Dirichlet series
| L(s) = 1 | − 7.23e3·7-s − 1.07e5·19-s − 1.27e6·25-s − 4.73e6·31-s + 6.80e6·37-s − 5.96e5·43-s + 1.65e7·49-s + 3.25e7·61-s − 2.59e7·67-s − 5.28e7·73-s − 1.29e8·79-s − 7.46e8·103-s + 1.33e8·109-s + 9.33e8·121-s + 127-s + 131-s + 7.76e8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.83e9·169-s + 173-s + 9.25e9·175-s + ⋯ |
| L(s) = 1 | − 3.01·7-s − 0.823·19-s − 3.27·25-s − 5.12·31-s + 3.62·37-s − 0.174·43-s + 2.87·49-s + 2.35·61-s − 1.29·67-s − 1.86·73-s − 3.33·79-s − 6.63·103-s + 0.942·109-s + 4.35·121-s + 2.48·133-s + 5.92·169-s + 9.86·175-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{40} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{40} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s+4)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{40} \cdot 3^{40} \cdot 7^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.69034\times 10^{40}\) |
| Root analytic conductor: | \(10.1320\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{40} \cdot 3^{40} \cdot 7^{20} ,\ ( \ : [4]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(8.368009666\) |
| \(L(\frac12)\) | \(\approx\) | \(8.368009666\) |
| \(L(5)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 7 | \( ( 1 + 517 p T + 231855 p^{2} T^{2} + 8886786 p^{4} T^{3} + 80716203 p^{7} T^{4} + 459922485 p^{10} T^{5} + 80716203 p^{15} T^{6} + 8886786 p^{20} T^{7} + 231855 p^{26} T^{8} + 517 p^{33} T^{9} + p^{40} T^{10} )^{2} \) | |
| good | 5 | \( 1 + 1278709 T^{2} + 633548303113 T^{4} + 56007103528131042 T^{6} - \)\(92\!\cdots\!66\)\( T^{8} - \)\(10\!\cdots\!94\)\( p T^{10} - \)\(60\!\cdots\!61\)\( p^{2} T^{12} - \)\(13\!\cdots\!37\)\( p^{5} T^{14} - \)\(30\!\cdots\!89\)\( p^{6} T^{16} + \)\(51\!\cdots\!56\)\( p^{12} T^{18} + \)\(81\!\cdots\!64\)\( p^{10} T^{20} + \)\(51\!\cdots\!56\)\( p^{28} T^{22} - \)\(30\!\cdots\!89\)\( p^{38} T^{24} - \)\(13\!\cdots\!37\)\( p^{53} T^{26} - \)\(60\!\cdots\!61\)\( p^{66} T^{28} - \)\(10\!\cdots\!94\)\( p^{81} T^{30} - \)\(92\!\cdots\!66\)\( p^{96} T^{32} + 56007103528131042 p^{112} T^{34} + 633548303113 p^{128} T^{36} + 1278709 p^{144} T^{38} + p^{160} T^{40} \) |
| 11 | \( 1 - 933900043 T^{2} + 36365903226462347 p T^{4} - \)\(83\!\cdots\!70\)\( T^{6} + \)\(31\!\cdots\!18\)\( T^{8} + \)\(16\!\cdots\!98\)\( T^{10} + \)\(30\!\cdots\!35\)\( T^{12} - \)\(32\!\cdots\!73\)\( T^{14} + \)\(59\!\cdots\!27\)\( p^{2} T^{16} - \)\(20\!\cdots\!20\)\( T^{18} - \)\(14\!\cdots\!96\)\( T^{20} - \)\(20\!\cdots\!20\)\( p^{16} T^{22} + \)\(59\!\cdots\!27\)\( p^{34} T^{24} - \)\(32\!\cdots\!73\)\( p^{48} T^{26} + \)\(30\!\cdots\!35\)\( p^{64} T^{28} + \)\(16\!\cdots\!98\)\( p^{80} T^{30} + \)\(31\!\cdots\!18\)\( p^{96} T^{32} - \)\(83\!\cdots\!70\)\( p^{112} T^{34} + 36365903226462347 p^{129} T^{36} - 933900043 p^{144} T^{38} + p^{160} T^{40} \) | |
