Dirichlet series
| L(s) = 1 | − 186·2-s + 8.55e4·4-s + 3.54e5·5-s − 8.88e6·8-s − 1.14e8·9-s − 6.58e7·10-s − 9.06e8·13-s + 2.12e9·16-s + 1.22e10·17-s + 2.13e10·18-s + 3.03e10·20-s − 1.05e12·25-s + 1.68e11·26-s + 3.27e11·29-s − 1.02e12·32-s − 2.27e12·34-s − 9.82e12·36-s − 8.14e12·37-s − 3.14e12·40-s − 2.55e13·41-s − 4.06e13·45-s + 2.19e14·49-s + 1.96e14·50-s − 7.75e13·52-s − 8.69e13·53-s − 6.09e13·58-s + 4.76e14·61-s + ⋯ |
| L(s) = 1 | − 0.726·2-s + 1.30·4-s + 0.906·5-s − 0.529·8-s − 8/3·9-s − 0.658·10-s − 1.11·13-s + 0.494·16-s + 1.75·17-s + 1.93·18-s + 1.18·20-s − 6.91·25-s + 0.807·26-s + 0.655·29-s − 0.930·32-s − 1.27·34-s − 3.48·36-s − 2.32·37-s − 0.479·40-s − 3.19·41-s − 2.41·45-s + 6.59·49-s + 5.02·50-s − 1.45·52-s − 1.39·53-s − 0.475·58-s + 2.48·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(17-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+8)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.29582\times 10^{20}\) |
| Root analytic conductor: | \(4.41349\) |
| Motivic weight: | \(16\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{16} ,\ ( \ : [8]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{17}{2})\) | \(\approx\) | \(0.5074757006\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5074757006\) |
| \(L(9)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + 93 p T - 12749 p^{2} T^{2} - 516399 p^{5} T^{3} + 3192589 p^{8} T^{4} + 212617953 p^{13} T^{5} + 636438355 p^{18} T^{6} - 2097643623 p^{25} T^{7} - 5081432363 p^{32} T^{8} - 2097643623 p^{41} T^{9} + 636438355 p^{50} T^{10} + 212617953 p^{61} T^{11} + 3192589 p^{72} T^{12} - 516399 p^{85} T^{13} - 12749 p^{98} T^{14} + 93 p^{113} T^{15} + p^{128} T^{16} \) |
| 3 | \( ( 1 + p^{15} T^{2} )^{8} \) | |
| good | 5 | \( ( 1 - 177072 T + 574433513464 T^{2} - 6872893181936016 p^{2} T^{3} + \)\(72\!\cdots\!84\)\( p^{2} T^{4} - \)\(20\!\cdots\!08\)\( p^{5} T^{5} + \)\(13\!\cdots\!96\)\( p^{5} T^{6} - \)\(35\!\cdots\!68\)\( p^{8} T^{7} + \)\(19\!\cdots\!46\)\( p^{8} T^{8} - \)\(35\!\cdots\!68\)\( p^{24} T^{9} + \)\(13\!\cdots\!96\)\( p^{37} T^{10} - \)\(20\!\cdots\!08\)\( p^{53} T^{11} + \)\(72\!\cdots\!84\)\( p^{66} T^{12} - 6872893181936016 p^{82} T^{13} + 574433513464 p^{96} T^{14} - 177072 p^{112} T^{15} + p^{128} T^{16} )^{2} \) |
