Dirichlet series
| L(s) = 1 | + 6.17e5·4-s − 2.92e5·5-s − 9.07e8·11-s + 1.78e11·16-s + 4.43e10·19-s − 1.80e11·20-s + 8.58e11·25-s + 5.34e12·29-s − 1.69e13·31-s + 4.87e12·41-s − 5.60e14·44-s + 1.90e15·49-s + 2.65e14·55-s − 2.88e15·59-s − 6.34e15·61-s + 3.35e16·64-s + 1.17e16·71-s + 2.73e16·76-s + 2.86e16·79-s − 5.22e16·80-s + 1.19e17·89-s − 1.29e16·95-s + 5.29e17·100-s − 4.19e17·101-s − 6.56e17·109-s + 3.30e18·116-s − 4.98e18·121-s + ⋯ |
| L(s) = 1 | + 4.70·4-s − 0.335·5-s − 1.27·11-s + 10.3·16-s + 0.599·19-s − 1.57·20-s + 1.12·25-s + 1.98·29-s − 3.56·31-s + 0.0953·41-s − 6.01·44-s + 8.18·49-s + 0.427·55-s − 2.55·59-s − 4.23·61-s + 14.8·64-s + 2.15·71-s + 2.82·76-s + 2.12·79-s − 3.47·80-s + 3.20·89-s − 0.200·95-s + 5.29·100-s − 3.85·101-s − 3.15·109-s + 9.34·116-s − 9.85·121-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(18-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+17/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{32} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.56084\times 10^{30}\) |
| Root analytic conductor: | \(9.08019\) |
| Motivic weight: | \(17\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{32} \cdot 5^{16} ,\ ( \ : [17/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(9)\) | \(\approx\) | \(3.961651664\) |
| \(L(\frac12)\) | \(\approx\) | \(3.961651664\) |
| \(L(\frac{19}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 58548 p T - 6182830996 p^{3} T^{2} - 5048706044676 p^{7} T^{3} + 86166204677924032 p^{9} T^{4} - 22884924495809073152 p^{13} T^{5} - \)\(30\!\cdots\!48\)\( p^{19} T^{6} + \)\(81\!\cdots\!84\)\( p^{26} T^{7} + \)\(12\!\cdots\!16\)\( p^{34} T^{8} + \)\(81\!\cdots\!84\)\( p^{43} T^{9} - \)\(30\!\cdots\!48\)\( p^{53} T^{10} - 22884924495809073152 p^{64} T^{11} + 86166204677924032 p^{77} T^{12} - 5048706044676 p^{92} T^{13} - 6182830996 p^{105} T^{14} + 58548 p^{120} T^{15} + p^{136} T^{16} \) | |
| good | 2 | \( 1 - 617227 T^{2} + 50662703229 p^{2} T^{4} - 758285590213331 p^{6} T^{6} + 581638750433410535 p^{14} T^{8} - \)\(25\!\cdots\!27\)\( p^{16} T^{10} + \)\(15\!\cdots\!15\)\( p^{24} T^{12} - \)\(93\!\cdots\!05\)\( p^{32} T^{14} + \)\(49\!\cdots\!21\)\( p^{40} T^{16} - \)\(93\!\cdots\!05\)\( p^{66} T^{18} + \)\(15\!\cdots\!15\)\( p^{92} T^{20} - \)\(25\!\cdots\!27\)\( p^{118} T^{22} + 581638750433410535 p^{150} T^{24} - 758285590213331 p^{176} T^{26} + 50662703229 p^{206} T^{28} - 617227 p^{238} T^{30} + p^{272} T^{32} \) |
