Dirichlet series
| L(s) = 1 | + 5.46e15·2-s + 1.59e25·3-s − 3.94e32·4-s + 2.47e37·5-s + 8.73e40·6-s − 9.35e44·7-s − 2.83e48·8-s − 3.09e51·9-s + 1.35e53·10-s + 7.53e55·11-s − 6.30e57·12-s + 3.86e59·13-s − 5.11e60·14-s + 3.95e62·15-s + 5.55e64·16-s + 5.57e65·17-s − 1.69e67·18-s + 8.58e68·19-s − 9.75e69·20-s − 1.49e70·21-s + 4.11e71·22-s − 1.00e73·23-s − 4.53e73·24-s − 1.42e75·25-s + 2.11e75·26-s − 5.28e76·27-s + 3.69e77·28-s + ⋯ |
| L(s) = 1 | + 0.429·2-s + 0.476·3-s − 2.43·4-s + 0.996·5-s + 0.204·6-s − 0.573·7-s − 1.37·8-s − 2.74·9-s + 0.427·10-s + 1.45·11-s − 1.15·12-s + 0.980·13-s − 0.246·14-s + 0.474·15-s + 2.10·16-s + 0.826·17-s − 1.17·18-s + 3.31·19-s − 2.42·20-s − 0.273·21-s + 0.624·22-s − 1.40·23-s − 0.653·24-s − 2.30·25-s + 0.420·26-s − 1.39·27-s + 1.39·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(108-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+53.5)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.53661\times 10^{16}\) |
| Root analytic conductor: | \(8.51491\) |
| Motivic weight: | \(107\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 1,\ (\ :[107/2]^{9}),\ 1)\) |
Particular Values
| \(L(54)\) | \(\approx\) | \(11.08286079\) |
| \(L(\frac12)\) | \(\approx\) | \(11.08286079\) |
| \(L(\frac{109}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| good | 2 | \( 1 - 683222451679437 p^{3} T + \)\(20\!\cdots\!81\)\( p^{11} T^{2} - \)\(38\!\cdots\!15\)\( p^{22} T^{3} + \)\(19\!\cdots\!57\)\( p^{39} T^{4} - \)\(41\!\cdots\!81\)\( p^{60} T^{5} + \)\(71\!\cdots\!39\)\( p^{81} T^{6} + \)\(96\!\cdots\!95\)\( p^{106} T^{7} + \)\(71\!\cdots\!09\)\( p^{138} T^{8} + \)\(16\!\cdots\!57\)\( p^{173} T^{9} + \)\(71\!\cdots\!09\)\( p^{245} T^{10} + \)\(96\!\cdots\!95\)\( p^{320} T^{11} + \)\(71\!\cdots\!39\)\( p^{402} T^{12} - \)\(41\!\cdots\!81\)\( p^{488} T^{13} + \)\(19\!\cdots\!57\)\( p^{574} T^{14} - \)\(38\!\cdots\!15\)\( p^{664} T^{15} + \)\(20\!\cdots\!81\)\( p^{760} T^{16} - 683222451679437 p^{859} T^{17} + p^{963} T^{18} \) |
| 3 | \( 1 - \)\(17\!\cdots\!68\)\( p^{2} T + \)\(51\!\cdots\!87\)\( p^{8} T^{2} - \)\(11\!\cdots\!40\)\( p^{16} T^{3} + \)\(17\!\cdots\!04\)\( p^{26} T^{4} - \)\(92\!\cdots\!68\)\( p^{42} T^{5} + \)\(11\!\cdots\!52\)\( p^{60} T^{6} - \)\(14\!\cdots\!80\)\( p^{80} T^{7} + \)\(16\!\cdots\!66\)\( p^{104} T^{8} - \)\(28\!\cdots\!08\)\( p^{130} T^{9} + \)\(16\!\cdots\!66\)\( p^{211} T^{10} - \)\(14\!\cdots\!80\)\( p^{294} T^{11} + \)\(11\!\cdots\!52\)\( p^{381} T^{12} - \)\(92\!\cdots\!68\)\( p^{470} T^{13} + \)\(17\!\cdots\!04\)\( p^{561} T^{14} - \)\(11\!\cdots\!40\)\( p^{658} T^{15} + \)\(51\!\cdots\!87\)\( p^{757} T^{16} - \)\(17\!\cdots\!68\)\( p^{858} T^{17} + p^{963} T^{18} \) | |