| 7 | \( 1 - 219219938074768 T^{2} + \)\(24\!\cdots\!12\)\( T^{4} - \)\(37\!\cdots\!96\)\( p^{2} T^{6} + \)\(43\!\cdots\!80\)\( p^{4} T^{8} - \)\(40\!\cdots\!80\)\( p^{6} T^{10} + \)\(65\!\cdots\!64\)\( p^{10} T^{12} - \)\(95\!\cdots\!64\)\( p^{14} T^{14} + \)\(13\!\cdots\!66\)\( p^{18} T^{16} - \)\(95\!\cdots\!64\)\( p^{46} T^{18} + \)\(65\!\cdots\!64\)\( p^{74} T^{20} - \)\(40\!\cdots\!80\)\( p^{102} T^{22} + \)\(43\!\cdots\!80\)\( p^{132} T^{24} - \)\(37\!\cdots\!96\)\( p^{162} T^{26} + \)\(24\!\cdots\!12\)\( p^{192} T^{28} - 219219938074768 p^{224} T^{30} + p^{256} T^{32} \) | |
| 11 | \( 1 - 195579060629182096 T^{2} + \)\(24\!\cdots\!48\)\( T^{4} - \)\(22\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!68\)\( p^{2} T^{8} - \)\(69\!\cdots\!64\)\( p^{4} T^{10} + \)\(31\!\cdots\!64\)\( p^{6} T^{12} - \)\(11\!\cdots\!24\)\( p^{9} T^{14} + \)\(49\!\cdots\!10\)\( p^{10} T^{16} - \)\(11\!\cdots\!24\)\( p^{41} T^{18} + \)\(31\!\cdots\!64\)\( p^{70} T^{20} - \)\(69\!\cdots\!64\)\( p^{100} T^{22} + \)\(13\!\cdots\!68\)\( p^{130} T^{24} - \)\(22\!\cdots\!20\)\( p^{160} T^{26} + \)\(24\!\cdots\!48\)\( p^{192} T^{28} - 195579060629182096 p^{224} T^{30} + p^{256} T^{32} \) | |
| 13 | \( ( 1 + 453209648 T + 2986640942699007480 T^{2} + \)\(34\!\cdots\!80\)\( p T^{3} + \)\(23\!\cdots\!36\)\( p^{2} T^{4} - \)\(22\!\cdots\!28\)\( p^{3} T^{5} + \)\(11\!\cdots\!84\)\( p^{4} T^{6} - \)\(29\!\cdots\!20\)\( p^{5} T^{7} + \)\(49\!\cdots\!86\)\( p^{6} T^{8} - \)\(29\!\cdots\!20\)\( p^{21} T^{9} + \)\(11\!\cdots\!84\)\( p^{36} T^{10} - \)\(22\!\cdots\!28\)\( p^{51} T^{11} + \)\(23\!\cdots\!36\)\( p^{66} T^{12} + \)\(34\!\cdots\!80\)\( p^{81} T^{13} + 2986640942699007480 p^{96} T^{14} + 453209648 p^{112} T^{15} + p^{128} T^{16} )^{2} \) | |
| 17 | \( ( 1 - 6120382800 T + \)\(24\!\cdots\!04\)\( T^{2} - \)\(17\!\cdots\!80\)\( T^{3} + \)\(29\!\cdots\!56\)\( T^{4} - \)\(21\!\cdots\!40\)\( T^{5} + \)\(23\!\cdots\!52\)\( T^{6} - \)\(95\!\cdots\!40\)\( p T^{7} + \)\(13\!\cdots\!78\)\( T^{8} - \)\(95\!\cdots\!40\)\( p^{17} T^{9} + \)\(23\!\cdots\!52\)\( p^{32} T^{10} - \)\(21\!\cdots\!40\)\( p^{48} T^{11} + \)\(29\!\cdots\!56\)\( p^{64} T^{12} - \)\(17\!\cdots\!80\)\( p^{80} T^{13} + \)\(24\!\cdots\!04\)\( p^{96} T^{14} - 6120382800 p^{112} T^{15} + p^{128} T^{16} )^{2} \) | |