| 13 | \( ( 1 - 2416330576 T^{2} + 2165542351299164958 T^{4} - \)\(69\!\cdots\!78\)\( p T^{6} + \)\(62\!\cdots\!29\)\( T^{8} - \)\(72\!\cdots\!36\)\( T^{10} + \)\(62\!\cdots\!29\)\( p^{16} T^{12} - \)\(69\!\cdots\!78\)\( p^{33} T^{14} + 2165542351299164958 p^{48} T^{16} - 2416330576 p^{64} T^{18} + p^{80} T^{20} )^{2} \) | |
| 17 | \( 1 + 26682030514 T^{2} + \)\(23\!\cdots\!67\)\( T^{4} + \)\(11\!\cdots\!34\)\( T^{6} + \)\(16\!\cdots\!95\)\( T^{8} + \)\(17\!\cdots\!92\)\( T^{10} + \)\(55\!\cdots\!46\)\( T^{12} + \)\(20\!\cdots\!44\)\( T^{14} + \)\(48\!\cdots\!41\)\( T^{16} + \)\(18\!\cdots\!06\)\( T^{18} - \)\(61\!\cdots\!19\)\( T^{20} + \)\(18\!\cdots\!06\)\( p^{16} T^{22} + \)\(48\!\cdots\!41\)\( p^{32} T^{24} + \)\(20\!\cdots\!44\)\( p^{48} T^{26} + \)\(55\!\cdots\!46\)\( p^{64} T^{28} + \)\(17\!\cdots\!92\)\( p^{80} T^{30} + \)\(16\!\cdots\!95\)\( p^{96} T^{32} + \)\(11\!\cdots\!34\)\( p^{112} T^{34} + \)\(23\!\cdots\!67\)\( p^{128} T^{36} + 26682030514 p^{144} T^{38} + p^{160} T^{40} \) | |
| 19 | \( ( 1 + 53628 T + 55051686614 T^{2} + 2900901146159208 T^{3} + \)\(15\!\cdots\!83\)\( T^{4} + \)\(81\!\cdots\!84\)\( T^{5} + \)\(30\!\cdots\!00\)\( T^{6} + \)\(15\!\cdots\!28\)\( T^{7} + \)\(52\!\cdots\!17\)\( T^{8} + \)\(21\!\cdots\!40\)\( T^{9} + \)\(87\!\cdots\!66\)\( T^{10} + \)\(21\!\cdots\!40\)\( p^{8} T^{11} + \)\(52\!\cdots\!17\)\( p^{16} T^{12} + \)\(15\!\cdots\!28\)\( p^{24} T^{13} + \)\(30\!\cdots\!00\)\( p^{32} T^{14} + \)\(81\!\cdots\!84\)\( p^{40} T^{15} + \)\(15\!\cdots\!83\)\( p^{48} T^{16} + 2900901146159208 p^{56} T^{17} + 55051686614 p^{64} T^{18} + 53628 p^{72} T^{19} + p^{80} T^{20} )^{2} \) | |
| 23 | \( 1 - 242939945302 T^{2} + \)\(19\!\cdots\!67\)\( T^{4} + \)\(60\!\cdots\!30\)\( T^{6} - \)\(21\!\cdots\!57\)\( T^{8} + \)\(96\!\cdots\!92\)\( T^{10} + \)\(87\!\cdots\!30\)\( T^{12} - \)\(11\!\cdots\!92\)\( T^{14} + \)\(37\!\cdots\!57\)\( T^{16} + \)\(23\!\cdots\!50\)\( T^{18} - \)\(32\!\cdots\!11\)\( T^{20} + \)\(23\!\cdots\!50\)\( p^{16} T^{22} + \)\(37\!\cdots\!57\)\( p^{32} T^{24} - \)\(11\!\cdots\!92\)\( p^{48} T^{26} + \)\(87\!\cdots\!30\)\( p^{64} T^{28} + \)\(96\!\cdots\!92\)\( p^{80} T^{30} - \)\(21\!\cdots\!57\)\( p^{96} T^{32} + \)\(60\!\cdots\!30\)\( p^{112} T^{34} + \)\(19\!\cdots\!67\)\( p^{128} T^{36} - 242939945302 p^{144} T^{38} + p^{160} T^{40} \) | |
