^{15} + \)\(12\!\cdots\!43\)\( p^{55} T^{16} - \)\(39\!\cdots\!16\)\( p^{66} T^{17} + \)\(76\!\cdots\!47\)\( p^{77} T^{18} - \)\(14\!\cdots\!32\)\( p^{88} T^{19} + 3107396728530397461 p^{99} T^{20} - 647370308 p^{110} T^{21} + p^{121} T^{22} \) | |
| 47 | \( 1 + 5436527248 T + 23185955642850302173 T^{2} + \)\(66\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!93\)\( T^{4} + \)\(36\!\cdots\!60\)\( T^{5} + \)\(75\!\cdots\!53\)\( T^{6} + \)\(14\!\cdots\!36\)\( T^{7} + \)\(27\!\cdots\!45\)\( T^{8} + \)\(48\!\cdots\!16\)\( T^{9} + \)\(82\!\cdots\!53\)\( T^{10} + \)\(13\!\cdots\!96\)\( T^{11} + \)\(82\!\cdots\!53\)\( p^{11} T^{12} + \)\(48\!\cdots\!16\)\( p^{22} T^{13} + \)\(27\!\cdots\!45\)\( p^{33} T^{14} + \)\(14\!\cdots\!36\)\( p^{44} T^{15} + \)\(75\!\cdots\!53\)\( p^{55} T^{16} + \)\(36\!\cdots\!60\)\( p^{66} T^{17} + \)\(16\!\cdots\!93\)\( p^{77} T^{18} + \)\(66\!\cdots\!20\)\( p^{88} T^{19} + 23185955642850302173 p^{99} T^{20} + 5436527248 p^{110} T^{21} + p^{121} T^{22} \) | |
| 53 | \( 1 + 3387203910 T + 61615423446485057775 T^{2} + \)\(19\!\cdots\!44\)\( T^{3} + \)\(18\!\cdots\!19\)\( T^{4} + \)\(52\!\cdots\!26\)\( T^{5} + \)\(35\!\cdots\!13\)\( T^{6} + \)\(16\!\cdots\!56\)\( p T^{7} + \)\(49\!\cdots\!30\)\( T^{8} + \)\(11\!\cdots\!00\)\( T^{9} + \)\(54\!\cdots\!10\)\( T^{10} + \)\(11\!\cdots\!40\)\( T^{11} + \)\(54\!\cdots\!10\)\( p^{11} T^{12} + \)\(11\!\cdots\!00\)\( p^{22} T^{13} + \)\(49\!\cdots\!30\)\( p^{33} T^{14} + \)\(16\!\cdots\!56\)\( p^{45} T^{15} + \)\(35\!\cdots\!13\)\( p^{55} T^{16} + \)\(52\!\cdots\!26\)\( p^{66} T^{17} + \)\(18\!\cdots\!19\)\( p^{77} T^{18} + \)\(19\!\cdots\!44\)\( p^{88} T^{19} + 61615423446485057775 p^{99} T^{20} + 3387203910 p^{110} T^{21} + p^{121} T^{22} \) | |
| 59 | \( 1 - 15113662084 T + \)\(27\!\cdots\!13\)\( T^{2} - \)\(55\!\cdots\!84\)\( p T^{3} + \)\(36\!\cdots\!83\)\( T^{4} - \)\(34\!\cdots\!28\)\( T^{5} + \)\(29\!\cdots\!15\)\( T^{6} - \)\(23\!\cdots\!56\)\( T^{7} + \)\(16\!\cdots\!50\)\( T^{8} - \)\(11\!\cdots\!16\)\( T^{9} + \)\(68\!\cdots\!26\)\( T^{10} - \)\(38\!\cdots\!80\)\( T^{11} + \)\(68\!\cdots\!26\)\( p^{11} T^{12} - \)\(11\!\cdots\!16\)\( p^{22} T^{13} + \)\(16\!\cdots\!50\)\( p^{33} T^{14} - \)\(23\!\cdots\!56\)\( p^{44} T^{15} + \)\(29\!\cdots\!15\)\( p^{55} T^{16} - \)\(34\!\cdots\!28\)\( p^{66} T^{17} + \)\(36\!\cdots\!83\)\( p^{77} T^{18} - \)\(55\!\cdots\!84\)\( p^{89} T^{19} + \)\(27\!\cdots\!13\)\( p^{99} T^{20} - 15113662084 p^{110} T^{21} + p^{121} T^{22} \) | |