| 19 | \( 1 - \)\(26\!\cdots\!08\)\( T^{2} + \)\(36\!\cdots\!32\)\( T^{4} - \)\(33\!\cdots\!04\)\( T^{6} + \)\(23\!\cdots\!60\)\( T^{8} - \)\(35\!\cdots\!60\)\( p^{2} T^{10} + \)\(57\!\cdots\!36\)\( T^{12} - \)\(21\!\cdots\!56\)\( T^{14} + \)\(67\!\cdots\!74\)\( T^{16} - \)\(21\!\cdots\!56\)\( p^{32} T^{18} + \)\(57\!\cdots\!36\)\( p^{64} T^{20} - \)\(35\!\cdots\!60\)\( p^{98} T^{22} + \)\(23\!\cdots\!60\)\( p^{128} T^{24} - \)\(33\!\cdots\!04\)\( p^{160} T^{26} + \)\(36\!\cdots\!32\)\( p^{192} T^{28} - \)\(26\!\cdots\!08\)\( p^{224} T^{30} + p^{256} T^{32} \) | |
| 23 | \( 1 - \)\(25\!\cdots\!36\)\( T^{2} + \)\(37\!\cdots\!12\)\( T^{4} - \)\(45\!\cdots\!08\)\( T^{6} + \)\(44\!\cdots\!28\)\( T^{8} - \)\(38\!\cdots\!72\)\( T^{10} + \)\(30\!\cdots\!04\)\( T^{12} - \)\(21\!\cdots\!68\)\( T^{14} + \)\(13\!\cdots\!74\)\( T^{16} - \)\(21\!\cdots\!68\)\( p^{32} T^{18} + \)\(30\!\cdots\!04\)\( p^{64} T^{20} - \)\(38\!\cdots\!72\)\( p^{96} T^{22} + \)\(44\!\cdots\!28\)\( p^{128} T^{24} - \)\(45\!\cdots\!08\)\( p^{160} T^{26} + \)\(37\!\cdots\!12\)\( p^{192} T^{28} - \)\(25\!\cdots\!36\)\( p^{224} T^{30} + p^{256} T^{32} \) | |
| 29 | \( ( 1 - 163839786864 T + \)\(88\!\cdots\!84\)\( T^{2} - \)\(26\!\cdots\!80\)\( T^{3} + \)\(45\!\cdots\!32\)\( T^{4} - \)\(16\!\cdots\!36\)\( T^{5} + \)\(16\!\cdots\!64\)\( T^{6} - \)\(64\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!06\)\( T^{8} - \)\(64\!\cdots\!00\)\( p^{16} T^{9} + \)\(16\!\cdots\!64\)\( p^{32} T^{10} - \)\(16\!\cdots\!36\)\( p^{48} T^{11} + \)\(45\!\cdots\!32\)\( p^{64} T^{12} - \)\(26\!\cdots\!80\)\( p^{80} T^{13} + \)\(88\!\cdots\!84\)\( p^{96} T^{14} - 163839786864 p^{112} T^{15} + p^{128} T^{16} )^{2} \) | |
| 31 | \( 1 - \)\(22\!\cdots\!48\)\( p T^{2} + \)\(25\!\cdots\!00\)\( T^{4} - \)\(61\!\cdots\!64\)\( T^{6} + \)\(11\!\cdots\!28\)\( T^{8} - \)\(15\!\cdots\!04\)\( T^{10} + \)\(18\!\cdots\!88\)\( T^{12} - \)\(17\!\cdots\!72\)\( T^{14} + \)\(13\!\cdots\!86\)\( T^{16} - \)\(17\!\cdots\!72\)\( p^{32} T^{18} + \)\(18\!\cdots\!88\)\( p^{64} T^{20} - \)\(15\!\cdots\!04\)\( p^{96} T^{22} + \)\(11\!\cdots\!28\)\( p^{128} T^{24} - \)\(61\!\cdots\!64\)\( p^{160} T^{26} + \)\(25\!\cdots\!00\)\( p^{192} T^{28} - \)\(22\!\cdots\!48\)\( p^{225} T^{30} + p^{256} T^{32} \) | |