| 29 | \( ( 1 - 168649415141 T^{2} + \)\(38\!\cdots\!76\)\( T^{4} + \)\(76\!\cdots\!57\)\( T^{6} + \)\(43\!\cdots\!43\)\( T^{8} + \)\(42\!\cdots\!28\)\( T^{10} + \)\(43\!\cdots\!43\)\( p^{16} T^{12} + \)\(76\!\cdots\!57\)\( p^{32} T^{14} + \)\(38\!\cdots\!76\)\( p^{48} T^{16} - 168649415141 p^{64} T^{18} + p^{80} T^{20} )^{2} \) | |
| 31 | \( ( 1 + 2367213 T + 6253527532550 T^{2} + 10381716569761638951 T^{3} + \)\(17\!\cdots\!78\)\( T^{4} + \)\(24\!\cdots\!67\)\( T^{5} + \)\(32\!\cdots\!32\)\( T^{6} + \)\(37\!\cdots\!87\)\( T^{7} + \)\(42\!\cdots\!17\)\( T^{8} + \)\(42\!\cdots\!42\)\( T^{9} + \)\(41\!\cdots\!44\)\( T^{10} + \)\(42\!\cdots\!42\)\( p^{8} T^{11} + \)\(42\!\cdots\!17\)\( p^{16} T^{12} + \)\(37\!\cdots\!87\)\( p^{24} T^{13} + \)\(32\!\cdots\!32\)\( p^{32} T^{14} + \)\(24\!\cdots\!67\)\( p^{40} T^{15} + \)\(17\!\cdots\!78\)\( p^{48} T^{16} + 10381716569761638951 p^{56} T^{17} + 6253527532550 p^{64} T^{18} + 2367213 p^{72} T^{19} + p^{80} T^{20} )^{2} \) | |
| 37 | \( ( 1 - 3401374 T - 2700074299274 T^{2} + 19044761008552857852 T^{3} - \)\(88\!\cdots\!69\)\( p T^{4} - \)\(22\!\cdots\!68\)\( T^{5} - \)\(88\!\cdots\!96\)\( T^{6} + \)\(10\!\cdots\!14\)\( T^{7} + \)\(46\!\cdots\!49\)\( T^{8} - \)\(43\!\cdots\!84\)\( T^{9} - \)\(93\!\cdots\!34\)\( T^{10} - \)\(43\!\cdots\!84\)\( p^{8} T^{11} + \)\(46\!\cdots\!49\)\( p^{16} T^{12} + \)\(10\!\cdots\!14\)\( p^{24} T^{13} - \)\(88\!\cdots\!96\)\( p^{32} T^{14} - \)\(22\!\cdots\!68\)\( p^{40} T^{15} - \)\(88\!\cdots\!69\)\( p^{49} T^{16} + 19044761008552857852 p^{56} T^{17} - 2700074299274 p^{64} T^{18} - 3401374 p^{72} T^{19} + p^{80} T^{20} )^{2} \) | |
| 41 | \( ( 1 - 32147455552054 T^{2} + \)\(44\!\cdots\!09\)\( T^{4} - \)\(43\!\cdots\!56\)\( T^{6} + \)\(46\!\cdots\!02\)\( T^{8} - \)\(43\!\cdots\!44\)\( T^{10} + \)\(46\!\cdots\!02\)\( p^{16} T^{12} - \)\(43\!\cdots\!56\)\( p^{32} T^{14} + \)\(44\!\cdots\!09\)\( p^{48} T^{16} - 32147455552054 p^{64} T^{18} + p^{80} T^{20} )^{2} \) | |
| 43 | \( ( 1 + 149032 T + 36020005994904 T^{2} + 68097711618602334996 T^{3} + \)\(52\!\cdots\!67\)\( T^{4} + \)\(15\!\cdots\!28\)\( T^{5} + \)\(52\!\cdots\!67\)\( p^{8} T^{6} + 68097711618602334996 p^{16} T^{7} + 36020005994904 p^{24} T^{8} + 149032 p^{32} T^{9} + p^{40} T^{10} )^{4} \) | |
| 47 | \( 1 + 117518895466222 T^{2} + \)\(71\!\cdots\!67\)\( T^{4} + \)\(28\!\cdots\!66\)\( T^{6} + \)\(75\!\cdots\!91\)\( T^{8} + \)\(11\!\cdots\!32\)\( T^{10} - \)\(47\!\cdots\!22\)\( T^{12} - \)\(68\!\cdots\!12\)\( T^{14} - \)\(24\!\cdots\!07\)\( T^{16} - \)\(62\!\cdots\!02\)\( T^{18} - \)\(14\!\cdots\!87\)\( T^{20} - \)\(62\!\cdots\!02\)\( p^{16} T^{22} - \)\(24\!\cdots\!07\)\( p^{32} T^{24} - \)\(68\!\cdots\!12\)\( p^{48} T^{26} - \)\(47\!\cdots\!22\)\( p^{64} T^{28} + \)\(11\!\cdots\!32\)\( p^{80} T^{30} + \)\(75\!\cdots\!91\)\( p^{96} T^{32} + \)\(28\!\cdots\!66\)\( p^{112} T^{34} + \)\(71\!\cdots\!67\)\( p^{128} T^{36} + 117518895466222 p^{144} T^{38} + p^{160} T^{40} \) | |