| 5 | \( 1 - \)\(39\!\cdots\!34\)\( p^{4} T + \)\(13\!\cdots\!33\)\( p^{6} T^{2} - \)\(14\!\cdots\!96\)\( p^{9} T^{3} + \)\(11\!\cdots\!16\)\( p^{16} T^{4} - \)\(43\!\cdots\!12\)\( p^{26} T^{5} + \)\(28\!\cdots\!48\)\( p^{38} T^{6} + \)\(23\!\cdots\!04\)\( p^{51} T^{7} + \)\(16\!\cdots\!82\)\( p^{65} T^{8} + \)\(27\!\cdots\!68\)\( p^{81} T^{9} + \)\(16\!\cdots\!82\)\( p^{172} T^{10} + \)\(23\!\cdots\!04\)\( p^{265} T^{11} + \)\(28\!\cdots\!48\)\( p^{359} T^{12} - \)\(43\!\cdots\!12\)\( p^{454} T^{13} + \)\(11\!\cdots\!16\)\( p^{551} T^{14} - \)\(14\!\cdots\!96\)\( p^{651} T^{15} + \)\(13\!\cdots\!33\)\( p^{755} T^{16} - \)\(39\!\cdots\!34\)\( p^{860} T^{17} + p^{963} T^{18} \) | |
| 7 | \( 1 + \)\(13\!\cdots\!92\)\( p T + \)\(43\!\cdots\!01\)\( p^{3} T^{2} + \)\(13\!\cdots\!00\)\( p^{7} T^{3} + \)\(10\!\cdots\!28\)\( p^{13} T^{4} + \)\(71\!\cdots\!24\)\( p^{20} T^{5} + \)\(14\!\cdots\!04\)\( p^{29} T^{6} + \)\(27\!\cdots\!00\)\( p^{40} T^{7} + \)\(12\!\cdots\!42\)\( p^{51} T^{8} + \)\(18\!\cdots\!36\)\( p^{62} T^{9} + \)\(12\!\cdots\!42\)\( p^{158} T^{10} + \)\(27\!\cdots\!00\)\( p^{254} T^{11} + \)\(14\!\cdots\!04\)\( p^{350} T^{12} + \)\(71\!\cdots\!24\)\( p^{448} T^{13} + \)\(10\!\cdots\!28\)\( p^{548} T^{14} + \)\(13\!\cdots\!00\)\( p^{649} T^{15} + \)\(43\!\cdots\!01\)\( p^{752} T^{16} + \)\(13\!\cdots\!92\)\( p^{857} T^{17} + p^{963} T^{18} \) | |
| 11 | \( 1 - \)\(68\!\cdots\!68\)\( p T + \)\(98\!\cdots\!73\)\( p^{3} T^{2} - \)\(44\!\cdots\!36\)\( p^{6} T^{3} + \)\(38\!\cdots\!60\)\( p^{9} T^{4} - \)\(14\!\cdots\!28\)\( p^{13} T^{5} + \)\(76\!\cdots\!84\)\( p^{18} T^{6} - \)\(18\!\cdots\!04\)\( p^{25} T^{7} + \)\(63\!\cdots\!58\)\( p^{33} T^{8} - \)\(12\!\cdots\!20\)\( p^{41} T^{9} + \)\(63\!\cdots\!58\)\( p^{140} T^{10} - \)\(18\!\cdots\!04\)\( p^{239} T^{11} + \)\(76\!\cdots\!84\)\( p^{339} T^{12} - \)\(14\!\cdots\!28\)\( p^{441} T^{13} + \)\(38\!\cdots\!60\)\( p^{544} T^{14} - \)\(44\!\cdots\!36\)\( p^{648} T^{15} + \)\(98\!\cdots\!73\)\( p^{752} T^{16} - \)\(68\!\cdots\!68\)\( p^{857} T^{17} + p^{963} T^{18} \) | |
