Dirichlet series
| L(s) = 1 | + 4.04e16·2-s + 1.03e28·3-s − 4.07e35·4-s + 3.84e40·5-s + 4.16e44·6-s − 3.39e49·7-s + 9.55e51·8-s − 1.32e56·9-s + 1.55e57·10-s − 1.88e61·11-s − 4.19e63·12-s − 9.85e64·13-s − 1.37e66·14-s + 3.96e68·15-s + 7.10e70·16-s − 6.80e71·17-s − 5.34e72·18-s − 1.60e74·19-s − 1.56e76·20-s − 3.49e77·21-s − 7.61e77·22-s − 5.81e79·23-s + 9.84e79·24-s − 2.35e82·25-s − 3.98e81·26-s − 1.34e84·27-s + 1.38e85·28-s + ⋯ |
| L(s) = 1 | + 0.0991·2-s + 1.26·3-s − 2.44·4-s + 0.495·5-s + 0.125·6-s − 1.23·7-s + 0.141·8-s − 1.98·9-s + 0.0491·10-s − 2.25·11-s − 3.09·12-s − 0.673·13-s − 0.122·14-s + 0.625·15-s + 2.57·16-s − 0.709·17-s − 0.197·18-s − 0.250·19-s − 1.21·20-s − 1.56·21-s − 0.223·22-s − 1.26·23-s + 0.178·24-s − 3.91·25-s − 0.0667·26-s − 2.48·27-s + 3.03·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(118-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+58.5)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(1\) |
| Sign: | $-1$ |
| Analytic conductor: | \(2.76479\times 10^{17}\) |
| Root analytic conductor: | \(9.31067\) |
| Motivic weight: | \(117\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 1,\ (\ :[117/2]^{9}),\ -1)\) |
Particular Values
| \(L(59)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{119}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| good | 2 | \( 1 - 1263032396411211 p^{5} T + \)\(24\!\cdots\!73\)\( p^{14} T^{2} - \)\(12\!\cdots\!35\)\( p^{25} T^{3} + \)\(44\!\cdots\!03\)\( p^{41} T^{4} - \)\(13\!\cdots\!59\)\( p^{63} T^{5} + \)\(60\!\cdots\!77\)\( p^{88} T^{6} - \)\(94\!\cdots\!55\)\( p^{115} T^{7} + \)\(74\!\cdots\!43\)\( p^{145} T^{8} - \)\(13\!\cdots\!93\)\( p^{182} T^{9} + \)\(74\!\cdots\!43\)\( p^{262} T^{10} - \)\(94\!\cdots\!55\)\( p^{349} T^{11} + \)\(60\!\cdots\!77\)\( p^{439} T^{12} - \)\(13\!\cdots\!59\)\( p^{531} T^{13} + \)\(44\!\cdots\!03\)\( p^{626} T^{14} - \)\(12\!\cdots\!35\)\( p^{727} T^{15} + \)\(24\!\cdots\!73\)\( p^{833} T^{16} - 1263032396411211 p^{941} T^{17} + p^{1053} T^{18} \) |
| 3 | \( 1 - \)\(42\!\cdots\!52\)\( p^{5} T + \)\(36\!\cdots\!03\)\( p^{8} T^{2} - \)\(19\!\cdots\!40\)\( p^{17} T^{3} + \)\(17\!\cdots\!64\)\( p^{30} T^{4} - \)\(10\!\cdots\!68\)\( p^{43} T^{5} + \)\(31\!\cdots\!76\)\( p^{61} T^{6} - \)\(28\!\cdots\!40\)\( p^{84} T^{7} + \)\(30\!\cdots\!78\)\( p^{107} T^{8} - \)\(29\!\cdots\!04\)\( p^{134} T^{9} + \)\(30\!\cdots\!78\)\( p^{224} T^{10} - \)\(28\!\cdots\!40\)\( p^{318} T^{11} + \)\(31\!\cdots\!76\)\( p^{412} T^{12} - \)\(10\!\cdots\!68\)\( p^{511} T^{13} + \)\(17\!\cdots\!64\)\( p^{615} T^{14} - \)\(19\!\cdots\!40\)\( p^{719} T^{15} + \)\(36\!\cdots\!03\)\( p^{827} T^{16} - \)\(42\!\cdots\!52\)\( p^{941} T^{17} + p^{1053} T^{18} \) | |