| 7 | \( 1 - 1903831896704536 T^{2} + \)\(18\!\cdots\!48\)\( T^{4} - \)\(37\!\cdots\!28\)\( p^{3} T^{6} + \)\(27\!\cdots\!88\)\( p^{4} T^{8} - \)\(33\!\cdots\!92\)\( p^{7} T^{10} + \)\(16\!\cdots\!00\)\( p^{8} T^{12} - \)\(19\!\cdots\!60\)\( p^{12} T^{14} + \)\(20\!\cdots\!26\)\( p^{16} T^{16} - \)\(19\!\cdots\!60\)\( p^{46} T^{18} + \)\(16\!\cdots\!00\)\( p^{76} T^{20} - \)\(33\!\cdots\!92\)\( p^{109} T^{22} + \)\(27\!\cdots\!88\)\( p^{140} T^{24} - \)\(37\!\cdots\!28\)\( p^{173} T^{26} + \)\(18\!\cdots\!48\)\( p^{204} T^{28} - 1903831896704536 p^{238} T^{30} + p^{272} T^{32} \) | |
| 11 | \( ( 1 + 453693072 T + 2799329255515249420 T^{2} + \)\(14\!\cdots\!44\)\( T^{3} + \)\(37\!\cdots\!76\)\( T^{4} + \)\(18\!\cdots\!92\)\( p T^{5} + \)\(26\!\cdots\!28\)\( p^{2} T^{6} + \)\(12\!\cdots\!76\)\( p^{3} T^{7} + \)\(13\!\cdots\!38\)\( p^{4} T^{8} + \)\(12\!\cdots\!76\)\( p^{20} T^{9} + \)\(26\!\cdots\!28\)\( p^{36} T^{10} + \)\(18\!\cdots\!92\)\( p^{52} T^{11} + \)\(37\!\cdots\!76\)\( p^{68} T^{12} + \)\(14\!\cdots\!44\)\( p^{85} T^{13} + 2799329255515249420 p^{102} T^{14} + 453693072 p^{119} T^{15} + p^{136} T^{16} )^{2} \) | |
| 13 | \( 1 - 58107256345704874648 T^{2} + \)\(17\!\cdots\!56\)\( T^{4} - \)\(17\!\cdots\!68\)\( p^{3} T^{6} + \)\(22\!\cdots\!80\)\( p^{4} T^{8} - \)\(18\!\cdots\!72\)\( p^{6} T^{10} + \)\(13\!\cdots\!00\)\( p^{8} T^{12} - \)\(81\!\cdots\!80\)\( p^{10} T^{14} + \)\(44\!\cdots\!06\)\( p^{12} T^{16} - \)\(81\!\cdots\!80\)\( p^{44} T^{18} + \)\(13\!\cdots\!00\)\( p^{76} T^{20} - \)\(18\!\cdots\!72\)\( p^{108} T^{22} + \)\(22\!\cdots\!80\)\( p^{140} T^{24} - \)\(17\!\cdots\!68\)\( p^{173} T^{26} + \)\(17\!\cdots\!56\)\( p^{204} T^{28} - 58107256345704874648 p^{238} T^{30} + p^{272} T^{32} \) | |
| 17 | \( 1 - \)\(82\!\cdots\!00\)\( T^{2} + \)\(34\!\cdots\!32\)\( T^{4} - \)\(93\!\cdots\!00\)\( T^{6} + \)\(18\!\cdots\!48\)\( T^{8} - \)\(30\!\cdots\!00\)\( T^{10} + \)\(39\!\cdots\!84\)\( T^{12} - \)\(42\!\cdots\!00\)\( T^{14} + \)\(37\!\cdots\!70\)\( T^{16} - \)\(42\!\cdots\!00\)\( p^{34} T^{18} + \)\(39\!\cdots\!84\)\( p^{68} T^{20} - \)\(30\!\cdots\!00\)\( p^{102} T^{22} + \)\(18\!\cdots\!48\)\( p^{136} T^{24} - \)\(93\!\cdots\!00\)\( p^{170} T^{26} + \)\(34\!\cdots\!32\)\( p^{204} T^{28} - \)\(82\!\cdots\!00\)\( p^{238} T^{30} + p^{272} T^{32} \) | |