| 53 | \( 1 - 226222978603411 T^{2} + \)\(30\!\cdots\!73\)\( T^{4} - \)\(18\!\cdots\!82\)\( T^{6} - \)\(91\!\cdots\!70\)\( T^{8} + \)\(17\!\cdots\!86\)\( T^{10} - \)\(18\!\cdots\!89\)\( T^{12} + \)\(10\!\cdots\!35\)\( T^{14} + \)\(12\!\cdots\!35\)\( T^{16} - \)\(46\!\cdots\!16\)\( T^{18} + \)\(43\!\cdots\!20\)\( T^{20} - \)\(46\!\cdots\!16\)\( p^{16} T^{22} + \)\(12\!\cdots\!35\)\( p^{32} T^{24} + \)\(10\!\cdots\!35\)\( p^{48} T^{26} - \)\(18\!\cdots\!89\)\( p^{64} T^{28} + \)\(17\!\cdots\!86\)\( p^{80} T^{30} - \)\(91\!\cdots\!70\)\( p^{96} T^{32} - \)\(18\!\cdots\!82\)\( p^{112} T^{34} + \)\(30\!\cdots\!73\)\( p^{128} T^{36} - 226222978603411 p^{144} T^{38} + p^{160} T^{40} \) | |
| 59 | \( 1 + 778896881123101 T^{2} + \)\(27\!\cdots\!81\)\( T^{4} + \)\(69\!\cdots\!94\)\( T^{6} + \)\(15\!\cdots\!34\)\( T^{8} + \)\(31\!\cdots\!94\)\( T^{10} + \)\(50\!\cdots\!15\)\( T^{12} + \)\(74\!\cdots\!15\)\( T^{14} + \)\(10\!\cdots\!15\)\( T^{16} + \)\(14\!\cdots\!80\)\( T^{18} + \)\(19\!\cdots\!80\)\( T^{20} + \)\(14\!\cdots\!80\)\( p^{16} T^{22} + \)\(10\!\cdots\!15\)\( p^{32} T^{24} + \)\(74\!\cdots\!15\)\( p^{48} T^{26} + \)\(50\!\cdots\!15\)\( p^{64} T^{28} + \)\(31\!\cdots\!94\)\( p^{80} T^{30} + \)\(15\!\cdots\!34\)\( p^{96} T^{32} + \)\(69\!\cdots\!94\)\( p^{112} T^{34} + \)\(27\!\cdots\!81\)\( p^{128} T^{36} + 778896881123101 p^{144} T^{38} + p^{160} T^{40} \) | |
| 61 | \( ( 1 - 16286886 T + 960867162985925 T^{2} - \)\(14\!\cdots\!98\)\( T^{3} + \)\(51\!\cdots\!35\)\( T^{4} - \)\(73\!\cdots\!48\)\( T^{5} + \)\(18\!\cdots\!58\)\( T^{6} - \)\(25\!\cdots\!24\)\( T^{7} + \)\(52\!\cdots\!81\)\( T^{8} - \)\(65\!\cdots\!54\)\( T^{9} + \)\(11\!\cdots\!55\)\( T^{10} - \)\(65\!\cdots\!54\)\( p^{8} T^{11} + \)\(52\!\cdots\!81\)\( p^{16} T^{12} - \)\(25\!\cdots\!24\)\( p^{24} T^{13} + \)\(18\!\cdots\!58\)\( p^{32} T^{14} - \)\(73\!\cdots\!48\)\( p^{40} T^{15} + \)\(51\!\cdots\!35\)\( p^{48} T^{16} - \)\(14\!\cdots\!98\)\( p^{56} T^{17} + 960867162985925 p^{64} T^{18} - 16286886 p^{72} T^{19} + p^{80} T^{20} )^{2} \) | |