| 5 | \( 1 - \)\(15\!\cdots\!66\)\( p^{2} T + \)\(64\!\cdots\!93\)\( p^{8} T^{2} - \)\(79\!\cdots\!04\)\( p^{16} T^{3} + \)\(59\!\cdots\!28\)\( p^{24} T^{4} - \)\(80\!\cdots\!72\)\( p^{32} T^{5} + \)\(15\!\cdots\!16\)\( p^{42} T^{6} - \)\(27\!\cdots\!96\)\( p^{57} T^{7} + \)\(49\!\cdots\!66\)\( p^{74} T^{8} - \)\(67\!\cdots\!44\)\( p^{92} T^{9} + \)\(49\!\cdots\!66\)\( p^{191} T^{10} - \)\(27\!\cdots\!96\)\( p^{291} T^{11} + \)\(15\!\cdots\!16\)\( p^{393} T^{12} - \)\(80\!\cdots\!72\)\( p^{500} T^{13} + \)\(59\!\cdots\!28\)\( p^{609} T^{14} - \)\(79\!\cdots\!04\)\( p^{718} T^{15} + \)\(64\!\cdots\!93\)\( p^{827} T^{16} - \)\(15\!\cdots\!66\)\( p^{938} T^{17} + p^{1053} T^{18} \) | |
| 7 | \( 1 + \)\(48\!\cdots\!44\)\( p T + \)\(16\!\cdots\!07\)\( p^{4} T^{2} + \)\(27\!\cdots\!00\)\( p^{9} T^{3} + \)\(15\!\cdots\!72\)\( p^{15} T^{4} + \)\(44\!\cdots\!32\)\( p^{22} T^{5} + \)\(55\!\cdots\!04\)\( p^{31} T^{6} + \)\(55\!\cdots\!00\)\( p^{42} T^{7} + \)\(18\!\cdots\!94\)\( p^{54} T^{8} + \)\(47\!\cdots\!48\)\( p^{68} T^{9} + \)\(18\!\cdots\!94\)\( p^{171} T^{10} + \)\(55\!\cdots\!00\)\( p^{276} T^{11} + \)\(55\!\cdots\!04\)\( p^{382} T^{12} + \)\(44\!\cdots\!32\)\( p^{490} T^{13} + \)\(15\!\cdots\!72\)\( p^{600} T^{14} + \)\(27\!\cdots\!00\)\( p^{711} T^{15} + \)\(16\!\cdots\!07\)\( p^{823} T^{16} + \)\(48\!\cdots\!44\)\( p^{937} T^{17} + p^{1053} T^{18} \) | |
| 11 | \( 1 + \)\(15\!\cdots\!12\)\( p^{2} T + \)\(33\!\cdots\!13\)\( p^{5} T^{2} + \)\(37\!\cdots\!84\)\( p^{8} T^{3} + \)\(46\!\cdots\!60\)\( p^{11} T^{4} + \)\(43\!\cdots\!52\)\( p^{14} T^{5} + \)\(35\!\cdots\!84\)\( p^{18} T^{6} + \)\(18\!\cdots\!96\)\( p^{25} T^{7} + \)\(84\!\cdots\!58\)\( p^{33} T^{8} + \)\(34\!\cdots\!80\)\( p^{41} T^{9} + \)\(84\!\cdots\!58\)\( p^{150} T^{10} + \)\(18\!\cdots\!96\)\( p^{259} T^{11} + \)\(35\!\cdots\!84\)\( p^{369} T^{12} + \)\(43\!\cdots\!52\)\( p^{482} T^{13} + \)\(46\!\cdots\!60\)\( p^{596} T^{14} + \)\(37\!\cdots\!84\)\( p^{710} T^{15} + \)\(33\!\cdots\!13\)\( p^{824} T^{16} + \)\(15\!\cdots\!12\)\( p^{938} T^{17} + p^{1053} T^{18} \) | |