| 13 | \( 1 - \)\(38\!\cdots\!82\)\( T + \)\(42\!\cdots\!89\)\( p T^{2} - \)\(90\!\cdots\!40\)\( p^{3} T^{3} + \)\(35\!\cdots\!04\)\( p^{6} T^{4} - \)\(40\!\cdots\!08\)\( p^{10} T^{5} + \)\(72\!\cdots\!76\)\( p^{15} T^{6} - \)\(55\!\cdots\!60\)\( p^{20} T^{7} + \)\(92\!\cdots\!02\)\( p^{25} T^{8} - \)\(50\!\cdots\!36\)\( p^{31} T^{9} + \)\(92\!\cdots\!02\)\( p^{132} T^{10} - \)\(55\!\cdots\!60\)\( p^{234} T^{11} + \)\(72\!\cdots\!76\)\( p^{336} T^{12} - \)\(40\!\cdots\!08\)\( p^{438} T^{13} + \)\(35\!\cdots\!04\)\( p^{541} T^{14} - \)\(90\!\cdots\!40\)\( p^{645} T^{15} + \)\(42\!\cdots\!89\)\( p^{750} T^{16} - \)\(38\!\cdots\!82\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 17 | \( 1 - \)\(55\!\cdots\!26\)\( T + \)\(12\!\cdots\!09\)\( p T^{2} - \)\(28\!\cdots\!60\)\( p^{3} T^{3} + \)\(91\!\cdots\!24\)\( p^{6} T^{4} - \)\(13\!\cdots\!28\)\( p^{9} T^{5} + \)\(17\!\cdots\!44\)\( p^{13} T^{6} - \)\(85\!\cdots\!40\)\( p^{18} T^{7} + \)\(52\!\cdots\!02\)\( p^{23} T^{8} - \)\(22\!\cdots\!56\)\( p^{28} T^{9} + \)\(52\!\cdots\!02\)\( p^{130} T^{10} - \)\(85\!\cdots\!40\)\( p^{232} T^{11} + \)\(17\!\cdots\!44\)\( p^{334} T^{12} - \)\(13\!\cdots\!28\)\( p^{437} T^{13} + \)\(91\!\cdots\!24\)\( p^{541} T^{14} - \)\(28\!\cdots\!60\)\( p^{645} T^{15} + \)\(12\!\cdots\!09\)\( p^{750} T^{16} - \)\(55\!\cdots\!26\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 19 | \( 1 - \)\(85\!\cdots\!60\)\( T + \)\(35\!\cdots\!29\)\( p T^{2} - \)\(97\!\cdots\!20\)\( p^{2} T^{3} + \)\(12\!\cdots\!36\)\( p^{4} T^{4} - \)\(70\!\cdots\!20\)\( p^{7} T^{5} + \)\(36\!\cdots\!96\)\( p^{10} T^{6} - \)\(87\!\cdots\!40\)\( p^{14} T^{7} + \)\(10\!\cdots\!54\)\( p^{19} T^{8} - \)\(11\!\cdots\!00\)\( p^{24} T^{9} + \)\(10\!\cdots\!54\)\( p^{126} T^{10} - \)\(87\!\cdots\!40\)\( p^{228} T^{11} + \)\(36\!\cdots\!96\)\( p^{331} T^{12} - \)\(70\!\cdots\!20\)\( p^{435} T^{13} + \)\(12\!\cdots\!36\)\( p^{539} T^{14} - \)\(97\!\cdots\!20\)\( p^{644} T^{15} + \)\(35\!\cdots\!29\)\( p^{750} T^{16} - \)\(85\!\cdots\!60\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 23 | \( 1 + \)\(10\!\cdots\!48\)\( T + \)\(11\!\cdots\!09\)\( p T^{2} + \)\(41\!\cdots\!20\)\( p^{2} T^{3} + \)\(13\!\cdots\!16\)\( p^{4} T^{4} + \)\(18\!\cdots\!52\)\( p^{6} T^{5} + \)\(19\!\cdots\!24\)\( p^{9} T^{6} + \)\(45\!\cdots\!20\)\( p^{13} T^{7} + \)\(17\!\cdots\!42\)\( p^{17} T^{8} + \)\(34\!\cdots\!36\)\( p^{21} T^{9} + \)\(17\!\cdots\!42\)\( p^{124} T^{10} + \)\(45\!\cdots\!20\)\( p^{227} T^{11} + \)\(19\!\cdots\!24\)\( p^{330} T^{12} + \)\(18\!\cdots\!52\)\( p^{434} T^{13} + \)\(13\!\cdots\!16\)\( p^{539} T^{14} + \)\(41\!\cdots\!20\)\( p^{644} T^{15} + \)\(11\!\cdots\!09\)\( p^{750} T^{16} + \)\(10\!\cdots\!48\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 29 | \( 1 - \)\(35\!\cdots\!90\)\( T + \)\(18\!\cdots\!81\)\( T^{2} - \)\(16\!