| 37 | \( ( 1 + 4074747374576 T + \)\(44\!\cdots\!16\)\( T^{2} + \)\(14\!\cdots\!44\)\( T^{3} + \)\(86\!\cdots\!20\)\( T^{4} + \)\(21\!\cdots\!68\)\( T^{5} + \)\(88\!\cdots\!40\)\( T^{6} + \)\(19\!\cdots\!52\)\( T^{7} + \)\(81\!\cdots\!86\)\( T^{8} + \)\(19\!\cdots\!52\)\( p^{16} T^{9} + \)\(88\!\cdots\!40\)\( p^{32} T^{10} + \)\(21\!\cdots\!68\)\( p^{48} T^{11} + \)\(86\!\cdots\!20\)\( p^{64} T^{12} + \)\(14\!\cdots\!44\)\( p^{80} T^{13} + \)\(44\!\cdots\!16\)\( p^{96} T^{14} + 4074747374576 p^{112} T^{15} + p^{128} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 311423470896 p T + \)\(31\!\cdots\!84\)\( T^{2} + \)\(29\!\cdots\!68\)\( T^{3} + \)\(46\!\cdots\!96\)\( T^{4} + \)\(37\!\cdots\!84\)\( T^{5} + \)\(46\!\cdots\!96\)\( T^{6} + \)\(31\!\cdots\!88\)\( T^{7} + \)\(33\!\cdots\!58\)\( T^{8} + \)\(31\!\cdots\!88\)\( p^{16} T^{9} + \)\(46\!\cdots\!96\)\( p^{32} T^{10} + \)\(37\!\cdots\!84\)\( p^{48} T^{11} + \)\(46\!\cdots\!96\)\( p^{64} T^{12} + \)\(29\!\cdots\!68\)\( p^{80} T^{13} + \)\(31\!\cdots\!84\)\( p^{96} T^{14} + 311423470896 p^{113} T^{15} + p^{128} T^{16} )^{2} \) | |
| 43 | \( 1 - \)\(10\!\cdots\!04\)\( T^{2} + \)\(57\!\cdots\!76\)\( T^{4} - \)\(21\!\cdots\!52\)\( T^{6} + \)\(63\!\cdots\!84\)\( T^{8} - \)\(15\!\cdots\!04\)\( T^{10} + \)\(30\!\cdots\!16\)\( T^{12} - \)\(51\!\cdots\!48\)\( T^{14} + \)\(75\!\cdots\!78\)\( T^{16} - \)\(51\!\cdots\!48\)\( p^{32} T^{18} + \)\(30\!\cdots\!16\)\( p^{64} T^{20} - \)\(15\!\cdots\!04\)\( p^{96} T^{22} + \)\(63\!\cdots\!84\)\( p^{128} T^{24} - \)\(21\!\cdots\!52\)\( p^{160} T^{26} + \)\(57\!\cdots\!76\)\( p^{192} T^{28} - \)\(10\!\cdots\!04\)\( p^{224} T^{30} + p^{256} T^{32} \) | |
| 47 | \( 1 - \)\(38\!\cdots\!04\)\( T^{2} + \)\(85\!\cdots\!88\)\( T^{4} - \)\(13\!\cdots\!32\)\( T^{6} + \)\(16\!\cdots\!04\)\( T^{8} - \)\(16\!\cdots\!12\)\( T^{10} + \)\(13\!\cdots\!36\)\( T^{12} - \)\(96\!\cdots\!76\)\( T^{14} + \)\(58\!\cdots\!30\)\( T^{16} - \)\(96\!\cdots\!76\)\( p^{32} T^{18} + \)\(13\!\cdots\!36\)\( p^{64} T^{20} - \)\(16\!\cdots\!12\)\( p^{96} T^{22} + \)\(16\!\cdots\!04\)\( p^{128} T^{24} - \)\(13\!\cdots\!32\)\( p^{160} T^{26} + \)\(85\!\cdots\!88\)\( p^{192} T^{28} - \)\(38\!\cdots\!04\)\( p^{224} T^{30} + p^{256} T^{32} \) | |
| 53 | \( ( 1 + 43464218314896 T + \)\(96\!\cdots\!76\)\( T^{2} + \)\(29\!\cdots\!64\)\( T^{3} + \)\(39\!\cdots\!40\)\( T^{4} + \)\(19\!\cdots\!68\)\( T^{5} + \)\(13\!\cdots\!20\)\( T^{6} - \)\(18\!\cdots\!88\)\( T^{7} + \)\(53\!\cdots\!66\)\( T^{8} - \)\(18\!\cdots\!88\)\( p^{16} T^{9} + \)\(13\!\cdots\!20\)\( p^{32} T^{10} + \)\(19\!\cdots\!68\)\( p^{48} T^{11} + \)\(39\!\cdots\!40\)\( p^{64} T^{12} + \)\(29\!\cdots\!64\)\( p^{80} T^{13} + \)\(96\!\cdots\!76\)\( p^{96} T^{14} + 43464218314896 p^{112} T^{15} + p^{128} T^{16} )^{2} \) | |