| 19 | \( ( 1 - 1167786256 p T + \)\(15\!\cdots\!68\)\( T^{2} - \)\(81\!\cdots\!16\)\( T^{3} + \)\(13\!\cdots\!88\)\( T^{4} - \)\(74\!\cdots\!24\)\( T^{5} + \)\(87\!\cdots\!80\)\( T^{6} - \)\(53\!\cdots\!40\)\( T^{7} + \)\(43\!\cdots\!46\)\( T^{8} - \)\(53\!\cdots\!40\)\( p^{17} T^{9} + \)\(87\!\cdots\!80\)\( p^{34} T^{10} - \)\(74\!\cdots\!24\)\( p^{51} T^{11} + \)\(13\!\cdots\!88\)\( p^{68} T^{12} - \)\(81\!\cdots\!16\)\( p^{85} T^{13} + \)\(15\!\cdots\!68\)\( p^{102} T^{14} - 1167786256 p^{120} T^{15} + p^{136} T^{16} )^{2} \) | |
| 23 | \( 1 - \)\(15\!\cdots\!44\)\( T^{2} + \)\(12\!\cdots\!68\)\( T^{4} - \)\(60\!\cdots\!56\)\( T^{6} + \)\(21\!\cdots\!28\)\( T^{8} - \)\(60\!\cdots\!44\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{12} - \)\(25\!\cdots\!40\)\( T^{14} + \)\(38\!\cdots\!86\)\( T^{16} - \)\(25\!\cdots\!40\)\( p^{34} T^{18} + \)\(13\!\cdots\!00\)\( p^{68} T^{20} - \)\(60\!\cdots\!44\)\( p^{102} T^{22} + \)\(21\!\cdots\!28\)\( p^{136} T^{24} - \)\(60\!\cdots\!56\)\( p^{170} T^{26} + \)\(12\!\cdots\!68\)\( p^{204} T^{28} - \)\(15\!\cdots\!44\)\( p^{238} T^{30} + p^{272} T^{32} \) | |
| 29 | \( ( 1 - 2674234302252 T + \)\(21\!\cdots\!36\)\( T^{2} - \)\(50\!\cdots\!64\)\( T^{3} + \)\(33\!\cdots\!60\)\( T^{4} - \)\(59\!\cdots\!12\)\( T^{5} + \)\(31\!\cdots\!80\)\( T^{6} - \)\(49\!\cdots\!60\)\( T^{7} + \)\(25\!\cdots\!86\)\( T^{8} - \)\(49\!\cdots\!60\)\( p^{17} T^{9} + \)\(31\!\cdots\!80\)\( p^{34} T^{10} - \)\(59\!\cdots\!12\)\( p^{51} T^{11} + \)\(33\!\cdots\!60\)\( p^{68} T^{12} - \)\(50\!\cdots\!64\)\( p^{85} T^{13} + \)\(21\!\cdots\!36\)\( p^{102} T^{14} - 2674234302252 p^{119} T^{15} + p^{136} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 8467982026112 T + \)\(10\!\cdots\!92\)\( T^{2} + \)\(70\!\cdots\!32\)\( T^{3} + \)\(56\!\cdots\!08\)\( T^{4} + \)\(32\!\cdots\!32\)\( T^{5} + \)\(20\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} + \)\(55\!\cdots\!06\)\( T^{8} + \)\(10\!\cdots\!00\)\( p^{17} T^{9} + \)\(20\!\cdots\!60\)\( p^{34} T^{10} + \)\(32\!\cdots\!32\)\( p^{51} T^{11} + \)\(56\!\cdots\!08\)\( p^{68} T^{12} + \)\(70\!\cdots\!32\)\( p^{85} T^{13} + \)\(10\!\cdots\!92\)\( p^{102} T^{14} + 8467982026112 p^{119} T^{15} + p^{136} T^{16} )^{2} \) | |