\cdots\!20\)\( p T^{3} + \)\(19\!\cdots\!76\)\( p^{2} T^{4} - \)\(48\!\cdots\!20\)\( p^{4} T^{5} + \)\(15\!\cdots\!16\)\( p^{6} T^{6} - \)\(11\!\cdots\!40\)\( p^{9} T^{7} + \)\(10\!\cdots\!46\)\( p^{12} T^{8} - \)\(67\!\cdots\!00\)\( p^{15} T^{9} + \)\(10\!\cdots\!46\)\( p^{119} T^{10} - \)\(11\!\cdots\!40\)\( p^{223} T^{11} + \)\(15\!\cdots\!16\)\( p^{327} T^{12} - \)\(48\!\cdots\!20\)\( p^{432} T^{13} + \)\(19\!\cdots\!76\)\( p^{537} T^{14} - \)\(16\!\cdots\!20\)\( p^{643} T^{15} + \)\(18\!\cdots\!81\)\( p^{749} T^{16} - \)\(35\!\cdots\!90\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 31 | \( 1 - \)\(80\!\cdots\!08\)\( T + \)\(65\!\cdots\!93\)\( p T^{2} - \)\(14\!\cdots\!96\)\( p^{2} T^{3} + \)\(58\!\cdots\!60\)\( p^{3} T^{4} - \)\(33\!\cdots\!68\)\( p^{5} T^{5} + \)\(30\!\cdots\!44\)\( p^{7} T^{6} - \)\(48\!\cdots\!64\)\( p^{10} T^{7} + \)\(12\!\cdots\!58\)\( p^{13} T^{8} - \)\(19\!\cdots\!20\)\( p^{16} T^{9} + \)\(12\!\cdots\!58\)\( p^{120} T^{10} - \)\(48\!\cdots\!64\)\( p^{224} T^{11} + \)\(30\!\cdots\!44\)\( p^{328} T^{12} - \)\(33\!\cdots\!68\)\( p^{433} T^{13} + \)\(58\!\cdots\!60\)\( p^{538} T^{14} - \)\(14\!\cdots\!96\)\( p^{644} T^{15} + \)\(65\!\cdots\!93\)\( p^{750} T^{16} - \)\(80\!\cdots\!08\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 37 | \( 1 + \)\(21\!\cdots\!34\)\( T + \)\(25\!\cdots\!73\)\( T^{2} + \)\(63\!\cdots\!60\)\( T^{3} + \)\(95\!\cdots\!48\)\( p T^{4} + \)\(71\!\cdots\!16\)\( p^{2} T^{5} + \)\(69\!\cdots\!76\)\( p^{3} T^{6} + \)\(14\!\cdots\!60\)\( p^{5} T^{7} + \)\(28\!\cdots\!02\)\( p^{7} T^{8} + \)\(56\!\cdots\!92\)\( p^{9} T^{9} + \)\(28\!\cdots\!02\)\( p^{114} T^{10} + \)\(14\!\cdots\!60\)\( p^{219} T^{11} + \)\(69\!\cdots\!76\)\( p^{324} T^{12} + \)\(71\!\cdots\!16\)\( p^{430} T^{13} + \)\(95\!\cdots\!48\)\( p^{536} T^{14} + \)\(63\!\cdots\!60\)\( p^{642} T^{15} + \)\(25\!\cdots\!73\)\( p^{749} T^{16} + \)\(21\!\cdots\!34\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 41 | \( 1 + \)\(15\!\cdots\!62\)\( T + \)\(21\!\cdots\!93\)\( T^{2} + \)\(84\!\cdots\!04\)\( p T^{3} + \)\(14\!\cdots\!60\)\( p^{2} T^{4} + \)\(52\!\cdots\!92\)\( p^{3} T^{5} + \)\(14\!\cdots\!24\)\( p^{5} T^{6} + \)\(11\!\cdots\!76\)\( p^{7} T^{7} + \)\(26\!\cdots\!38\)\( p^{9} T^{8} + \)\(18\!\cdots\!80\)\( p^{11} T^{9} + \)\(26\!\cdots\!38\)\( p^{116} T^{10} + \)\(11\!\cdots\!76\)\( p^{221} T^{11} + \)\(14\!\cdots\!24\)\( p^{326} T^{12} + \)\(52\!\cdots\!92\)\( p^{431} T^{13} + \)\(14\!\cdots\!60\)\( p^{537} T^{14} + \)\(84\!\cdots\!04\)\( p^{643} T^{15} + \)\(21\!