| 61 | \( 1 - 23895772578 T + \)\(53\!\cdots\!19\)\( T^{2} - \)\(75\!\cdots\!24\)\( T^{3} + \)\(10\!\cdots\!95\)\( T^{4} - \)\(10\!\cdots\!62\)\( T^{5} + \)\(11\!\cdots\!89\)\( T^{6} - \)\(96\!\cdots\!52\)\( T^{7} + \)\(82\!\cdots\!10\)\( T^{8} - \)\(62\!\cdots\!44\)\( T^{9} + \)\(46\!\cdots\!38\)\( T^{10} - \)\(30\!\cdots\!44\)\( T^{11} + \)\(46\!\cdots\!38\)\( p^{11} T^{12} - \)\(62\!\cdots\!44\)\( p^{22} T^{13} + \)\(82\!\cdots\!10\)\( p^{33} T^{14} - \)\(96\!\cdots\!52\)\( p^{44} T^{15} + \)\(11\!\cdots\!89\)\( p^{55} T^{16} - \)\(10\!\cdots\!62\)\( p^{66} T^{17} + \)\(10\!\cdots\!95\)\( p^{77} T^{18} - \)\(75\!\cdots\!24\)\( p^{88} T^{19} + \)\(53\!\cdots\!19\)\( p^{99} T^{20} - 23895772578 p^{110} T^{21} + p^{121} T^{22} \) | |
| 67 | \( 1 - 46806014468 T + \)\(16\!\cdots\!01\)\( T^{2} - \)\(40\!\cdots\!08\)\( T^{3} + \)\(87\!\cdots\!47\)\( T^{4} - \)\(15\!\cdots\!72\)\( T^{5} + \)\(26\!\cdots\!71\)\( T^{6} - \)\(39\!\cdots\!96\)\( T^{7} + \)\(55\!\cdots\!50\)\( T^{8} - \)\(71\!\cdots\!72\)\( T^{9} + \)\(87\!\cdots\!82\)\( T^{10} - \)\(99\!\cdots\!32\)\( T^{11} + \)\(87\!\cdots\!82\)\( p^{11} T^{12} - \)\(71\!\cdots\!72\)\( p^{22} T^{13} + \)\(55\!\cdots\!50\)\( p^{33} T^{14} - \)\(39\!\cdots\!96\)\( p^{44} T^{15} + \)\(26\!\cdots\!71\)\( p^{55} T^{16} - \)\(15\!\cdots\!72\)\( p^{66} T^{17} + \)\(87\!\cdots\!47\)\( p^{77} T^{18} - \)\(40\!\cdots\!08\)\( p^{88} T^{19} + \)\(16\!\cdots\!01\)\( p^{99} T^{20} - 46806014468 p^{110} T^{21} + p^{121} T^{22} \) | |
| 71 | \( 1 - 45541532768 T + \)\(23\!\cdots\!17\)\( T^{2} - \)\(63\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!45\)\( T^{4} - \)\(34\!\cdots\!64\)\( T^{5} + \)\(74\!\cdots\!21\)\( T^{6} - \)\(10\!\cdots\!12\)\( T^{7} + \)\(19\!\cdots\!09\)\( T^{8} - \)\(22\!\cdots\!80\)\( T^{9} + \)\(41\!\cdots\!33\)\( T^{10} - \)\(46\!\cdots\!40\)\( T^{11} + \)\(41\!\cdots\!33\)\( p^{11} T^{12} - \)\(22\!\cdots\!80\)\( p^{22} T^{13} + \)\(19\!\cdots\!09\)\( p^{33} T^{14} - \)\(10\!\cdots\!12\)\( p^{44} T^{15} + \)\(74\!\cdots\!21\)\( p^{55} T^{16} - \)\(34\!\cdots\!64\)\( p^{66} T^{17} + \)\(18\!\cdots\!45\)\( p^{77} T^{18} - \)\(63\!\cdots\!60\)\( p^{88} T^{19} + \)\(23\!\cdots\!17\)\( p^{99} T^{20} - 45541532768 p^{110} T^{21} + p^{121} T^{22} \) | |
| 73 | \( 1 - 63786612542 T + \)\(38\!\cdots\!59\)\( T^{2} - \)\(15\!\cdots\!04\)\( T^{3} + \)\(58\!\cdots\!53\)\( T^{4} - \)\(18\!\cdots\!74\)\( T^{5} + \)\(52\!\cdots\!07\)\( T^{6} - \)\(13\!\cdots\!80\)\( T^{7} + \)\(31\!\cdots\!97\)\( T^{8} - \)\(66\!\cdots\!74\)\( T^{9} + \)\(13\!\cdots\!51\)\( T^{10} - \)\(24\!\cdots\!80\)\( T^{11} + \)\(13\!\cdots\!51\)\( p^{11} T^{12} - \)\(66\!\cdots\!74\)\( p^{22} T^{13} + \)\(31\!\cdots\!97\)\( p^{33} T^{14} - \)\(13\!\cdots\!80\)\( p^{44} T^{15} + \)\(52\!\cdots\!07\)\( p^{55} T^{16} - \)\(18\!\cdots\!74\)\( p^{66} T^{17} + \)\(58\!\cdots\!53\)\( p^{77} T^{18} - \)\(15\!\cdots\!04\)\( p^{88} T^{19} + \)\(38\!\cdots\!59\)\( p^{99} T^{20} - 63786612542 p^{110} T^{21} + p^{121} T^{22} \) | |