| 13 | \( 1 + \)\(75\!\cdots\!18\)\( p T + \)\(32\!\cdots\!09\)\( p^{3} T^{2} + \)\(69\!\cdots\!40\)\( p^{6} T^{3} + \)\(16\!\cdots\!24\)\( p^{10} T^{4} - \)\(25\!\cdots\!64\)\( p^{14} T^{5} + \)\(40\!\cdots\!32\)\( p^{18} T^{6} - \)\(85\!\cdots\!40\)\( p^{23} T^{7} + \)\(36\!\cdots\!42\)\( p^{29} T^{8} - \)\(86\!\cdots\!76\)\( p^{36} T^{9} + \)\(36\!\cdots\!42\)\( p^{146} T^{10} - \)\(85\!\cdots\!40\)\( p^{257} T^{11} + \)\(40\!\cdots\!32\)\( p^{369} T^{12} - \)\(25\!\cdots\!64\)\( p^{482} T^{13} + \)\(16\!\cdots\!24\)\( p^{595} T^{14} + \)\(69\!\cdots\!40\)\( p^{708} T^{15} + \)\(32\!\cdots\!09\)\( p^{822} T^{16} + \)\(75\!\cdots\!18\)\( p^{937} T^{17} + p^{1053} T^{18} \) | |
| 17 | \( 1 + \)\(40\!\cdots\!34\)\( p T + \)\(10\!\cdots\!89\)\( p^{3} T^{2} + \)\(33\!\cdots\!20\)\( p^{5} T^{3} + \)\(32\!\cdots\!12\)\( p^{7} T^{4} + \)\(11\!\cdots\!84\)\( p^{9} T^{5} + \)\(40\!\cdots\!72\)\( p^{12} T^{6} + \)\(25\!\cdots\!40\)\( p^{17} T^{7} + \)\(25\!\cdots\!94\)\( p^{22} T^{8} + \)\(80\!\cdots\!68\)\( p^{28} T^{9} + \)\(25\!\cdots\!94\)\( p^{139} T^{10} + \)\(25\!\cdots\!40\)\( p^{251} T^{11} + \)\(40\!\cdots\!72\)\( p^{363} T^{12} + \)\(11\!\cdots\!84\)\( p^{477} T^{13} + \)\(32\!\cdots\!12\)\( p^{592} T^{14} + \)\(33\!\cdots\!20\)\( p^{707} T^{15} + \)\(10\!\cdots\!89\)\( p^{822} T^{16} + \)\(40\!\cdots\!34\)\( p^{937} T^{17} + p^{1053} T^{18} \) | |
| 19 | \( 1 + \)\(84\!\cdots\!60\)\( p T + \)\(68\!\cdots\!91\)\( p^{2} T^{2} + \)\(46\!\cdots\!80\)\( p^{4} T^{3} + \)\(33\!\cdots\!04\)\( p^{7} T^{4} + \)\(14\!\cdots\!20\)\( p^{10} T^{5} + \)\(29\!\cdots\!76\)\( p^{14} T^{6} + \)\(68\!\cdots\!60\)\( p^{18} T^{7} + \)\(96\!\cdots\!06\)\( p^{22} T^{8} + \)\(20\!\cdots\!00\)\( p^{26} T^{9} + \)\(96\!\cdots\!06\)\( p^{139} T^{10} + \)\(68\!\cdots\!60\)\( p^{252} T^{11} + \)\(29\!\cdots\!76\)\( p^{365} T^{12} + \)\(14\!\cdots\!20\)\( p^{478} T^{13} + \)\(33\!\cdots\!04\)\( p^{592} T^{14} + \)\(46\!\cdots\!80\)\( p^{706} T^{15} + \)\(68\!\cdots\!91\)\( p^{821} T^{16} + \)\(84\!\cdots\!60\)\( p^{937} T^{17} + p^{1053} T^{18} \) | |
| 23 | \( 1 + \)\(25\!\cdots\!48\)\( p T + \)\(19\!\cdots\!47\)\( p^{2} T^{2} + \)\(18\!\cdots\!40\)\( p^{4} T^{3} + \)\(35\!\cdots\!44\)\( p^{6} T^{4} + \)\(12\!\cdots\!88\)\( p^{9} T^{5} + \)\(81\!\cdots\!68\)\( p^{12} T^{6} + \)\(26\!\cdots\!40\)\( p^{15} T^{7} + \)\(14\!\cdots\!34\)\( p^{18} T^{8} + \)\(42\!\cdots\!28\)\( p^{21} T^{9} + \)\(14\!\cdots\!34\)\( p^{135} T^{10} + \)\(26\!\cdots\!40\)\( p^{249} T^{11} + \)\(81\!\cdots\!68\)\( p^{363} T^{12} + \)\(12\!\cdots\!88\)\( p^{477} T^{13} + \)\(35\!\cdots\!44\)\( p^{591} T^{14} + \)\(18\!\cdots\!40\)\( p^{706} T^{15} + \)\(19\!\cdots\!47\)\( p^{821} T^{16} + \)\(25\!\cdots\!48\)\( p^{937} T^{17} + p^{1053} T^{18} \) | |
