Dirichlet series
| L(s) = 1 | + 666·2-s − 4.72e5·3-s − 2.82e6·4-s − 3.14e8·6-s + 1.34e8·7-s − 1.27e9·8-s + 1.25e11·9-s − 1.44e10·11-s + 1.33e12·12-s − 3.28e11·13-s + 8.92e10·14-s + 3.48e12·16-s − 5.71e12·17-s + 8.35e13·18-s + 7.59e13·19-s − 6.33e13·21-s − 9.59e12·22-s + 4.97e13·23-s + 6.02e14·24-s − 2.18e14·26-s − 2.47e16·27-s − 3.78e14·28-s + 2.11e15·29-s + 5.67e15·31-s + 2.34e15·32-s + 6.80e15·33-s − 3.80e15·34-s + ⋯ |
| L(s) = 1 | + 0.459·2-s − 4.61·3-s − 1.34·4-s − 2.12·6-s + 0.179·7-s − 0.419·8-s + 12·9-s − 0.167·11-s + 6.22·12-s − 0.660·13-s + 0.0824·14-s + 0.792·16-s − 0.687·17-s + 5.51·18-s + 2.84·19-s − 0.828·21-s − 0.0770·22-s + 0.250·23-s + 1.93·24-s − 0.303·26-s − 23.0·27-s − 0.241·28-s + 0.931·29-s + 1.24·31-s + 0.367·32-s + 0.773·33-s − 0.316·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(22-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+21/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.72617\times 10^{18}\) |
| Root analytic conductor: | \(14.4778\) |
| Motivic weight: | \(21\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 5^{16} ,\ ( \ : [21/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(11)\) | \(\approx\) | \(1.108694901\) |
| \(L(\frac12)\) | \(\approx\) | \(1.108694901\) |
| \(L(\frac{23}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p^{10} T )^{8} \) |
| 5 | \( 1 \) | |
| good | 2 | \( 1 - 333 p T + 1634717 p T^{2} - 87004017 p^{5} T^{3} + 211248995987 p^{5} T^{4} - 8030305999071 p^{10} T^{5} + 844527680959737 p^{14} T^{6} - 6803978045500863 p^{22} T^{7} + 336623799319168707 p^{26} T^{8} - 6803978045500863 p^{43} T^{9} + 844527680959737 p^{56} T^{10} - 8030305999071 p^{73} T^{11} + 211248995987 p^{89} T^{12} - 87004017 p^{110} T^{13} + 1634717 p^{127} T^{14} - 333 p^{148} T^{15} + p^{168} T^{16} \) |
| 7 | \( 1 - 134034472 T + 1656716378457111060 T^{2} - \)\(12\!\cdots\!04\)\( p^{2} T^{3} + \)\(32\!\cdots\!14\)\( p^{2} T^{4} + \)\(14\!\cdots\!12\)\( p^{4} T^{5} + \)\(52\!\cdots\!36\)\( p^{4} T^{6} + \)\(21\!\cdots\!00\)\( p^{5} T^{7} + \)\(67\!\cdots\!91\)\( p^{6} T^{8} + \)\(21\!\cdots\!00\)\( p^{26} T^{9} + \)\(52\!\cdots\!36\)\( p^{46} T^{10} + \)\(14\!\cdots\!12\)\( p^{67} T^{11} + \)\(32\!\cdots\!14\)\( p^{86} T^{12} - \)\(12\!\cdots\!04\)\( p^{107} T^{13} + 1656716378457111060 p^{126} T^{14} - 134034472 p^{147} T^{15} + p^{168} T^{16} \) | |
| 11 | \( 1 + 1310015088 p T + \)\(24\!\cdots\!40\)\( T^{2} + \)\(58\!\cdots\!56\)\( T^{3} + \)\(32\!\cdots\!96\)\( p T^{4} + \)\(47\!\cdots\!68\)\( p^{2} T^{5} + \)\(29\!\cdots\!88\)\( p^{3} T^{6} + \)\(36\!\cdots\!24\)\( p^{4} T^{7} + \)\(19\!\cdots\!58\)\( p^{5} T^{8} + \)\(36\!\cdots\!24\)\( p^{25} T^{9} + \)\(29\!\cdots\!88\)\( p^{45} T^{10} + \)\(47\!\cdots\!68\)\( p^{65} T^{11} + \)\(32\!\cdots\!96\)\( p^{85} T^{12} + \)\(58\!\cdots\!56\)\( p^{105} T^{13} + \)\(24\!\cdots\!40\)\( p^{126} T^{14} + 1310015088 p^{148} T^{15} + p^{168} T^{16} \) | |