| 67 | \( ( 1 + 12999652 T - 1149127808873528 T^{2} - \)\(26\!\cdots\!12\)\( T^{3} + \)\(67\!\cdots\!17\)\( T^{4} + \)\(21\!\cdots\!84\)\( T^{5} - \)\(94\!\cdots\!90\)\( T^{6} - \)\(10\!\cdots\!40\)\( T^{7} - \)\(93\!\cdots\!67\)\( T^{8} + \)\(18\!\cdots\!52\)\( T^{9} + \)\(69\!\cdots\!62\)\( T^{10} + \)\(18\!\cdots\!52\)\( p^{8} T^{11} - \)\(93\!\cdots\!67\)\( p^{16} T^{12} - \)\(10\!\cdots\!40\)\( p^{24} T^{13} - \)\(94\!\cdots\!90\)\( p^{32} T^{14} + \)\(21\!\cdots\!84\)\( p^{40} T^{15} + \)\(67\!\cdots\!17\)\( p^{48} T^{16} - \)\(26\!\cdots\!12\)\( p^{56} T^{17} - 1149127808873528 p^{64} T^{18} + 12999652 p^{72} T^{19} + p^{80} T^{20} )^{2} \) | |
| 71 | \( ( 1 + 4247341285962358 T^{2} + \)\(87\!\cdots\!85\)\( T^{4} + \)\(11\!\cdots\!60\)\( T^{6} + \)\(11\!\cdots\!18\)\( T^{8} + \)\(83\!\cdots\!92\)\( T^{10} + \)\(11\!\cdots\!18\)\( p^{16} T^{12} + \)\(11\!\cdots\!60\)\( p^{32} T^{14} + \)\(87\!\cdots\!85\)\( p^{48} T^{16} + 4247341285962358 p^{64} T^{18} + p^{80} T^{20} )^{2} \) | |
| 73 | \( ( 1 + 26449806 T + 3361360208782412 T^{2} + \)\(82\!\cdots\!00\)\( T^{3} + \)\(58\!\cdots\!97\)\( T^{4} + \)\(11\!\cdots\!68\)\( T^{5} + \)\(99\!\cdots\!62\)\( p T^{6} + \)\(12\!\cdots\!54\)\( T^{7} + \)\(74\!\cdots\!21\)\( T^{8} + \)\(11\!\cdots\!52\)\( T^{9} + \)\(64\!\cdots\!66\)\( T^{10} + \)\(11\!\cdots\!52\)\( p^{8} T^{11} + \)\(74\!\cdots\!21\)\( p^{16} T^{12} + \)\(12\!\cdots\!54\)\( p^{24} T^{13} + \)\(99\!\cdots\!62\)\( p^{33} T^{14} + \)\(11\!\cdots\!68\)\( p^{40} T^{15} + \)\(58\!\cdots\!97\)\( p^{48} T^{16} + \)\(82\!\cdots\!00\)\( p^{56} T^{17} + 3361360208782412 p^{64} T^{18} + 26449806 p^{72} T^{19} + p^{80} T^{20} )^{2} \) | |
| 79 | \( ( 1 + 64972765 T - 1361882569079678 T^{2} - \)\(21\!\cdots\!25\)\( T^{3} - \)\(25\!\cdots\!86\)\( T^{4} + \)\(24\!\cdots\!55\)\( T^{5} + \)\(78\!\cdots\!80\)\( T^{6} + \)\(31\!\cdots\!35\)\( T^{7} - \)\(22\!\cdots\!03\)\( T^{8} - \)\(20\!\cdots\!30\)\( T^{9} - \)\(12\!\cdots\!68\)\( T^{10} - \)\(20\!\cdots\!30\)\( p^{8} T^{11} - \)\(22\!\cdots\!03\)\( p^{16} T^{12} + \)\(31\!\cdots\!35\)\( p^{24} T^{13} + \)\(78\!\cdots\!80\)\( p^{32} T^{14} + \)\(24\!\cdots\!55\)\( p^{40} T^{15} - \)\(25\!\cdots\!86\)\( p^{48} T^{16} - \)\(21\!\cdots\!25\)\( p^{56} T^{17} - 1361882569079678 p^{64} T^{18} + 64972765 p^{72} T^{19} + p^{80} T^{20} )^{2} \) | |
| 83 | \( ( 1 - 15666856688583229 T^{2} + \)\(11\!\cdots\!04\)\( T^{4} - \)\(57\!\cdots\!11\)\( T^{6} + \)\(19\!\cdots\!67\)\( T^{8} - \)\(51\!\cdots\!44\)\( T^{10} + \)\(19\!\cdots\!67\)\( p^{16} T^{12} - \)\(57\!\cdots\!11\)\( p^{32} T^{14} + \)\(11\!\cdots\!04\)\( p^{48} T^{16} - 15666856688583229 p^{64} T^{18} + p^{80} T^{20} )^{2} \) | |