| 29 | \( 1 + \)\(16\!\cdots\!90\)\( p T + \)\(59\!\cdots\!41\)\( p^{2} T^{2} + \)\(25\!\cdots\!20\)\( p^{4} T^{3} + \)\(18\!\cdots\!96\)\( p^{6} T^{4} + \)\(56\!\cdots\!80\)\( p^{8} T^{5} + \)\(29\!\cdots\!36\)\( p^{10} T^{6} + \)\(45\!\cdots\!40\)\( p^{12} T^{7} + \)\(25\!\cdots\!06\)\( p^{14} T^{8} + \)\(78\!\cdots\!00\)\( p^{16} T^{9} + \)\(25\!\cdots\!06\)\( p^{131} T^{10} + \)\(45\!\cdots\!40\)\( p^{246} T^{11} + \)\(29\!\cdots\!36\)\( p^{361} T^{12} + \)\(56\!\cdots\!80\)\( p^{476} T^{13} + \)\(18\!\cdots\!96\)\( p^{591} T^{14} + \)\(25\!\cdots\!20\)\( p^{706} T^{15} + \)\(59\!\cdots\!41\)\( p^{821} T^{16} + \)\(16\!\cdots\!90\)\( p^{937} T^{17} + p^{1053} T^{18} \) | |
| 31 | \( 1 + \)\(62\!\cdots\!92\)\( T + \)\(97\!\cdots\!83\)\( T^{2} + \)\(43\!\cdots\!24\)\( p T^{3} + \)\(57\!\cdots\!60\)\( p^{2} T^{4} + \)\(11\!\cdots\!92\)\( p^{4} T^{5} + \)\(30\!\cdots\!64\)\( p^{6} T^{6} + \)\(53\!\cdots\!96\)\( p^{8} T^{7} + \)\(44\!\cdots\!38\)\( p^{11} T^{8} + \)\(20\!\cdots\!80\)\( p^{14} T^{9} + \)\(44\!\cdots\!38\)\( p^{128} T^{10} + \)\(53\!\cdots\!96\)\( p^{242} T^{11} + \)\(30\!\cdots\!64\)\( p^{357} T^{12} + \)\(11\!\cdots\!92\)\( p^{472} T^{13} + \)\(57\!\cdots\!60\)\( p^{587} T^{14} + \)\(43\!\cdots\!24\)\( p^{703} T^{15} + \)\(97\!\cdots\!83\)\( p^{819} T^{16} + \)\(62\!\cdots\!92\)\( p^{936} T^{17} + p^{1053} T^{18} \) | |
| 37 | \( 1 + \)\(94\!\cdots\!14\)\( p T + \)\(11\!\cdots\!53\)\( p^{2} T^{2} + \)\(98\!\cdots\!40\)\( p^{3} T^{3} + \)\(18\!\cdots\!48\)\( p^{5} T^{4} + \)\(39\!\cdots\!76\)\( p^{7} T^{5} + \)\(51\!\cdots\!16\)\( p^{9} T^{6} + \)\(28\!\cdots\!40\)\( p^{12} T^{7} + \)\(79\!\cdots\!02\)\( p^{15} T^{8} + \)\(40\!\cdots\!92\)\( p^{18} T^{9} + \)\(79\!\cdots\!02\)\( p^{132} T^{10} + \)\(28\!\cdots\!40\)\( p^{246} T^{11} + \)\(51\!\cdots\!16\)\( p^{360} T^{12} + \)\(39\!\cdots\!76\)\( p^{475} T^{13} + \)\(18\!\cdots\!48\)\( p^{590} T^{14} + \)\(98\!\cdots\!40\)\( p^{705} T^{15} + \)\(11\!\cdots\!53\)\( p^{821} T^{16} + \)\(94\!\cdots\!14\)\( p^{937} T^{17} + p^{1053} T^{18} \) | |
| 41 | \( 1 - \)\(10\!\cdots\!38\)\( T + \)\(34\!\cdots\!93\)\( T^{2} - \)\(65\!\cdots\!96\)\( p T^{3} + \)\(33\!\cdots\!60\)\( p^{2} T^{4} - \)\(50\!\cdots\!08\)\( p^{3} T^{5} + \)\(20\!\cdots\!84\)\( p^{4} T^{6} - \)\(61\!\cdots\!84\)\( p^{6} T^{7} + \)\(50\!\cdots\!58\)\( p^{8} T^{8} - \)\(12\!\cdots\!20\)\( p^{10} T^{9} + \)\(50\!\cdots\!58\)\( p^{125} T^{10} - \)\(61\!\cdots\!84\)\( p^{240} T^{11} + \)\(20\!\cdots\!84\)\( p^{355} T^{12} - \)\(50\!\cdots\!08\)\( p^{471} T^{13} + \)\(33\!\cdots\!60\)\( p^{587} T^{14} - \)\(65\!\cdots\!96\)\( p^{703} T^{15} + \)\(34\!\cdots\!93\)\( p^{819} T^{16} - \)\(10\!\cdots\!38\)\( p^{936} T^{17} + p^{1053} T^{18} \) | |