| 13 | \( 1 + 328226481176 T + \)\(11\!\cdots\!52\)\( T^{2} + \)\(30\!\cdots\!76\)\( T^{3} + \)\(70\!\cdots\!62\)\( T^{4} + \)\(11\!\cdots\!08\)\( p T^{5} + \)\(16\!\cdots\!24\)\( p^{2} T^{6} + \)\(24\!\cdots\!16\)\( p^{3} T^{7} + \)\(28\!\cdots\!67\)\( p^{4} T^{8} + \)\(24\!\cdots\!16\)\( p^{24} T^{9} + \)\(16\!\cdots\!24\)\( p^{44} T^{10} + \)\(11\!\cdots\!08\)\( p^{64} T^{11} + \)\(70\!\cdots\!62\)\( p^{84} T^{12} + \)\(30\!\cdots\!76\)\( p^{105} T^{13} + \)\(11\!\cdots\!52\)\( p^{126} T^{14} + 328226481176 p^{147} T^{15} + p^{168} T^{16} \) | |
| 17 | \( 1 + 5718214953936 T + \)\(84\!\cdots\!84\)\( p T^{2} + \)\(79\!\cdots\!08\)\( p^{2} T^{3} + \)\(35\!\cdots\!92\)\( p^{3} T^{4} + \)\(29\!\cdots\!36\)\( p^{4} T^{5} + \)\(16\!\cdots\!20\)\( p^{5} T^{6} + \)\(77\!\cdots\!04\)\( p^{6} T^{7} + \)\(47\!\cdots\!38\)\( p^{7} T^{8} + \)\(77\!\cdots\!04\)\( p^{27} T^{9} + \)\(16\!\cdots\!20\)\( p^{47} T^{10} + \)\(29\!\cdots\!36\)\( p^{67} T^{11} + \)\(35\!\cdots\!92\)\( p^{87} T^{12} + \)\(79\!\cdots\!08\)\( p^{107} T^{13} + \)\(84\!\cdots\!84\)\( p^{127} T^{14} + 5718214953936 p^{147} T^{15} + p^{168} T^{16} \) | |
| 19 | \( 1 - 75919698170296 T + \)\(50\!\cdots\!28\)\( T^{2} - \)\(11\!\cdots\!16\)\( p T^{3} + \)\(23\!\cdots\!78\)\( p^{2} T^{4} - \)\(36\!\cdots\!64\)\( p^{3} T^{5} + \)\(54\!\cdots\!80\)\( p^{4} T^{6} - \)\(69\!\cdots\!40\)\( p^{5} T^{7} + \)\(10\!\cdots\!71\)\( p^{6} T^{8} - \)\(69\!\cdots\!40\)\( p^{26} T^{9} + \)\(54\!\cdots\!80\)\( p^{46} T^{10} - \)\(36\!\cdots\!64\)\( p^{66} T^{11} + \)\(23\!\cdots\!78\)\( p^{86} T^{12} - \)\(11\!\cdots\!16\)\( p^{106} T^{13} + \)\(50\!\cdots\!28\)\( p^{126} T^{14} - 75919698170296 p^{147} T^{15} + p^{168} T^{16} \) | |
| 23 | \( 1 - 49712781537936 T + \)\(97\!\cdots\!00\)\( T^{2} - \)\(16\!\cdots\!72\)\( T^{3} + \)\(57\!\cdots\!96\)\( T^{4} - \)\(18\!\cdots\!84\)\( T^{5} + \)\(30\!\cdots\!04\)\( T^{6} - \)\(14\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!34\)\( T^{8} - \)\(14\!\cdots\!40\)\( p^{21} T^{9} + \)\(30\!\cdots\!04\)\( p^{42} T^{10} - \)\(18\!\cdots\!84\)\( p^{63} T^{11} + \)\(57\!\cdots\!96\)\( p^{84} T^{12} - \)\(16\!\cdots\!72\)\( p^{105} T^{13} + \)\(97\!\cdots\!00\)\( p^{126} T^{14} - 49712781537936 p^{147} T^{15} + p^{168} T^{16} \) | |