| 37 | \( 1 - \)\(17\!\cdots\!96\)\( T^{2} + \)\(14\!\cdots\!32\)\( p^{2} T^{4} - \)\(18\!\cdots\!84\)\( T^{6} + \)\(14\!\cdots\!48\)\( T^{8} - \)\(98\!\cdots\!16\)\( T^{10} + \)\(58\!\cdots\!80\)\( T^{12} - \)\(31\!\cdots\!20\)\( T^{14} + \)\(15\!\cdots\!46\)\( T^{16} - \)\(31\!\cdots\!20\)\( p^{34} T^{18} + \)\(58\!\cdots\!80\)\( p^{68} T^{20} - \)\(98\!\cdots\!16\)\( p^{102} T^{22} + \)\(14\!\cdots\!48\)\( p^{136} T^{24} - \)\(18\!\cdots\!84\)\( p^{170} T^{26} + \)\(14\!\cdots\!32\)\( p^{206} T^{28} - \)\(17\!\cdots\!96\)\( p^{238} T^{30} + p^{272} T^{32} \) | |
| 41 | \( ( 1 - 2436672389592 T + \)\(14\!\cdots\!32\)\( T^{2} + \)\(19\!\cdots\!28\)\( T^{3} + \)\(10\!\cdots\!68\)\( T^{4} + \)\(34\!\cdots\!88\)\( T^{5} + \)\(44\!\cdots\!20\)\( T^{6} + \)\(17\!\cdots\!40\)\( T^{7} + \)\(13\!\cdots\!26\)\( T^{8} + \)\(17\!\cdots\!40\)\( p^{17} T^{9} + \)\(44\!\cdots\!20\)\( p^{34} T^{10} + \)\(34\!\cdots\!88\)\( p^{51} T^{11} + \)\(10\!\cdots\!68\)\( p^{68} T^{12} + \)\(19\!\cdots\!28\)\( p^{85} T^{13} + \)\(14\!\cdots\!32\)\( p^{102} T^{14} - 2436672389592 p^{119} T^{15} + p^{136} T^{16} )^{2} \) | |
| 43 | \( 1 - \)\(45\!\cdots\!00\)\( T^{2} + \)\(98\!\cdots\!92\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!28\)\( T^{8} - \)\(10\!\cdots\!00\)\( T^{10} + \)\(70\!\cdots\!44\)\( T^{12} - \)\(43\!\cdots\!00\)\( T^{14} + \)\(25\!\cdots\!70\)\( T^{16} - \)\(43\!\cdots\!00\)\( p^{34} T^{18} + \)\(70\!\cdots\!44\)\( p^{68} T^{20} - \)\(10\!\cdots\!00\)\( p^{102} T^{22} + \)\(13\!\cdots\!28\)\( p^{136} T^{24} - \)\(13\!\cdots\!00\)\( p^{170} T^{26} + \)\(98\!\cdots\!92\)\( p^{204} T^{28} - \)\(45\!\cdots\!00\)\( p^{238} T^{30} + p^{272} T^{32} \) | |
| 47 | \( 1 - \)\(19\!\cdots\!80\)\( T^{2} + \)\(18\!\cdots\!52\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{6} + \)\(71\!\cdots\!08\)\( T^{8} - \)\(31\!\cdots\!80\)\( T^{10} + \)\(11\!\cdots\!04\)\( T^{12} - \)\(38\!\cdots\!00\)\( T^{14} + \)\(11\!\cdots\!70\)\( T^{16} - \)\(38\!\cdots\!00\)\( p^{34} T^{18} + \)\(11\!\cdots\!04\)\( p^{68} T^{20} - \)\(31\!\cdots\!80\)\( p^{102} T^{22} + \)\(71\!\cdots\!08\)\( p^{136} T^{24} - \)\(13\!\cdots\!40\)\( p^{170} T^{26} + \)\(18\!\cdots\!52\)\( p^{204} T^{28} - \)\(19\!\cdots\!80\)\( p^{238} T^{30} + p^{272} T^{32} \) | |