\cdots\!93\)\( p^{749} T^{16} + \)\(15\!\cdots\!62\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 43 | \( 1 - \)\(58\!\cdots\!92\)\( T + \)\(33\!\cdots\!07\)\( T^{2} - \)\(26\!\cdots\!00\)\( p T^{3} + \)\(28\!\cdots\!04\)\( p^{2} T^{4} - \)\(13\!\cdots\!76\)\( p^{3} T^{5} + \)\(38\!\cdots\!04\)\( p^{5} T^{6} - \)\(29\!\cdots\!00\)\( p^{7} T^{7} + \)\(90\!\cdots\!42\)\( p^{9} T^{8} - \)\(58\!\cdots\!36\)\( p^{11} T^{9} + \)\(90\!\cdots\!42\)\( p^{116} T^{10} - \)\(29\!\cdots\!00\)\( p^{221} T^{11} + \)\(38\!\cdots\!04\)\( p^{326} T^{12} - \)\(13\!\cdots\!76\)\( p^{431} T^{13} + \)\(28\!\cdots\!04\)\( p^{537} T^{14} - \)\(26\!\cdots\!00\)\( p^{643} T^{15} + \)\(33\!\cdots\!07\)\( p^{749} T^{16} - \)\(58\!\cdots\!92\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 47 | \( 1 - \)\(41\!\cdots\!36\)\( T + \)\(13\!\cdots\!89\)\( p T^{2} - \)\(10\!\cdots\!80\)\( p^{2} T^{3} + \)\(19\!\cdots\!32\)\( p^{3} T^{4} - \)\(26\!\cdots\!48\)\( p^{5} T^{5} + \)\(70\!\cdots\!56\)\( p^{7} T^{6} - \)\(83\!\cdots\!20\)\( p^{9} T^{7} + \)\(17\!\cdots\!62\)\( p^{11} T^{8} - \)\(17\!\cdots\!88\)\( p^{13} T^{9} + \)\(17\!\cdots\!62\)\( p^{118} T^{10} - \)\(83\!\cdots\!20\)\( p^{223} T^{11} + \)\(70\!\cdots\!56\)\( p^{328} T^{12} - \)\(26\!\cdots\!48\)\( p^{433} T^{13} + \)\(19\!\cdots\!32\)\( p^{538} T^{14} - \)\(10\!\cdots\!80\)\( p^{644} T^{15} + \)\(13\!\cdots\!89\)\( p^{750} T^{16} - \)\(41\!\cdots\!36\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 53 | \( 1 + \)\(10\!\cdots\!46\)\( p T + \)\(64\!\cdots\!73\)\( p^{2} T^{2} + \)\(44\!\cdots\!80\)\( p^{3} T^{3} + \)\(21\!\cdots\!36\)\( p^{4} T^{4} + \)\(11\!\cdots\!16\)\( p^{5} T^{5} + \)\(45\!\cdots\!88\)\( p^{6} T^{6} + \)\(19\!\cdots\!60\)\( p^{7} T^{7} + \)\(69\!\cdots\!86\)\( p^{8} T^{8} + \)\(24\!\cdots\!76\)\( p^{9} T^{9} + \)\(69\!\cdots\!86\)\( p^{115} T^{10} + \)\(19\!\cdots\!60\)\( p^{221} T^{11} + \)\(45\!\cdots\!88\)\( p^{327} T^{12} + \)\(11\!\cdots\!16\)\( p^{433} T^{13} + \)\(21\!\cdots\!36\)\( p^{539} T^{14} + \)\(44\!\cdots\!80\)\( p^{645} T^{15} + \)\(64\!\cdots\!73\)\( p^{751} T^{16} + \)\(10\!\cdots\!46\)\( p^{857} T^{17} + p^{963} T^{18} \) | |