| 79 | \( 1 - 21847812496 T + \)\(46\!\cdots\!81\)\( T^{2} - \)\(57\!\cdots\!48\)\( T^{3} + \)\(10\!\cdots\!63\)\( T^{4} - \)\(58\!\cdots\!24\)\( T^{5} + \)\(14\!\cdots\!11\)\( T^{6} - \)\(36\!\cdots\!96\)\( p T^{7} + \)\(16\!\cdots\!14\)\( T^{8} - \)\(66\!\cdots\!52\)\( T^{9} + \)\(15\!\cdots\!10\)\( T^{10} + \)\(79\!\cdots\!24\)\( T^{11} + \)\(15\!\cdots\!10\)\( p^{11} T^{12} - \)\(66\!\cdots\!52\)\( p^{22} T^{13} + \)\(16\!\cdots\!14\)\( p^{33} T^{14} - \)\(36\!\cdots\!96\)\( p^{45} T^{15} + \)\(14\!\cdots\!11\)\( p^{55} T^{16} - \)\(58\!\cdots\!24\)\( p^{66} T^{17} + \)\(10\!\cdots\!63\)\( p^{77} T^{18} - \)\(57\!\cdots\!48\)\( p^{88} T^{19} + \)\(46\!\cdots\!81\)\( p^{99} T^{20} - 21847812496 p^{110} T^{21} + p^{121} T^{22} \) | |
| 83 | \( 1 - 40153340788 T + \)\(74\!\cdots\!85\)\( T^{2} - \)\(31\!\cdots\!12\)\( T^{3} + \)\(27\!\cdots\!99\)\( T^{4} - \)\(11\!\cdots\!72\)\( T^{5} + \)\(68\!\cdots\!59\)\( T^{6} - \)\(25\!\cdots\!28\)\( T^{7} + \)\(12\!\cdots\!10\)\( T^{8} - \)\(44\!\cdots\!12\)\( T^{9} + \)\(19\!\cdots\!66\)\( T^{10} - \)\(63\!\cdots\!56\)\( T^{11} + \)\(19\!\cdots\!66\)\( p^{11} T^{12} - \)\(44\!\cdots\!12\)\( p^{22} T^{13} + \)\(12\!\cdots\!10\)\( p^{33} T^{14} - \)\(25\!\cdots\!28\)\( p^{44} T^{15} + \)\(68\!\cdots\!59\)\( p^{55} T^{16} - \)\(11\!\cdots\!72\)\( p^{66} T^{17} + \)\(27\!\cdots\!99\)\( p^{77} T^{18} - \)\(31\!\cdots\!12\)\( p^{88} T^{19} + \)\(74\!\cdots\!85\)\( p^{99} T^{20} - 40153340788 p^{110} T^{21} + p^{121} T^{22} \) | |
| 89 | \( 1 - 37300228382 T + \)\(22\!\cdots\!31\)\( p T^{2} - \)\(86\!\cdots\!96\)\( T^{3} + \)\(19\!\cdots\!67\)\( T^{4} - \)\(91\!\cdots\!98\)\( T^{5} + \)\(11\!\cdots\!73\)\( T^{6} - \)\(58\!\cdots\!72\)\( T^{7} + \)\(55\!\cdots\!78\)\( T^{8} - \)\(25\!\cdots\!44\)\( T^{9} + \)\(19\!\cdots\!70\)\( T^{10} - \)\(82\!\cdots\!96\)\( T^{11} + \)\(19\!\cdots\!70\)\( p^{11} T^{12} - \)\(25\!\cdots\!44\)\( p^{22} T^{13} + \)\(55\!\cdots\!78\)\( p^{33} T^{14} - \)\(58\!\cdots\!72\)\( p^{44} T^{15} + \)\(11\!\cdots\!73\)\( p^{55} T^{16} - \)\(91\!\cdots\!98\)\( p^{66} T^{17} + \)\(19\!\cdots\!67\)\( p^{77} T^{18} - \)\(86\!\cdots\!96\)\( p^{88} T^{19} + \)\(22\!\cdots\!31\)\( p^{100} T^{20} - 37300228382 p^{110} T^{21} + p^{121} T^{22} \) | |