| 43 | \( 1 - \)\(50\!\cdots\!56\)\( T + \)\(59\!\cdots\!43\)\( T^{2} - \)\(56\!\cdots\!00\)\( p T^{3} + \)\(10\!\cdots\!04\)\( p^{2} T^{4} - \)\(70\!\cdots\!68\)\( p^{3} T^{5} + \)\(12\!\cdots\!28\)\( p^{4} T^{6} - \)\(14\!\cdots\!00\)\( p^{6} T^{7} + \)\(62\!\cdots\!06\)\( p^{8} T^{8} - \)\(57\!\cdots\!64\)\( p^{10} T^{9} + \)\(62\!\cdots\!06\)\( p^{125} T^{10} - \)\(14\!\cdots\!00\)\( p^{240} T^{11} + \)\(12\!\cdots\!28\)\( p^{355} T^{12} - \)\(70\!\cdots\!68\)\( p^{471} T^{13} + \)\(10\!\cdots\!04\)\( p^{587} T^{14} - \)\(56\!\cdots\!00\)\( p^{703} T^{15} + \)\(59\!\cdots\!43\)\( p^{819} T^{16} - \)\(50\!\cdots\!56\)\( p^{936} T^{17} + p^{1053} T^{18} \) | |
| 47 | \( 1 - \)\(16\!\cdots\!12\)\( T + \)\(39\!\cdots\!81\)\( p T^{2} - \)\(38\!\cdots\!60\)\( p^{2} T^{3} + \)\(16\!\cdots\!92\)\( p^{3} T^{4} - \)\(24\!\cdots\!52\)\( p^{4} T^{5} + \)\(50\!\cdots\!36\)\( p^{5} T^{6} - \)\(18\!\cdots\!60\)\( p^{7} T^{7} + \)\(56\!\cdots\!38\)\( p^{9} T^{8} - \)\(19\!\cdots\!64\)\( p^{11} T^{9} + \)\(56\!\cdots\!38\)\( p^{126} T^{10} - \)\(18\!\cdots\!60\)\( p^{241} T^{11} + \)\(50\!\cdots\!36\)\( p^{356} T^{12} - \)\(24\!\cdots\!52\)\( p^{472} T^{13} + \)\(16\!\cdots\!92\)\( p^{588} T^{14} - \)\(38\!\cdots\!60\)\( p^{704} T^{15} + \)\(39\!\cdots\!81\)\( p^{820} T^{16} - \)\(16\!\cdots\!12\)\( p^{936} T^{17} + p^{1053} T^{18} \) | |
| 53 | \( 1 + \)\(40\!\cdots\!38\)\( p T + \)\(14\!\cdots\!37\)\( p^{2} T^{2} + \)\(32\!\cdots\!40\)\( p^{3} T^{3} + \)\(71\!\cdots\!56\)\( p^{4} T^{4} + \)\(12\!\cdots\!48\)\( p^{5} T^{5} + \)\(41\!\cdots\!44\)\( p^{7} T^{6} + \)\(11\!\cdots\!20\)\( p^{9} T^{7} + \)\(34\!\cdots\!38\)\( p^{11} T^{8} + \)\(88\!\cdots\!88\)\( p^{13} T^{9} + \)\(34\!\cdots\!38\)\( p^{128} T^{10} + \)\(11\!\cdots\!20\)\( p^{243} T^{11} + \)\(41\!\cdots\!44\)\( p^{358} T^{12} + \)\(12\!\cdots\!48\)\( p^{473} T^{13} + \)\(71\!\cdots\!56\)\( p^{589} T^{14} + \)\(32\!\cdots\!40\)\( p^{705} T^{15} + \)\(14\!\cdots\!37\)\( p^{821} T^{16} + \)\(40\!\cdots\!38\)\( p^{937} T^{17} + p^{1053} T^{18} \) | |