| 59 | \( 1 + \)\(17\!\cdots\!20\)\( T + \)\(18\!\cdots\!71\)\( T^{2} + \)\(27\!\cdots\!40\)\( T^{3} + \)\(27\!\cdots\!44\)\( p T^{4} + \)\(58\!\cdots\!60\)\( p^{2} T^{5} + \)\(45\!\cdots\!64\)\( p^{3} T^{6} + \)\(13\!\cdots\!20\)\( p^{5} T^{7} + \)\(52\!\cdots\!54\)\( p^{5} T^{8} + \)\(78\!\cdots\!00\)\( p^{6} T^{9} + \)\(52\!\cdots\!54\)\( p^{112} T^{10} + \)\(13\!\cdots\!20\)\( p^{219} T^{11} + \)\(45\!\cdots\!64\)\( p^{324} T^{12} + \)\(58\!\cdots\!60\)\( p^{430} T^{13} + \)\(27\!\cdots\!44\)\( p^{536} T^{14} + \)\(27\!\cdots\!40\)\( p^{642} T^{15} + \)\(18\!\cdots\!71\)\( p^{749} T^{16} + \)\(17\!\cdots\!20\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 61 | \( 1 - \)\(99\!\cdots\!98\)\( T + \)\(10\!\cdots\!13\)\( T^{2} - \)\(67\!\cdots\!96\)\( T^{3} + \)\(41\!\cdots\!60\)\( T^{4} - \)\(33\!\cdots\!88\)\( p T^{5} + \)\(25\!\cdots\!24\)\( p^{2} T^{6} - \)\(16\!\cdots\!84\)\( p^{3} T^{7} + \)\(10\!\cdots\!78\)\( p^{4} T^{8} - \)\(57\!\cdots\!20\)\( p^{5} T^{9} + \)\(10\!\cdots\!78\)\( p^{111} T^{10} - \)\(16\!\cdots\!84\)\( p^{217} T^{11} + \)\(25\!\cdots\!24\)\( p^{323} T^{12} - \)\(33\!\cdots\!88\)\( p^{429} T^{13} + \)\(41\!\cdots\!60\)\( p^{535} T^{14} - \)\(67\!\cdots\!96\)\( p^{642} T^{15} + \)\(10\!\cdots\!13\)\( p^{749} T^{16} - \)\(99\!\cdots\!98\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 67 | \( 1 - \)\(67\!\cdots\!76\)\( T + \)\(17\!\cdots\!03\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(21\!\cdots\!68\)\( p T^{4} - \)\(18\!\cdots\!44\)\( p^{2} T^{5} + \)\(24\!\cdots\!56\)\( p^{3} T^{6} - \)\(17\!\cdots\!60\)\( p^{4} T^{7} + \)\(18\!\cdots\!18\)\( p^{5} T^{8} - \)\(11\!\cdots\!84\)\( p^{6} T^{9} + \)\(18\!\cdots\!18\)\( p^{112} T^{10} - \)\(17\!\cdots\!60\)\( p^{218} T^{11} + \)\(24\!\cdots\!56\)\( p^{324} T^{12} - \)\(18\!\cdots\!44\)\( p^{430} T^{13} + \)\(21\!\cdots\!68\)\( p^{536} T^{14} - \)\(10\!\cdots\!80\)\( p^{642} T^{15} + \)\(17\!\cdots\!03\)\( p^{749} T^{16} - \)\(67\!\cdots\!76\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 71 | \( 1 - \)\(55\!\cdots\!28\)\( T + \)\(30\!\cdots\!23\)\( T^{2} - \)\(13\!\cdots\!56\)\( p T^{3} + \)\(10\!\cdots\!60\)\( p^{2} T^{4} - \)\(63\!\cdots\!88\)\( p^{3} T^{5} + \)\(35\!\cdots\!24\)\( p^{4} T^{6} - \)\(13\!\cdots\!84\)\( p^{5} T^{7} + \)\(94\!\cdots\!78\)\( p^{6} T^{8} - \)\(20\!\cdots\!20\)\( p^{7} T^{9} + \)\(94\!\cdots\!78\)\( p^{113} T^{10} - \)\(13\!\cdots\!84\)\( p^{219} T^{11} + \)\(35\!\cdots\!24\)\( p^{325} T^{12} - \)\(63\!\cdots\!88\)\( p^{431} T^{13} + \)\(10\!\cdots\!60\)\( p^{537} T^{14} - \)\(13\!\cdots\!56\)\( p^{643} T^{15} + \)\(30\!\cdots\!23\)\( p^{749} T^{16} - \)\(55\!\cdots\!28\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 73 | \( 1 - \)\(13\!\cdots\!02\)\( T + \)\(21\!\cdots\!57\)\( T^{2} - \)\(19\!\cdots\!20\)\( T^{3} + \)\(25\!\cdots\!72\)\( p T^{4} - \)\(24\!\cdots\!68\)\( p^{2} T^{5} + \)\(22\!