| 97 | \( 1 + 243602730 T + \)\(47\!\cdots\!11\)\( T^{2} - \)\(23\!\cdots\!08\)\( T^{3} + \)\(12\!\cdots\!95\)\( p T^{4} - \)\(93\!\cdots\!26\)\( T^{5} + \)\(19\!\cdots\!49\)\( T^{6} - \)\(18\!\cdots\!52\)\( T^{7} + \)\(23\!\cdots\!62\)\( T^{8} - \)\(22\!\cdots\!40\)\( T^{9} + \)\(21\!\cdots\!82\)\( T^{10} - \)\(19\!\cdots\!32\)\( T^{11} + \)\(21\!\cdots\!82\)\( p^{11} T^{12} - \)\(22\!\cdots\!40\)\( p^{22} T^{13} + \)\(23\!\cdots\!62\)\( p^{33} T^{14} - \)\(18\!\cdots\!52\)\( p^{44} T^{15} + \)\(19\!\cdots\!49\)\( p^{55} T^{16} - \)\(93\!\cdots\!26\)\( p^{66} T^{17} + \)\(12\!\cdots\!95\)\( p^{78} T^{18} - \)\(23\!\cdots\!08\)\( p^{88} T^{19} + \)\(47\!\cdots\!11\)\( p^{99} T^{20} + 243602730 p^{110} T^{21} + p^{121} T^{22} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{22} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.91638655469529011677625270946, −4.66757376858630682808923244572, −4.18003504441529912615311393460, −4.06550177869592047515341454021, −3.68160251755383744375597886098, −3.63669267560571451309218530662, −3.52792555824017801664203452047, −3.44991719782822630315690890244, −3.29750733362717178726142283994, −3.29657558971662269959298331680, −3.06737697485531784348868406772, −2.45946709252111215422519490880, −2.40911408076291246536585308166, −2.25599259947542572295831592342, −2.10045803532283701518918520372, −1.99035502406725664211712595178, −1.52349034281012250807956305261, −1.38164459323889650486326966852, −1.20727324447892347666792247959, −1.02477922814437543175871281175, −0.78624375597842130475081254158, −0.63039921227013152568507738084, −0.62104388475514199664334276475, −0.42104002175365072270312971029, −0.37920356752326696246101170198, 0.37920356752326696246101170198, 0.42104002175365072270312971029, 0.62104388475514199664334276475, 0.63039921227013152568507738084, 0.78624375597842130475081254158, 1.02477922814437543175871281175, 1.20727324447892347666792247959, 1.38164459323889650486326966852, 1.52349034281012250807956305261, 1.99035502406725664211712595178, 2.10045803532283701518918520372, 2.25599259947542572295831592342, 2.40911408076291246536585308166, 2.45946709252111215422519490880, 3.06737697485531784348868406772, 3.29657558971662269959298331680, 3.29750733362717178726142283994, 3.44991719782822630315690890244, 3.52792555824017801664203452047, 3.63669267560571451309218530662, 3.68160251755383744375597886098, 4.06550177869592047515341454021, 4.18003504441529912615311393460, 4.66757376858630682808923244572, 4.91638655469529011677625270946
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.