| 59 | \( 1 - \)\(11\!\cdots\!08\)\( T^{2} + \)\(77\!\cdots\!32\)\( T^{4} - \)\(34\!\cdots\!64\)\( T^{6} + \)\(11\!\cdots\!20\)\( T^{8} - \)\(29\!\cdots\!40\)\( T^{10} + \)\(59\!\cdots\!16\)\( T^{12} - \)\(10\!\cdots\!96\)\( T^{14} + \)\(20\!\cdots\!34\)\( T^{16} - \)\(10\!\cdots\!96\)\( p^{32} T^{18} + \)\(59\!\cdots\!16\)\( p^{64} T^{20} - \)\(29\!\cdots\!40\)\( p^{96} T^{22} + \)\(11\!\cdots\!20\)\( p^{128} T^{24} - \)\(34\!\cdots\!64\)\( p^{160} T^{26} + \)\(77\!\cdots\!32\)\( p^{192} T^{28} - \)\(11\!\cdots\!08\)\( p^{224} T^{30} + p^{256} T^{32} \) | |
| 61 | \( ( 1 - 238014298234000 T + \)\(17\!\cdots\!00\)\( T^{2} - \)\(44\!\cdots\!28\)\( T^{3} + \)\(16\!\cdots\!80\)\( T^{4} - \)\(37\!\cdots\!08\)\( T^{5} + \)\(10\!\cdots\!96\)\( T^{6} - \)\(19\!\cdots\!20\)\( T^{7} + \)\(45\!\cdots\!94\)\( T^{8} - \)\(19\!\cdots\!20\)\( p^{16} T^{9} + \)\(10\!\cdots\!96\)\( p^{32} T^{10} - \)\(37\!\cdots\!08\)\( p^{48} T^{11} + \)\(16\!\cdots\!80\)\( p^{64} T^{12} - \)\(44\!\cdots\!28\)\( p^{80} T^{13} + \)\(17\!\cdots\!00\)\( p^{96} T^{14} - 238014298234000 p^{112} T^{15} + p^{128} T^{16} )^{2} \) | |
| 67 | \( 1 - \)\(15\!\cdots\!08\)\( T^{2} + \)\(10\!\cdots\!92\)\( T^{4} - \)\(49\!\cdots\!64\)\( T^{6} + \)\(16\!\cdots\!00\)\( T^{8} - \)\(38\!\cdots\!20\)\( T^{10} + \)\(74\!\cdots\!76\)\( T^{12} - \)\(12\!\cdots\!36\)\( T^{14} + \)\(19\!\cdots\!94\)\( T^{16} - \)\(12\!\cdots\!36\)\( p^{32} T^{18} + \)\(74\!\cdots\!76\)\( p^{64} T^{20} - \)\(38\!\cdots\!20\)\( p^{96} T^{22} + \)\(16\!\cdots\!00\)\( p^{128} T^{24} - \)\(49\!\cdots\!64\)\( p^{160} T^{26} + \)\(10\!\cdots\!92\)\( p^{192} T^{28} - \)\(15\!\cdots\!08\)\( p^{224} T^{30} + p^{256} T^{32} \) | |
| 71 | \( 1 - \)\(31\!\cdots\!28\)\( T^{2} + \)\(50\!\cdots\!64\)\( T^{4} - \)\(55\!\cdots\!72\)\( T^{6} + \)\(46\!\cdots\!20\)\( T^{8} - \)\(32\!\cdots\!80\)\( T^{10} + \)\(19\!\cdots\!68\)\( T^{12} - \)\(97\!\cdots\!28\)\( T^{14} + \)\(43\!\cdots\!50\)\( T^{16} - \)\(97\!\cdots\!28\)\( p^{32} T^{18} + \)\(19\!\cdots\!68\)\( p^{64} T^{20} - \)\(32\!\cdots\!80\)\( p^{96} T^{22} + \)\(46\!\cdots\!20\)\( p^{128} T^{24} - \)\(55\!\cdots\!72\)\( p^{160} T^{26} + \)\(50\!\cdots\!64\)\( p^{192} T^{28} - \)\(31\!\cdots\!28\)\( p^{224} T^{30} + p^{256} T^{32} \) | |