| 59 | \( 1 + \)\(77\!\cdots\!20\)\( T + \)\(87\!\cdots\!71\)\( T^{2} + \)\(11\!\cdots\!40\)\( T^{3} + \)\(37\!\cdots\!96\)\( T^{4} + \)\(70\!\cdots\!60\)\( T^{5} + \)\(17\!\cdots\!84\)\( p T^{6} + \)\(65\!\cdots\!80\)\( p^{2} T^{7} + \)\(10\!\cdots\!74\)\( p^{3} T^{8} + \)\(36\!\cdots\!00\)\( p^{4} T^{9} + \)\(10\!\cdots\!74\)\( p^{120} T^{10} + \)\(65\!\cdots\!80\)\( p^{236} T^{11} + \)\(17\!\cdots\!84\)\( p^{352} T^{12} + \)\(70\!\cdots\!60\)\( p^{468} T^{13} + \)\(37\!\cdots\!96\)\( p^{585} T^{14} + \)\(11\!\cdots\!40\)\( p^{702} T^{15} + \)\(87\!\cdots\!71\)\( p^{819} T^{16} + \)\(77\!\cdots\!20\)\( p^{936} T^{17} + p^{1053} T^{18} \) | |
| 61 | \( 1 + \)\(22\!\cdots\!02\)\( T + \)\(45\!\cdots\!33\)\( p T^{2} + \)\(63\!\cdots\!04\)\( T^{3} + \)\(43\!\cdots\!60\)\( T^{4} + \)\(15\!\cdots\!12\)\( p T^{5} + \)\(13\!\cdots\!24\)\( p^{2} T^{6} + \)\(45\!\cdots\!16\)\( p^{3} T^{7} + \)\(36\!\cdots\!78\)\( p^{4} T^{8} + \)\(10\!\cdots\!80\)\( p^{5} T^{9} + \)\(36\!\cdots\!78\)\( p^{121} T^{10} + \)\(45\!\cdots\!16\)\( p^{237} T^{11} + \)\(13\!\cdots\!24\)\( p^{353} T^{12} + \)\(15\!\cdots\!12\)\( p^{469} T^{13} + \)\(43\!\cdots\!60\)\( p^{585} T^{14} + \)\(63\!\cdots\!04\)\( p^{702} T^{15} + \)\(45\!\cdots\!33\)\( p^{820} T^{16} + \)\(22\!\cdots\!02\)\( p^{936} T^{17} + p^{1053} T^{18} \) | |
| 67 | \( 1 - \)\(10\!\cdots\!72\)\( T + \)\(26\!\cdots\!07\)\( T^{2} - \)\(29\!\cdots\!60\)\( T^{3} + \)\(38\!\cdots\!76\)\( T^{4} - \)\(53\!\cdots\!56\)\( p T^{5} + \)\(79\!\cdots\!28\)\( p^{2} T^{6} - \)\(93\!\cdots\!40\)\( p^{3} T^{7} + \)\(11\!\cdots\!46\)\( p^{4} T^{8} - \)\(11\!\cdots\!16\)\( p^{5} T^{9} + \)\(11\!\cdots\!46\)\( p^{121} T^{10} - \)\(93\!\cdots\!40\)\( p^{237} T^{11} + \)\(79\!\cdots\!28\)\( p^{353} T^{12} - \)\(53\!\cdots\!56\)\( p^{469} T^{13} + \)\(38\!\cdots\!76\)\( p^{585} T^{14} - \)\(29\!\cdots\!60\)\( p^{702} T^{15} + \)\(26\!\cdots\!07\)\( p^{819} T^{16} - \)\(10\!\cdots\!72\)\( p^{936} T^{17} + p^{1053} T^{18} \) | |
| 71 | \( 1 - \)\(15\!\cdots\!28\)\( T + \)\(16\!\cdots\!23\)\( T^{2} - \)\(20\!\cdots\!76\)\( T^{3} + \)\(15\!\cdots\!60\)\( T^{4} - \)\(26\!\cdots\!08\)\( p T^{5} + \)\(21\!\cdots\!84\)\( p^{2} T^{6} - \)\(32\!\cdots\!44\)\( p^{3} T^{7} + \)\(21\!\cdots\!98\)\( p^{4} T^{8} - \)\(29\!\cdots\!20\)\( p^{5} T^{9} + \)\(21\!\cdots\!98\)\( p^{121} T^{10} - \)\(32\!\cdots\!44\)\( p^{237} T^{11} + \)\(21\!\cdots\!84\)\( p^{353} T^{12} - \)\(26\!\cdots\!08\)\( p^{469} T^{13} + \)\(15\!\cdots\!60\)\( p^{585} T^{14} - \)\(20\!\cdots\!76\)\( p^{702} T^{15} + \)\(16\!\cdots\!23\)\( p^{819} T^{16} - \)\(15\!\cdots\!28\)\( p^{936} T^{17} + p^{1053} T^{18} \) | |