| 53 | \( 1 - \)\(98\!\cdots\!64\)\( T^{2} + \)\(63\!\cdots\!48\)\( T^{4} - \)\(29\!\cdots\!76\)\( T^{6} + \)\(11\!\cdots\!68\)\( T^{8} - \)\(36\!\cdots\!24\)\( T^{10} + \)\(10\!\cdots\!60\)\( T^{12} - \)\(25\!\cdots\!20\)\( T^{14} + \)\(55\!\cdots\!06\)\( T^{16} - \)\(25\!\cdots\!20\)\( p^{34} T^{18} + \)\(10\!\cdots\!60\)\( p^{68} T^{20} - \)\(36\!\cdots\!24\)\( p^{102} T^{22} + \)\(11\!\cdots\!68\)\( p^{136} T^{24} - \)\(29\!\cdots\!76\)\( p^{170} T^{26} + \)\(63\!\cdots\!48\)\( p^{204} T^{28} - \)\(98\!\cdots\!64\)\( p^{238} T^{30} + p^{272} T^{32} \) | |
| 59 | \( ( 1 + 1441181478551136 T + \)\(53\!\cdots\!08\)\( T^{2} + \)\(42\!\cdots\!44\)\( T^{3} + \)\(10\!\cdots\!88\)\( T^{4} + \)\(12\!\cdots\!16\)\( T^{5} + \)\(73\!\cdots\!40\)\( T^{6} - \)\(10\!\cdots\!40\)\( T^{7} + \)\(24\!\cdots\!86\)\( T^{8} - \)\(10\!\cdots\!40\)\( p^{17} T^{9} + \)\(73\!\cdots\!40\)\( p^{34} T^{10} + \)\(12\!\cdots\!16\)\( p^{51} T^{11} + \)\(10\!\cdots\!88\)\( p^{68} T^{12} + \)\(42\!\cdots\!44\)\( p^{85} T^{13} + \)\(53\!\cdots\!08\)\( p^{102} T^{14} + 1441181478551136 p^{119} T^{15} + p^{136} T^{16} )^{2} \) | |
| 61 | \( ( 1 + 3170473554101840 T + \)\(10\!\cdots\!72\)\( T^{2} + \)\(24\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!08\)\( T^{4} + \)\(90\!\cdots\!60\)\( T^{5} + \)\(15\!\cdots\!84\)\( T^{6} + \)\(23\!\cdots\!00\)\( T^{7} + \)\(35\!\cdots\!70\)\( T^{8} + \)\(23\!\cdots\!00\)\( p^{17} T^{9} + \)\(15\!\cdots\!84\)\( p^{34} T^{10} + \)\(90\!\cdots\!60\)\( p^{51} T^{11} + \)\(50\!\cdots\!08\)\( p^{68} T^{12} + \)\(24\!\cdots\!00\)\( p^{85} T^{13} + \)\(10\!\cdots\!72\)\( p^{102} T^{14} + 3170473554101840 p^{119} T^{15} + p^{136} T^{16} )^{2} \) | |
| 67 | \( 1 - \)\(10\!\cdots\!40\)\( T^{2} + \)\(51\!\cdots\!32\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(47\!\cdots\!48\)\( T^{8} - \)\(97\!\cdots\!40\)\( T^{10} + \)\(16\!\cdots\!84\)\( T^{12} - \)\(23\!\cdots\!00\)\( T^{14} + \)\(28\!\cdots\!70\)\( T^{16} - \)\(23\!\cdots\!00\)\( p^{34} T^{18} + \)\(16\!\cdots\!84\)\( p^{68} T^{20} - \)\(97\!\cdots\!40\)\( p^{102} T^{22} + \)\(47\!\cdots\!48\)\( p^{136} T^{24} - \)\(17\!\cdots\!20\)\( p^{170} T^{26} + \)\(51\!\cdots\!32\)\( p^{204} T^{28} - \)\(10\!\cdots\!40\)\( p^{238} T^{30} + p^{272} T^{32} \) | |
| 71 | \( ( 1 - 5869642874949504 T + \)\(21\!\cdots\!60\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!60\)\( T^{4} - \)\(83\!\cdots\!72\)\( T^{5} + \)\(16\!\cdots\!28\)\( p T^{6} - \)\(38\!\cdots\!80\)\( T^{7} + \)\(42\!\cdots\!90\)\( T^{8} - \)\(38\!\cdots\!80\)\( p^{17} T^{9} + \)\(16\!\cdots\!28\)\( p^{35} T^{10} - \)\(83\!\cdots\!72\)\( p^{51} T^{11} + \)\(20\!\cdots\!60\)\( p^{68} T^{12} - \)\(10\!\cdots\!80\)\( p^{85} T^{13} + \)\(21\!\cdots\!60\)\( p^{102} T^{14} - 5869642874949504 p^{119} T^{15} + p^{136} T^{16} )^{2} \) | |