| 73 | \( ( 1 + 184343450950640 T + \)\(33\!\cdots\!68\)\( T^{2} + \)\(99\!\cdots\!60\)\( T^{3} + \)\(54\!\cdots\!04\)\( T^{4} + \)\(19\!\cdots\!60\)\( T^{5} + \)\(56\!\cdots\!64\)\( T^{6} + \)\(20\!\cdots\!80\)\( T^{7} + \)\(42\!\cdots\!86\)\( T^{8} + \)\(20\!\cdots\!80\)\( p^{16} T^{9} + \)\(56\!\cdots\!64\)\( p^{32} T^{10} + \)\(19\!\cdots\!60\)\( p^{48} T^{11} + \)\(54\!\cdots\!04\)\( p^{64} T^{12} + \)\(99\!\cdots\!60\)\( p^{80} T^{13} + \)\(33\!\cdots\!68\)\( p^{96} T^{14} + 184343450950640 p^{112} T^{15} + p^{128} T^{16} )^{2} \) | |
| 79 | \( 1 - \)\(17\!\cdots\!20\)\( T^{2} + \)\(16\!\cdots\!24\)\( T^{4} - \)\(13\!\cdots\!60\)\( p T^{6} + \)\(53\!\cdots\!76\)\( T^{8} - \)\(21\!\cdots\!00\)\( T^{10} + \)\(70\!\cdots\!12\)\( T^{12} - \)\(20\!\cdots\!60\)\( T^{14} + \)\(49\!\cdots\!58\)\( T^{16} - \)\(20\!\cdots\!60\)\( p^{32} T^{18} + \)\(70\!\cdots\!12\)\( p^{64} T^{20} - \)\(21\!\cdots\!00\)\( p^{96} T^{22} + \)\(53\!\cdots\!76\)\( p^{128} T^{24} - \)\(13\!\cdots\!60\)\( p^{161} T^{26} + \)\(16\!\cdots\!24\)\( p^{192} T^{28} - \)\(17\!\cdots\!20\)\( p^{224} T^{30} + p^{256} T^{32} \) | |
| 83 | \( 1 - \)\(34\!\cdots\!44\)\( p T^{2} + \)\(47\!\cdots\!32\)\( T^{4} - \)\(56\!\cdots\!32\)\( T^{6} + \)\(53\!\cdots\!64\)\( T^{8} - \)\(42\!\cdots\!68\)\( T^{10} + \)\(29\!\cdots\!72\)\( T^{12} - \)\(17\!\cdots\!60\)\( T^{14} + \)\(95\!\cdots\!30\)\( T^{16} - \)\(17\!\cdots\!60\)\( p^{32} T^{18} + \)\(29\!\cdots\!72\)\( p^{64} T^{20} - \)\(42\!\cdots\!68\)\( p^{96} T^{22} + \)\(53\!\cdots\!64\)\( p^{128} T^{24} - \)\(56\!\cdots\!32\)\( p^{160} T^{26} + \)\(47\!\cdots\!32\)\( p^{192} T^{28} - \)\(34\!\cdots\!44\)\( p^{225} T^{30} + p^{256} T^{32} \) | |
| 89 | \( ( 1 + 2855533837100784 T + \)\(62\!\cdots\!76\)\( T^{2} + \)\(26\!\cdots\!84\)\( T^{3} + \)\(21\!\cdots\!44\)\( T^{4} + \)\(98\!\cdots\!52\)\( T^{5} + \)\(53\!\cdots\!48\)\( T^{6} + \)\(21\!\cdots\!32\)\( T^{7} + \)\(10\!\cdots\!66\)\( T^{8} + \)\(21\!\cdots\!32\)\( p^{16} T^{9} + \)\(53\!\cdots\!48\)\( p^{32} T^{10} + \)\(98\!\cdots\!52\)\( p^{48} T^{11} + \)\(21\!\cdots\!44\)\( p^{64} T^{12} + \)\(26\!\cdots\!84\)\( p^{80} T^{13} + \)\(62\!\cdots\!76\)\( p^{96} T^{14} + 2855533837100784 p^{112} T^{15} + p^{128} T^{16} )^{2} \) | |