| 29 | \( 1 - 2111332005818352 T + \)\(47\!\cdots\!24\)\( p T^{2} - \)\(68\!\cdots\!44\)\( T^{3} + \)\(80\!\cdots\!20\)\( T^{4} + \)\(33\!\cdots\!68\)\( T^{5} + \)\(35\!\cdots\!20\)\( T^{6} + \)\(81\!\cdots\!40\)\( T^{7} + \)\(94\!\cdots\!46\)\( T^{8} + \)\(81\!\cdots\!40\)\( p^{21} T^{9} + \)\(35\!\cdots\!20\)\( p^{42} T^{10} + \)\(33\!\cdots\!68\)\( p^{63} T^{11} + \)\(80\!\cdots\!20\)\( p^{84} T^{12} - \)\(68\!\cdots\!44\)\( p^{105} T^{13} + \)\(47\!\cdots\!24\)\( p^{127} T^{14} - 2111332005818352 p^{147} T^{15} + p^{168} T^{16} \) | |
| 31 | \( 1 - 5677451410844968 T + \)\(10\!\cdots\!32\)\( T^{2} - \)\(53\!\cdots\!08\)\( T^{3} + \)\(55\!\cdots\!78\)\( T^{4} - \)\(25\!\cdots\!68\)\( T^{5} + \)\(19\!\cdots\!60\)\( T^{6} - \)\(77\!\cdots\!00\)\( T^{7} + \)\(48\!\cdots\!11\)\( T^{8} - \)\(77\!\cdots\!00\)\( p^{21} T^{9} + \)\(19\!\cdots\!60\)\( p^{42} T^{10} - \)\(25\!\cdots\!68\)\( p^{63} T^{11} + \)\(55\!\cdots\!78\)\( p^{84} T^{12} - \)\(53\!\cdots\!08\)\( p^{105} T^{13} + \)\(10\!\cdots\!32\)\( p^{126} T^{14} - 5677451410844968 p^{147} T^{15} + p^{168} T^{16} \) | |
| 37 | \( 1 + 26538401323409648 T + \)\(34\!\cdots\!80\)\( T^{2} + \)\(89\!\cdots\!64\)\( T^{3} + \)\(59\!\cdots\!56\)\( T^{4} + \)\(13\!\cdots\!32\)\( T^{5} + \)\(65\!\cdots\!56\)\( T^{6} + \)\(14\!\cdots\!80\)\( T^{7} + \)\(59\!\cdots\!74\)\( T^{8} + \)\(14\!\cdots\!80\)\( p^{21} T^{9} + \)\(65\!\cdots\!56\)\( p^{42} T^{10} + \)\(13\!\cdots\!32\)\( p^{63} T^{11} + \)\(59\!\cdots\!56\)\( p^{84} T^{12} + \)\(89\!\cdots\!64\)\( p^{105} T^{13} + \)\(34\!\cdots\!80\)\( p^{126} T^{14} + 26538401323409648 p^{147} T^{15} + p^{168} T^{16} \) | |
| 41 | \( 1 + 9719778050982432 T + \)\(47\!\cdots\!12\)\( T^{2} + \)\(43\!\cdots\!32\)\( T^{3} + \)\(10\!\cdots\!28\)\( T^{4} + \)\(86\!\cdots\!72\)\( T^{5} + \)\(14\!\cdots\!80\)\( T^{6} + \)\(24\!\cdots\!00\)\( p T^{7} + \)\(12\!\cdots\!06\)\( T^{8} + \)\(24\!\cdots\!00\)\( p^{22} T^{9} + \)\(14\!\cdots\!80\)\( p^{42} T^{10} + \)\(86\!\cdots\!72\)\( p^{63} T^{11} + \)\(10\!\cdots\!28\)\( p^{84} T^{12} + \)\(43\!\cdots\!32\)\( p^{105} T^{13} + \)\(47\!\cdots\!12\)\( p^{126} T^{14} + 9719778050982432 p^{147} T^{15} + p^{168} T^{16} \) | |
| 43 | \( 1 + 20888744714565080 T + \)\(94\!\cdots\!40\)\( T^{2} + \)\(24\!\cdots\!40\)\( T^{3} + \)\(48\!\cdots\!46\)\( T^{4} + \)\(12\!\cdots\!60\)\( T^{5} + \)\(16\!\cdots\!80\)\( T^{6} + \)\(40\!\cdots\!80\)\( T^{7} + \)\(38\!\cdots\!31\)\( T^{8} + \)\(40\!\cdots\!80\)\( p^{21} T^{9} + \)\(16\!\cdots\!80\)\( p^{42} T^{10} + \)\(12\!\cdots\!60\)\( p^{63} T^{11} + \)\(48\!\cdots\!46\)\( p^{84} T^{12} + \)\(24\!\cdots\!40\)\( p^{105} T^{13} + \)\(94\!\cdots\!40\)\( p^{126} T^{14} + 20888744714565080 p^{147} T^{15} + p^{168} T^{16} \) | |