| 73 | \( 1 - \)\(34\!\cdots\!84\)\( T^{2} + \)\(53\!\cdots\!28\)\( T^{4} - \)\(50\!\cdots\!16\)\( T^{6} + \)\(31\!\cdots\!28\)\( T^{8} - \)\(14\!\cdots\!24\)\( T^{10} + \)\(63\!\cdots\!60\)\( T^{12} - \)\(30\!\cdots\!20\)\( T^{14} + \)\(15\!\cdots\!46\)\( T^{16} - \)\(30\!\cdots\!20\)\( p^{34} T^{18} + \)\(63\!\cdots\!60\)\( p^{68} T^{20} - \)\(14\!\cdots\!24\)\( p^{102} T^{22} + \)\(31\!\cdots\!28\)\( p^{136} T^{24} - \)\(50\!\cdots\!16\)\( p^{170} T^{26} + \)\(53\!\cdots\!28\)\( p^{204} T^{28} - \)\(34\!\cdots\!84\)\( p^{238} T^{30} + p^{272} T^{32} \) | |
| 79 | \( ( 1 - 14327637370181920 T + \)\(54\!\cdots\!72\)\( T^{2} - \)\(35\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!68\)\( T^{4} + \)\(34\!\cdots\!80\)\( T^{5} + \)\(62\!\cdots\!24\)\( T^{6} + \)\(33\!\cdots\!00\)\( T^{7} - \)\(77\!\cdots\!30\)\( T^{8} + \)\(33\!\cdots\!00\)\( p^{17} T^{9} + \)\(62\!\cdots\!24\)\( p^{34} T^{10} + \)\(34\!\cdots\!80\)\( p^{51} T^{11} + \)\(11\!\cdots\!68\)\( p^{68} T^{12} - \)\(35\!\cdots\!60\)\( p^{85} T^{13} + \)\(54\!\cdots\!72\)\( p^{102} T^{14} - 14327637370181920 p^{119} T^{15} + p^{136} T^{16} )^{2} \) | |
| 83 | \( 1 - \)\(43\!\cdots\!28\)\( T^{2} + \)\(90\!\cdots\!96\)\( T^{4} - \)\(12\!\cdots\!76\)\( T^{6} + \)\(11\!\cdots\!00\)\( T^{8} - \)\(90\!\cdots\!08\)\( T^{10} + \)\(55\!\cdots\!00\)\( T^{12} - \)\(28\!\cdots\!60\)\( T^{14} + \)\(13\!\cdots\!26\)\( T^{16} - \)\(28\!\cdots\!60\)\( p^{34} T^{18} + \)\(55\!\cdots\!00\)\( p^{68} T^{20} - \)\(90\!\cdots\!08\)\( p^{102} T^{22} + \)\(11\!\cdots\!00\)\( p^{136} T^{24} - \)\(12\!\cdots\!76\)\( p^{170} T^{26} + \)\(90\!\cdots\!96\)\( p^{204} T^{28} - \)\(43\!\cdots\!28\)\( p^{238} T^{30} + p^{272} T^{32} \) | |
| 89 | \( ( 1 - 59540116605688656 T + \)\(32\!\cdots\!88\)\( T^{2} - \)\(19\!\cdots\!04\)\( T^{3} + \)\(11\!\cdots\!48\)\( T^{4} - \)\(50\!\cdots\!56\)\( T^{5} + \)\(22\!\cdots\!20\)\( T^{6} - \)\(92\!\cdots\!40\)\( T^{7} + \)\(37\!\cdots\!26\)\( T^{8} - \)\(92\!\cdots\!40\)\( p^{17} T^{9} + \)\(22\!\cdots\!20\)\( p^{34} T^{10} - \)\(50\!\cdots\!56\)\( p^{51} T^{11} + \)\(11\!\cdots\!48\)\( p^{68} T^{12} - \)\(19\!\cdots\!04\)\( p^{85} T^{13} + \)\(32\!\cdots\!88\)\( p^{102} T^{14} - 59540116605688656 p^{119} T^{15} + p^{136} T^{16} )^{2} \) | |