| 97 | \( ( 1 + 6782475780280304 T + \)\(41\!\cdots\!12\)\( T^{2} + \)\(26\!\cdots\!68\)\( p T^{3} + \)\(76\!\cdots\!72\)\( T^{4} + \)\(42\!\cdots\!08\)\( T^{5} + \)\(84\!\cdots\!24\)\( T^{6} + \)\(40\!\cdots\!96\)\( T^{7} + \)\(62\!\cdots\!38\)\( T^{8} + \)\(40\!\cdots\!96\)\( p^{16} T^{9} + \)\(84\!\cdots\!24\)\( p^{32} T^{10} + \)\(42\!\cdots\!08\)\( p^{48} T^{11} + \)\(76\!\cdots\!72\)\( p^{64} T^{12} + \)\(26\!\cdots\!68\)\( p^{81} T^{13} + \)\(41\!\cdots\!12\)\( p^{96} T^{14} + 6782475780280304 p^{112} T^{15} + p^{128} T^{16} )^{2} \) | |
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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
−3.29542566844191670389385202213, −2.98786979893338980473056719325, −2.72852938485569419770076332942, −2.70727303487181833561538748145, −2.62902984121383472971299881439, −2.45264226776900564594363950765, −2.35516042657287899353635821793, −2.34588293343594323782995749667, −2.15956597324740932130136324824, −2.04935157563990595325427594418, −1.89773771260177539758327761056, −1.68597768775810568360603934721, −1.68541957405731364780160598864, −1.62514902593241156704497031745, −1.60511341059975435931289263189, −1.34278651652105894491583883023, −1.16203593641942243829007164462, −0.855987303459895895914255798529, −0.801009426968903661628570182694, −0.65933761585762836218873009392, −0.55950069651480980350766947969, −0.52316511956028171752043080997, −0.18863335643725837725540176627, −0.11601814435952020806809787462, −0.11104524376596599128944118466, 0.11104524376596599128944118466, 0.11601814435952020806809787462, 0.18863335643725837725540176627, 0.52316511956028171752043080997, 0.55950069651480980350766947969, 0.65933761585762836218873009392, 0.801009426968903661628570182694, 0.855987303459895895914255798529, 1.16203593641942243829007164462, 1.34278651652105894491583883023, 1.60511341059975435931289263189, 1.62514902593241156704497031745, 1.68541957405731364780160598864, 1.68597768775810568360603934721, 1.89773771260177539758327761056, 2.04935157563990595325427594418, 2.15956597324740932130136324824, 2.34588293343594323782995749667, 2.35516042657287899353635821793, 2.45264226776900564594363950765, 2.62902984121383472971299881439, 2.70727303487181833561538748145, 2.72852938485569419770076332942, 2.98786979893338980473056719325, 3.29542566844191670389385202213
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.