| 47 | \( 1 - 405141072730526160 T + \)\(53\!\cdots\!40\)\( T^{2} - \)\(22\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!36\)\( T^{4} - \)\(64\!\cdots\!20\)\( T^{5} + \)\(28\!\cdots\!80\)\( T^{6} - \)\(11\!\cdots\!40\)\( T^{7} + \)\(42\!\cdots\!86\)\( T^{8} - \)\(11\!\cdots\!40\)\( p^{21} T^{9} + \)\(28\!\cdots\!80\)\( p^{42} T^{10} - \)\(64\!\cdots\!20\)\( p^{63} T^{11} + \)\(14\!\cdots\!36\)\( p^{84} T^{12} - \)\(22\!\cdots\!20\)\( p^{105} T^{13} + \)\(53\!\cdots\!40\)\( p^{126} T^{14} - 405141072730526160 p^{147} T^{15} + p^{168} T^{16} \) | |
| 53 | \( 1 - 1890950875516930176 T + \)\(94\!\cdots\!80\)\( T^{2} - \)\(14\!\cdots\!12\)\( T^{3} + \)\(43\!\cdots\!76\)\( T^{4} - \)\(56\!\cdots\!64\)\( T^{5} + \)\(12\!\cdots\!64\)\( T^{6} - \)\(13\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!74\)\( T^{8} - \)\(13\!\cdots\!00\)\( p^{21} T^{9} + \)\(12\!\cdots\!64\)\( p^{42} T^{10} - \)\(56\!\cdots\!64\)\( p^{63} T^{11} + \)\(43\!\cdots\!76\)\( p^{84} T^{12} - \)\(14\!\cdots\!12\)\( p^{105} T^{13} + \)\(94\!\cdots\!80\)\( p^{126} T^{14} - 1890950875516930176 p^{147} T^{15} + p^{168} T^{16} \) | |
| 59 | \( 1 + 3496737905134765776 T + \)\(68\!\cdots\!08\)\( T^{2} + \)\(30\!\cdots\!64\)\( T^{3} + \)\(25\!\cdots\!68\)\( T^{4} + \)\(11\!\cdots\!76\)\( T^{5} + \)\(64\!\cdots\!80\)\( T^{6} + \)\(26\!\cdots\!60\)\( T^{7} + \)\(11\!\cdots\!66\)\( T^{8} + \)\(26\!\cdots\!60\)\( p^{21} T^{9} + \)\(64\!\cdots\!80\)\( p^{42} T^{10} + \)\(11\!\cdots\!76\)\( p^{63} T^{11} + \)\(25\!\cdots\!68\)\( p^{84} T^{12} + \)\(30\!\cdots\!64\)\( p^{105} T^{13} + \)\(68\!\cdots\!08\)\( p^{126} T^{14} + 3496737905134765776 p^{147} T^{15} + p^{168} T^{16} \) | |
| 61 | \( 1 - 17508733151770015960 T + \)\(24\!\cdots\!52\)\( T^{2} - \)\(24\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!98\)\( T^{4} - \)\(16\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!64\)\( T^{6} - \)\(76\!\cdots\!00\)\( T^{7} + \)\(44\!\cdots\!95\)\( T^{8} - \)\(76\!\cdots\!00\)\( p^{21} T^{9} + \)\(12\!\cdots\!64\)\( p^{42} T^{10} - \)\(16\!\cdots\!40\)\( p^{63} T^{11} + \)\(22\!\cdots\!98\)\( p^{84} T^{12} - \)\(24\!\cdots\!00\)\( p^{105} T^{13} + \)\(24\!\cdots\!52\)\( p^{126} T^{14} - 17508733151770015960 p^{147} T^{15} + p^{168} T^{16} \) | |
| 67 | \( 1 + 29911607071855004216 T + \)\(10\!\cdots\!48\)\( T^{2} + \)\(23\!\cdots\!52\)\( T^{3} + \)\(58\!\cdots\!86\)\( T^{4} + \)\(10\!\cdots\!76\)\( T^{5} + \)\(21\!\cdots\!00\)\( T^{6} + \)\(32\!\cdots\!16\)\( T^{7} + \)\(55\!\cdots\!79\)\( T^{8} + \)\(32\!\cdots\!16\)\( p^{21} T^{9} + \)\(21\!\cdots\!00\)\( p^{42} T^{10} + \)\(10\!\cdots\!76\)\( p^{63} T^{11} + \)\(58\!\cdots\!86\)\( p^{84} T^{12} + \)\(23\!\cdots\!52\)\( p^{105} T^{13} + \)\(10\!\cdots\!48\)\( p^{126} T^{14} + 29911607071855004216 p^{147} T^{15} + p^{168} T^{16} \) | |