| 73 | \( 1 + \)\(36\!\cdots\!54\)\( T + \)\(66\!\cdots\!13\)\( T^{2} + \)\(23\!\cdots\!40\)\( T^{3} + \)\(29\!\cdots\!92\)\( p T^{4} + \)\(13\!\cdots\!36\)\( p^{2} T^{5} + \)\(11\!\cdots\!84\)\( p^{3} T^{6} + \)\(49\!\cdots\!80\)\( p^{4} T^{7} + \)\(30\!\cdots\!22\)\( p^{5} T^{8} + \)\(11\!\cdots\!96\)\( p^{6} T^{9} + \)\(30\!\cdots\!22\)\( p^{122} T^{10} + \)\(49\!\cdots\!80\)\( p^{238} T^{11} + \)\(11\!\cdots\!84\)\( p^{354} T^{12} + \)\(13\!\cdots\!36\)\( p^{470} T^{13} + \)\(29\!\cdots\!92\)\( p^{586} T^{14} + \)\(23\!\cdots\!40\)\( p^{702} T^{15} + \)\(66\!\cdots\!13\)\( p^{819} T^{16} + \)\(36\!\cdots\!54\)\( p^{936} T^{17} + p^{1053} T^{18} \) | |
| 79 | \( 1 + \)\(22\!\cdots\!60\)\( T + \)\(85\!\cdots\!31\)\( T^{2} + \)\(15\!\cdots\!20\)\( T^{3} + \)\(41\!\cdots\!04\)\( p T^{4} + \)\(78\!\cdots\!80\)\( p^{2} T^{5} + \)\(15\!\cdots\!24\)\( p^{3} T^{6} + \)\(24\!\cdots\!40\)\( p^{4} T^{7} + \)\(37\!\cdots\!14\)\( p^{5} T^{8} + \)\(62\!\cdots\!00\)\( p^{7} T^{9} + \)\(37\!\cdots\!14\)\( p^{122} T^{10} + \)\(24\!\cdots\!40\)\( p^{238} T^{11} + \)\(15\!\cdots\!24\)\( p^{354} T^{12} + \)\(78\!\cdots\!80\)\( p^{470} T^{13} + \)\(41\!\cdots\!04\)\( p^{586} T^{14} + \)\(15\!\cdots\!20\)\( p^{702} T^{15} + \)\(85\!\cdots\!31\)\( p^{819} T^{16} + \)\(22\!\cdots\!60\)\( p^{936} T^{17} + p^{1053} T^{18} \) | |
| 83 | \( 1 + \)\(26\!\cdots\!24\)\( T + \)\(19\!\cdots\!03\)\( T^{2} + \)\(55\!\cdots\!20\)\( T^{3} + \)\(24\!\cdots\!32\)\( p T^{4} + \)\(77\!\cdots\!56\)\( p^{2} T^{5} + \)\(24\!\cdots\!44\)\( p^{3} T^{6} + \)\(66\!\cdots\!40\)\( p^{4} T^{7} + \)\(17\!\cdots\!82\)\( p^{5} T^{8} + \)\(38\!\cdots\!16\)\( p^{6} T^{9} + \)\(17\!\cdots\!82\)\( p^{122} T^{10} + \)\(66\!\cdots\!40\)\( p^{238} T^{11} + \)\(24\!\cdots\!44\)\( p^{354} T^{12} + \)\(77\!\cdots\!56\)\( p^{470} T^{13} + \)\(24\!\cdots\!32\)\( p^{586} T^{14} + \)\(55\!\cdots\!20\)\( p^{702} T^{15} + \)\(19\!\cdots\!03\)\( p^{819} T^{16} + \)\(26\!\cdots\!24\)\( p^{936} T^{17} + p^{1053} T^{18} \) | |
| 89 | \( 1 + \)\(43\!\cdots\!30\)\( T + \)\(14\!\cdots\!61\)\( T^{2} + \)\(39\!\cdots\!40\)\( p T^{3} + \)\(93\!\cdots\!56\)\( p^{2} T^{4} + \)\(18\!\cdots\!60\)\( p^{3} T^{5} + \)\(32\!\cdots\!36\)\( p^{4} T^{6} + \)\(51\!\cdots\!80\)\( p^{5} T^{7} + \)\(73\!\cdots\!46\)\( p^{6} T^{8} + \)\(94\!\cdots\!00\)\( p^{7} T^{9} + \)\(73\!\cdots\!46\)\( p^{123} T^{10} + \)\(51\!\cdots\!80\)\( p^{239} T^{11} + \)\(32\!\cdots\!36\)\( p^{355} T^{12} + \)\(18\!\cdots\!60\)\( p^{471} T^{13} + \)\(93\!\cdots\!56\)\( p^{587} T^{14} + \)\(39\!\cdots\!40\)\( p^{703} T^{15} + \)\(14\!\cdots\!61\)\( p^{819} T^{16} + \)\(43\!\cdots\!30\)\( p^{936} T^{17} + p^{1053} T^{18} \) | |
