Dirichlet series
| L(s) = 1 | + 326·3-s − 2.56e3·4-s − 7.89e4·9-s − 8.34e5·12-s + 2.92e6·16-s − 9.93e4·17-s + 6.25e6·23-s − 1.87e7·25-s − 3.34e7·27-s + 5.42e6·29-s + 2.02e8·36-s + 5.32e7·43-s + 9.52e8·48-s − 3.96e8·49-s − 3.23e7·51-s − 3.64e7·53-s + 5.36e8·61-s − 2.04e9·64-s + 2.54e8·68-s + 2.03e9·69-s − 6.11e9·75-s + 1.55e9·79-s + 3.33e9·81-s + 1.76e9·87-s − 1.60e10·92-s + 4.80e10·100-s − 3.67e9·101-s + ⋯ |
| L(s) = 1 | + 2.32·3-s − 5.00·4-s − 4.01·9-s − 11.6·12-s + 11.1·16-s − 0.288·17-s + 4.65·23-s − 9.60·25-s − 12.0·27-s + 1.42·29-s + 20.0·36-s + 2.37·43-s + 25.8·48-s − 9.82·49-s − 0.670·51-s − 0.634·53-s + 4.96·61-s − 15.2·64-s + 1.44·68-s + 10.8·69-s − 22.3·75-s + 4.49·79-s + 8.60·81-s + 3.30·87-s − 23.3·92-s + 48.0·100-s − 3.51·101-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{40}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(13^{40}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.22992\times 10^{38}\) |
| Root analytic conductor: | \(9.32957\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 13^{40} ,\ ( \ : [9/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(0.04923141791\) |
| \(L(\frac12)\) | \(\approx\) | \(0.04923141791\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 + 2561 T^{2} + 909265 p^{2} T^{4} + 7566965 p^{9} T^{6} + 54406481057 p^{6} T^{8} + 10840550729209 p^{8} T^{10} + 488114341991933 p^{12} T^{12} + 10073627325839483 p^{17} T^{14} + 385231428691215799 p^{21} T^{16} + 27421398175067442169 p^{24} T^{18} + \)\(90\!\cdots\!61\)\( p^{28} T^{20} + 27421398175067442169 p^{42} T^{22} + 385231428691215799 p^{57} T^{24} + 10073627325839483 p^{71} T^{26} + 488114341991933 p^{84} T^{28} + 10840550729209 p^{98} T^{30} + 54406481057 p^{114} T^{32} + 7566965 p^{135} T^{34} + 909265 p^{146} T^{36} + 2561 p^{162} T^{38} + p^{180} T^{40} \) |
| 3 | \( ( 1 - 163 T + 79318 T^{2} - 4474865 p T^{3} + 383805682 p^{2} T^{4} - 6256779041 p^{4} T^{5} + 403662559972 p^{5} T^{6} - 17084366468795 p^{6} T^{7} + 2827169147415389 p^{6} T^{8} - 112858897457614078 p^{7} T^{9} + 6024779946139796012 p^{8} T^{10} - 112858897457614078 p^{16} T^{11} + 2827169147415389 p^{24} T^{12} - 17084366468795 p^{33} T^{13} + 403662559972 p^{41} T^{14} - 6256779041 p^{49} T^{15} + 383805682 p^{56} T^{16} - 4474865 p^{64} T^{17} + 79318 p^{72} T^{18} - 163 p^{81} T^{19} + p^{90} T^{20} )^{2} \) | |