\cdots\!36\)\( p^{3} T^{6} - \)\(18\!\cdots\!40\)\( p^{4} T^{7} + \)\(14\!\cdots\!82\)\( p^{5} T^{8} - \)\(96\!\cdots\!48\)\( p^{6} T^{9} + \)\(14\!\cdots\!82\)\( p^{112} T^{10} - \)\(18\!\cdots\!40\)\( p^{218} T^{11} + \)\(22\!\cdots\!36\)\( p^{324} T^{12} - \)\(24\!\cdots\!68\)\( p^{430} T^{13} + \)\(25\!\cdots\!72\)\( p^{536} T^{14} - \)\(19\!\cdots\!20\)\( p^{642} T^{15} + \)\(21\!\cdots\!57\)\( p^{749} T^{16} - \)\(13\!\cdots\!02\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 79 | \( 1 - \)\(17\!\cdots\!40\)\( T + \)\(21\!\cdots\!31\)\( T^{2} - \)\(24\!\cdots\!20\)\( p T^{3} + \)\(22\!\cdots\!76\)\( p^{2} T^{4} - \)\(16\!\cdots\!80\)\( p^{3} T^{5} + \)\(10\!\cdots\!56\)\( p^{4} T^{6} - \)\(62\!\cdots\!40\)\( p^{5} T^{7} + \)\(31\!\cdots\!66\)\( p^{6} T^{8} - \)\(14\!\cdots\!00\)\( p^{7} T^{9} + \)\(31\!\cdots\!66\)\( p^{113} T^{10} - \)\(62\!\cdots\!40\)\( p^{219} T^{11} + \)\(10\!\cdots\!56\)\( p^{325} T^{12} - \)\(16\!\cdots\!80\)\( p^{431} T^{13} + \)\(22\!\cdots\!76\)\( p^{537} T^{14} - \)\(24\!\cdots\!20\)\( p^{643} T^{15} + \)\(21\!\cdots\!31\)\( p^{749} T^{16} - \)\(17\!\cdots\!40\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 83 | \( 1 - \)\(10\!\cdots\!72\)\( T + \)\(17\!\cdots\!29\)\( p T^{2} - \)\(15\!\cdots\!40\)\( p^{2} T^{3} + \)\(16\!\cdots\!48\)\( p^{3} T^{4} - \)\(11\!\cdots\!12\)\( p^{4} T^{5} + \)\(96\!\cdots\!44\)\( p^{5} T^{6} - \)\(60\!\cdots\!80\)\( p^{6} T^{7} + \)\(41\!\cdots\!58\)\( p^{7} T^{8} - \)\(22\!\cdots\!32\)\( p^{8} T^{9} + \)\(41\!\cdots\!58\)\( p^{114} T^{10} - \)\(60\!\cdots\!80\)\( p^{220} T^{11} + \)\(96\!\cdots\!44\)\( p^{326} T^{12} - \)\(11\!\cdots\!12\)\( p^{432} T^{13} + \)\(16\!\cdots\!48\)\( p^{538} T^{14} - \)\(15\!\cdots\!40\)\( p^{644} T^{15} + \)\(17\!\cdots\!29\)\( p^{750} T^{16} - \)\(10\!\cdots\!72\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 89 | \( 1 + \)\(88\!\cdots\!30\)\( T + \)\(62\!\cdots\!49\)\( p T^{2} + \)\(33\!\cdots\!60\)\( p^{2} T^{3} + \)\(14\!\cdots\!04\)\( p^{3} T^{4} + \)\(61\!\cdots\!60\)\( p^{5} T^{5} + \)\(17\!\cdots\!24\)\( p^{5} T^{6} + \)\(52\!\cdots\!20\)\( p^{6} T^{7} + \)\(13\!\cdots\!14\)\( p^{7} T^{8} + \)\(31\!\cdots\!00\)\( p^{8} T^{9} + \)\(13\!\cdots\!14\)\( p^{114} T^{10} + \)\(52\!\cdots\!20\)\( p^{220} T^{11} + \)\(17\!\cdots\!24\)\( p^{326} T^{12} + \)\(61\!\cdots\!60\)\( p^{433} T^{13} + \)\(14\!\cdots\!04\)\( p^{538} T^{14} + \)\(33\!\cdots\!60\)\( p^{644} T^{15} + \)\(62\!\cdots\!49\)\( p^{750} T^{16} + \)\(88\!\cdots\!30\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