| 89 | \( 1 + 12887110889384086 T^{2} + \)\(61\!\cdots\!71\)\( T^{4} + \)\(52\!\cdots\!94\)\( T^{6} - \)\(80\!\cdots\!81\)\( T^{8} - \)\(44\!\cdots\!56\)\( T^{10} - \)\(16\!\cdots\!10\)\( T^{12} - \)\(54\!\cdots\!60\)\( T^{14} - \)\(59\!\cdots\!35\)\( T^{16} + \)\(14\!\cdots\!30\)\( T^{18} + \)\(91\!\cdots\!05\)\( T^{20} + \)\(14\!\cdots\!30\)\( p^{16} T^{22} - \)\(59\!\cdots\!35\)\( p^{32} T^{24} - \)\(54\!\cdots\!60\)\( p^{48} T^{26} - \)\(16\!\cdots\!10\)\( p^{64} T^{28} - \)\(44\!\cdots\!56\)\( p^{80} T^{30} - \)\(80\!\cdots\!81\)\( p^{96} T^{32} + \)\(52\!\cdots\!94\)\( p^{112} T^{34} + \)\(61\!\cdots\!71\)\( p^{128} T^{36} + 12887110889384086 p^{144} T^{38} + p^{160} T^{40} \) | |
| 97 | \( ( 1 - 13097093693109961 T^{2} + \)\(22\!\cdots\!48\)\( T^{4} - \)\(23\!\cdots\!19\)\( T^{6} + \)\(22\!\cdots\!19\)\( T^{8} - \)\(19\!\cdots\!96\)\( T^{10} + \)\(22\!\cdots\!19\)\( p^{16} T^{12} - \)\(23\!\cdots\!19\)\( p^{32} T^{14} + \)\(22\!\cdots\!48\)\( p^{48} T^{16} - 13097093693109961 p^{64} T^{18} + p^{80} T^{20} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.60528455661905149762455479193, −1.57725410790620373047722597464, −1.55941041827540262187317333608, −1.43275586785936592847665150138, −1.38591230774000573913440527970, −1.33578131319554111590991491192, −1.31279196362500433831298728120, −1.27544265101498562708443493332, −1.03894431110472344517144617378, −0.962642779204216216156463226564, −0.961181233616489548704671099272, −0.928919000427370479703371172084, −0.912428643462302275313990994659, −0.78242123502789241017917321594, −0.74266534286686090713106249887, −0.58062588308573346339374647507, −0.35612688395861300901702705508, −0.33380594044679743788394784446, −0.33065221784918580809798818465, −0.30194462883120642584371418615, −0.27669855124380444785497904210, −0.23613230515611778599389030922, −0.21759076237133009288812105211, −0.17855162442598271233925402780, −0.13002770557864717646810456958, 0.13002770557864717646810456958, 0.17855162442598271233925402780, 0.21759076237133009288812105211, 0.23613230515611778599389030922, 0.27669855124380444785497904210, 0.30194462883120642584371418615, 0.33065221784918580809798818465, 0.33380594044679743788394784446, 0.35612688395861300901702705508, 0.58062588308573346339374647507, 0.74266534286686090713106249887, 0.78242123502789241017917321594, 0.912428643462302275313990994659, 0.928919000427370479703371172084, 0.961181233616489548704671099272, 0.962642779204216216156463226564, 1.03894431110472344517144617378, 1.27544265101498562708443493332, 1.31279196362500433831298728120, 1.33578131319554111590991491192, 1.38591230774000573913440527970, 1.43275586785936592847665150138, 1.55941041827540262187317333608, 1.57725410790620373047722597464, 1.60528455661905149762455479193
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.