| 97 | \( 1 - \)\(27\!\cdots\!80\)\( T^{2} + \)\(45\!\cdots\!52\)\( T^{4} - \)\(50\!\cdots\!40\)\( T^{6} + \)\(42\!\cdots\!08\)\( T^{8} - \)\(26\!\cdots\!80\)\( T^{10} + \)\(13\!\cdots\!04\)\( T^{12} - \)\(53\!\cdots\!00\)\( T^{14} + \)\(24\!\cdots\!70\)\( T^{16} - \)\(53\!\cdots\!00\)\( p^{34} T^{18} + \)\(13\!\cdots\!04\)\( p^{68} T^{20} - \)\(26\!\cdots\!80\)\( p^{102} T^{22} + \)\(42\!\cdots\!08\)\( p^{136} T^{24} - \)\(50\!\cdots\!40\)\( p^{170} T^{26} + \)\(45\!\cdots\!52\)\( p^{204} T^{28} - \)\(27\!\cdots\!80\)\( p^{238} T^{30} + p^{272} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.31713501224025495200919330503, −2.15164927762369689620245621469, −1.88840881670398595543940744089, −1.80981045919609643578421683579, −1.79235882045364164457136244992, −1.74341864688830742073479506868, −1.64975789094416944733527323851, −1.63125279710149402400921777682, −1.53726740139738184119878993512, −1.51097163463291151690053554946, −1.49446766680274916921638637343, −1.29926200435922551305197527087, −1.11064596872210881871020726041, −1.09940568009520692636908660389, −0.946996136612534130035825708060, −0.74529541717746556617313864529, −0.67809330158203121344736537028, −0.66953942053083577167983598154, −0.64646148034065240078699373961, −0.63296619126749243515277128602, −0.58139106561039919871457589023, −0.44557504365249700474556576151, −0.10441754686176901230238652689, −0.086255224065109531491829343290, −0.06221153331782688138429344230, 0.06221153331782688138429344230, 0.086255224065109531491829343290, 0.10441754686176901230238652689, 0.44557504365249700474556576151, 0.58139106561039919871457589023, 0.63296619126749243515277128602, 0.64646148034065240078699373961, 0.66953942053083577167983598154, 0.67809330158203121344736537028, 0.74529541717746556617313864529, 0.946996136612534130035825708060, 1.09940568009520692636908660389, 1.11064596872210881871020726041, 1.29926200435922551305197527087, 1.49446766680274916921638637343, 1.51097163463291151690053554946, 1.53726740139738184119878993512, 1.63125279710149402400921777682, 1.64975789094416944733527323851, 1.74341864688830742073479506868, 1.79235882045364164457136244992, 1.80981045919609643578421683579, 1.88840881670398595543940744089, 2.15164927762369689620245621469, 2.31713501224025495200919330503
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.