| 97 | \( 1 + \)\(29\!\cdots\!38\)\( T + \)\(16\!\cdots\!81\)\( p T^{2} + \)\(41\!\cdots\!40\)\( p^{2} T^{3} + \)\(14\!\cdots\!92\)\( p^{3} T^{4} + \)\(31\!\cdots\!48\)\( p^{4} T^{5} + \)\(82\!\cdots\!36\)\( p^{5} T^{6} + \)\(15\!\cdots\!80\)\( p^{6} T^{7} + \)\(33\!\cdots\!42\)\( p^{7} T^{8} + \)\(54\!\cdots\!28\)\( p^{8} T^{9} + \)\(33\!\cdots\!42\)\( p^{124} T^{10} + \)\(15\!\cdots\!80\)\( p^{240} T^{11} + \)\(82\!\cdots\!36\)\( p^{356} T^{12} + \)\(31\!\cdots\!48\)\( p^{472} T^{13} + \)\(14\!\cdots\!92\)\( p^{588} T^{14} + \)\(41\!\cdots\!40\)\( p^{704} T^{15} + \)\(16\!\cdots\!81\)\( p^{820} T^{16} + \)\(29\!\cdots\!38\)\( p^{936} T^{17} + p^{1053} T^{18} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.55517495542043454825302704589, −4.35264451308832202408857342812, −4.25286081921700665394395127677, −4.14885874010406987951235497989, −3.94413044967518093813253795304, −3.66653307308572113135144513267, −3.43236070035097728801493097985, −3.32583011947617435297976952020, −3.19917691136389881709213785971, −3.18364495187382794127371731700, −3.15630870689706360231223709679, −3.08029273520848102044919707110, −2.49800976588505039494979511779, −2.38872698826637916404811803779, −2.38237177115846581823803131084, −2.28238491608509172464717261844, −2.24417974729927893389916508838, −1.89959076524503662772047926533, −1.81127454281176306960033876410, −1.67245587722051495666385498863, −1.59285980763853953176535287657, −1.03782551078879549704895614971, −0.970894232838823648354091164091, −0.953036884019109512333656165982, −0.950580776337677638989911294733, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.950580776337677638989911294733, 0.953036884019109512333656165982, 0.970894232838823648354091164091, 1.03782551078879549704895614971, 1.59285980763853953176535287657, 1.67245587722051495666385498863, 1.81127454281176306960033876410, 1.89959076524503662772047926533, 2.24417974729927893389916508838, 2.28238491608509172464717261844, 2.38237177115846581823803131084, 2.38872698826637916404811803779, 2.49800976588505039494979511779, 3.08029273520848102044919707110, 3.15630870689706360231223709679, 3.18364495187382794127371731700, 3.19917691136389881709213785971, 3.32583011947617435297976952020, 3.43236070035097728801493097985, 3.66653307308572113135144513267, 3.94413044967518093813253795304, 4.14885874010406987951235497989, 4.25286081921700665394395127677, 4.35264451308832202408857342812, 4.55517495542043454825302704589
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.