| 71 | \( 1 - 52204030316256846816 T + \)\(39\!\cdots\!80\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!80\)\( T^{4} - \)\(16\!\cdots\!08\)\( T^{5} + \)\(49\!\cdots\!68\)\( T^{6} - \)\(11\!\cdots\!60\)\( T^{7} + \)\(33\!\cdots\!50\)\( T^{8} - \)\(11\!\cdots\!60\)\( p^{21} T^{9} + \)\(49\!\cdots\!68\)\( p^{42} T^{10} - \)\(16\!\cdots\!08\)\( p^{63} T^{11} + \)\(59\!\cdots\!80\)\( p^{84} T^{12} - \)\(14\!\cdots\!00\)\( p^{105} T^{13} + \)\(39\!\cdots\!80\)\( p^{126} T^{14} - 52204030316256846816 p^{147} T^{15} + p^{168} T^{16} \) | |
| 73 | \( 1 + 39697032757192143344 T + \)\(80\!\cdots\!40\)\( T^{2} + \)\(23\!\cdots\!48\)\( T^{3} + \)\(28\!\cdots\!96\)\( T^{4} + \)\(66\!\cdots\!36\)\( T^{5} + \)\(63\!\cdots\!64\)\( T^{6} + \)\(11\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!54\)\( T^{8} + \)\(11\!\cdots\!60\)\( p^{21} T^{9} + \)\(63\!\cdots\!64\)\( p^{42} T^{10} + \)\(66\!\cdots\!36\)\( p^{63} T^{11} + \)\(28\!\cdots\!96\)\( p^{84} T^{12} + \)\(23\!\cdots\!48\)\( p^{105} T^{13} + \)\(80\!\cdots\!40\)\( p^{126} T^{14} + 39697032757192143344 p^{147} T^{15} + p^{168} T^{16} \) | |
| 79 | \( 1 + \)\(27\!\cdots\!00\)\( T + \)\(72\!\cdots\!32\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!48\)\( T^{4} + \)\(24\!\cdots\!00\)\( T^{5} + \)\(28\!\cdots\!84\)\( T^{6} + \)\(27\!\cdots\!00\)\( T^{7} + \)\(25\!\cdots\!70\)\( T^{8} + \)\(27\!\cdots\!00\)\( p^{21} T^{9} + \)\(28\!\cdots\!84\)\( p^{42} T^{10} + \)\(24\!\cdots\!00\)\( p^{63} T^{11} + \)\(19\!\cdots\!48\)\( p^{84} T^{12} + \)\(12\!\cdots\!00\)\( p^{105} T^{13} + \)\(72\!\cdots\!32\)\( p^{126} T^{14} + \)\(27\!\cdots\!00\)\( p^{147} T^{15} + p^{168} T^{16} \) | |
| 83 | \( 1 + \)\(27\!\cdots\!12\)\( T + \)\(11\!\cdots\!76\)\( T^{2} + \)\(23\!\cdots\!52\)\( T^{3} + \)\(63\!\cdots\!64\)\( T^{4} + \)\(10\!\cdots\!88\)\( T^{5} + \)\(21\!\cdots\!12\)\( T^{6} + \)\(30\!\cdots\!76\)\( T^{7} + \)\(51\!\cdots\!98\)\( T^{8} + \)\(30\!\cdots\!76\)\( p^{21} T^{9} + \)\(21\!\cdots\!12\)\( p^{42} T^{10} + \)\(10\!\cdots\!88\)\( p^{63} T^{11} + \)\(63\!\cdots\!64\)\( p^{84} T^{12} + \)\(23\!\cdots\!52\)\( p^{105} T^{13} + \)\(11\!\cdots\!76\)\( p^{126} T^{14} + \)\(27\!\cdots\!12\)\( p^{147} T^{15} + p^{168} T^{16} \) | |