| 97 | \( 1 - \)\(18\!\cdots\!86\)\( T + \)\(31\!\cdots\!89\)\( p T^{2} - \)\(50\!\cdots\!80\)\( p^{2} T^{3} + \)\(46\!\cdots\!32\)\( p^{3} T^{4} - \)\(62\!\cdots\!56\)\( p^{4} T^{5} + \)\(41\!\cdots\!04\)\( p^{5} T^{6} - \)\(46\!\cdots\!60\)\( p^{6} T^{7} + \)\(24\!\cdots\!22\)\( p^{7} T^{8} - \)\(23\!\cdots\!16\)\( p^{8} T^{9} + \)\(24\!\cdots\!22\)\( p^{114} T^{10} - \)\(46\!\cdots\!60\)\( p^{220} T^{11} + \)\(41\!\cdots\!04\)\( p^{326} T^{12} - \)\(62\!\cdots\!56\)\( p^{432} T^{13} + \)\(46\!\cdots\!32\)\( p^{538} T^{14} - \)\(50\!\cdots\!80\)\( p^{644} T^{15} + \)\(31\!\cdots\!89\)\( p^{750} T^{16} - \)\(18\!\cdots\!86\)\( p^{856} T^{17} + p^{963} T^{18} \) | |
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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
−4.00807649857750845239722251941, −3.63027758635446743160351293587, −3.55168723766922161405068228353, −3.54092832235692618119050593636, −3.46936195921916403428326429912, −3.42238202500047433983181792502, −3.13322748712685785694912478442, −2.84761275705642239284518638591, −2.72124733121092692387955586261, −2.65416958064672393860877212481, −2.41207729420044933011581987461, −2.24867207721836840473228012595, −1.93969587283282521293714230788, −1.92192209136424795998717121472, −1.82364244676335413843629367495, −1.50405217199365304520646661676, −1.33234315892344639662017015994, −1.18690891918065881811111971990, −0.863441791992397739346135241278, −0.837147607785070088334852904639, −0.70134721888787551763708532858, −0.66690155952333327256954876059, −0.28352548999709803256134297094, −0.27506538166052387127385127231, −0.26730979185041636452216199213, 0.26730979185041636452216199213, 0.27506538166052387127385127231, 0.28352548999709803256134297094, 0.66690155952333327256954876059, 0.70134721888787551763708532858, 0.837147607785070088334852904639, 0.863441791992397739346135241278, 1.18690891918065881811111971990, 1.33234315892344639662017015994, 1.50405217199365304520646661676, 1.82364244676335413843629367495, 1.92192209136424795998717121472, 1.93969587283282521293714230788, 2.24867207721836840473228012595, 2.41207729420044933011581987461, 2.65416958064672393860877212481, 2.72124733121092692387955586261, 2.84761275705642239284518638591, 3.13322748712685785694912478442, 3.42238202500047433983181792502, 3.46936195921916403428326429912, 3.54092832235692618119050593636, 3.55168723766922161405068228353, 3.63027758635446743160351293587, 4.00807649857750845239722251941
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.