| 89 | \( 1 + \)\(31\!\cdots\!84\)\( T + \)\(53\!\cdots\!88\)\( T^{2} + \)\(12\!\cdots\!16\)\( T^{3} + \)\(12\!\cdots\!68\)\( T^{4} + \)\(22\!\cdots\!04\)\( T^{5} + \)\(17\!\cdots\!80\)\( T^{6} + \)\(26\!\cdots\!60\)\( T^{7} + \)\(18\!\cdots\!46\)\( T^{8} + \)\(26\!\cdots\!60\)\( p^{21} T^{9} + \)\(17\!\cdots\!80\)\( p^{42} T^{10} + \)\(22\!\cdots\!04\)\( p^{63} T^{11} + \)\(12\!\cdots\!68\)\( p^{84} T^{12} + \)\(12\!\cdots\!16\)\( p^{105} T^{13} + \)\(53\!\cdots\!88\)\( p^{126} T^{14} + \)\(31\!\cdots\!84\)\( p^{147} T^{15} + p^{168} T^{16} \) | |
| 97 | \( 1 - \)\(45\!\cdots\!64\)\( T + \)\(15\!\cdots\!68\)\( T^{2} - \)\(10\!\cdots\!28\)\( T^{3} + \)\(17\!\cdots\!66\)\( T^{4} - \)\(12\!\cdots\!24\)\( T^{5} + \)\(12\!\cdots\!00\)\( T^{6} - \)\(98\!\cdots\!44\)\( T^{7} + \)\(73\!\cdots\!19\)\( T^{8} - \)\(98\!\cdots\!44\)\( p^{21} T^{9} + \)\(12\!\cdots\!00\)\( p^{42} T^{10} - \)\(12\!\cdots\!24\)\( p^{63} T^{11} + \)\(17\!\cdots\!66\)\( p^{84} T^{12} - \)\(10\!\cdots\!28\)\( p^{105} T^{13} + \)\(15\!\cdots\!68\)\( p^{126} T^{14} - \)\(45\!\cdots\!64\)\( p^{147} T^{15} + p^{168} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.76279578943287492064145729767, −3.39632857401460406604528075568, −3.20736019544651136488502175940, −3.15904255633060398457059751210, −3.15776416608912715301114158597, −3.08297701714679269683032751140, −2.70333638464915024684553673196, −2.55608803886146437239972207108, −2.32832395440030470039805838238, −2.15883525952747318924400556781, −2.12911228690545830844956581896, −1.88428314701229523235116282577, −1.49525516333199497988170751836, −1.44691291187077919619540540673, −1.37275454631632053781901796651, −1.31479010177546473980100264157, −1.23763431782469498862300454648, −1.05489555069702029919504359314, −0.78850668214242117767840919149, −0.73854942913207828854717027817, −0.60786254441458650013198837257, −0.42800879008688174743616721589, −0.27028682365900038080695486010, −0.24274216492944101092101369077, −0.17631403898533137505773880277, 0.17631403898533137505773880277, 0.24274216492944101092101369077, 0.27028682365900038080695486010, 0.42800879008688174743616721589, 0.60786254441458650013198837257, 0.73854942913207828854717027817, 0.78850668214242117767840919149, 1.05489555069702029919504359314, 1.23763431782469498862300454648, 1.31479010177546473980100264157, 1.37275454631632053781901796651, 1.44691291187077919619540540673, 1.49525516333199497988170751836, 1.88428314701229523235116282577, 2.12911228690545830844956581896, 2.15883525952747318924400556781, 2.32832395440030470039805838238, 2.55608803886146437239972207108, 2.70333638464915024684553673196, 3.08297701714679269683032751140, 3.15776416608912715301114158597, 3.15904255633060398457059751210, 3.20736019544651136488502175940, 3.39632857401460406604528075568, 3.76279578943287492064145729767
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.