| 5 | \( 1 + 18766613 T^{2} + 182328407955298 T^{4} + \)\(12\!\cdots\!43\)\( T^{6} + \)\(60\!\cdots\!38\)\( T^{8} + \)\(24\!\cdots\!07\)\( T^{10} + \)\(66\!\cdots\!96\)\( p^{3} T^{12} + \)\(38\!\cdots\!71\)\( p^{4} T^{14} + \)\(39\!\cdots\!21\)\( p^{6} T^{16} + \)\(35\!\cdots\!14\)\( p^{8} T^{18} + \)\(29\!\cdots\!16\)\( p^{10} T^{20} + \)\(35\!\cdots\!14\)\( p^{26} T^{22} + \)\(39\!\cdots\!21\)\( p^{42} T^{24} + \)\(38\!\cdots\!71\)\( p^{58} T^{26} + \)\(66\!\cdots\!96\)\( p^{75} T^{28} + \)\(24\!\cdots\!07\)\( p^{90} T^{30} + \)\(60\!\cdots\!38\)\( p^{108} T^{32} + \)\(12\!\cdots\!43\)\( p^{126} T^{34} + 182328407955298 p^{144} T^{36} + 18766613 p^{162} T^{38} + p^{180} T^{40} \) | |
| 7 | \( 1 + 396508421 T^{2} + 78250729165264630 T^{4} + \)\(21\!\cdots\!47\)\( p^{2} T^{6} + \)\(43\!\cdots\!78\)\( p^{4} T^{8} + \)\(72\!\cdots\!67\)\( p^{6} T^{10} + \)\(10\!\cdots\!44\)\( p^{8} T^{12} + \)\(12\!\cdots\!19\)\( p^{10} T^{14} + \)\(19\!\cdots\!27\)\( p^{13} T^{16} + \)\(13\!\cdots\!70\)\( p^{14} T^{18} + \)\(23\!\cdots\!12\)\( p^{18} T^{20} + \)\(13\!\cdots\!70\)\( p^{32} T^{22} + \)\(19\!\cdots\!27\)\( p^{49} T^{24} + \)\(12\!\cdots\!19\)\( p^{64} T^{26} + \)\(10\!\cdots\!44\)\( p^{80} T^{28} + \)\(72\!\cdots\!67\)\( p^{96} T^{30} + \)\(43\!\cdots\!78\)\( p^{112} T^{32} + \)\(21\!\cdots\!47\)\( p^{128} T^{34} + 78250729165264630 p^{144} T^{36} + 396508421 p^{162} T^{38} + p^{180} T^{40} \) | |
| 11 | \( 1 + 1116156895 p T^{2} + 84568073197684469518 T^{4} + \)\(41\!\cdots\!43\)\( T^{6} + \)\(16\!\cdots\!02\)\( T^{8} + \)\(55\!\cdots\!03\)\( T^{10} + \)\(16\!\cdots\!36\)\( T^{12} + \)\(42\!\cdots\!91\)\( T^{14} + \)\(10\!\cdots\!65\)\( T^{16} + \)\(23\!\cdots\!94\)\( T^{18} + \)\(54\!\cdots\!48\)\( T^{20} + \)\(23\!\cdots\!94\)\( p^{18} T^{22} + \)\(10\!\cdots\!65\)\( p^{36} T^{24} + \)\(42\!\cdots\!91\)\( p^{54} T^{26} + \)\(16\!\cdots\!36\)\( p^{72} T^{28} + \)\(55\!\cdots\!03\)\( p^{90} T^{30} + \)\(16\!\cdots\!02\)\( p^{108} T^{32} + \)\(41\!\cdots\!43\)\( p^{126} T^{34} + 84568073197684469518 p^{144} T^{36} + 1116156895 p^{163} T^{38} + p^{180} T^{40} \) | |
| 17 | \( ( 1 + 49656 T + 534818052869 T^{2} - 31315884659192520 T^{3} + \)\(14\!\cdots\!39\)\( T^{4} - \)\(17\!\cdots\!24\)\( T^{5} + \)\(29\!\cdots\!38\)\( T^{6} - \)\(41\!\cdots\!96\)\( T^{7} + \)\(45\!\cdots\!57\)\( T^{8} - \)\(69\!\cdots\!92\)\( T^{9} + \)\(58\!\cdots\!75\)\( T^{10} - \)\(69\!\cdots\!92\)\( p^{9} T^{11} + \)\(45\!\cdots\!57\)\( p^{18} T^{12} - \)\(41\!\cdots\!96\)\( p^{27} T^{13} + \)\(29\!\cdots\!38\)\( p^{36} T^{14} - \)\(17\!\cdots\!24\)\( p^{45} T^{15} + \)\(14\!\cdots\!39\)\( p^{54} T^{16} - 31315884659192520 p^{63} T^{17} + 534818052869 p^{72} T^{18} + 49656 p^{81} T^{19} + p^{90} T^{20} )^{2} \) | |
| 19 | \( 1 + 548575019 p^{3} T^{2} + \)\(70\!\cdots\!10\)\( T^{4} + \)\(87\!\cdots\!91\)\( T^{6} + \)\(81\!\cdots\!38\)\( T^{8} + \)\(59\!\cdots\!43\)\( T^{10} + \)\(35\!\cdots\!28\)\( T^{12} + \)\(18\!\cdots\!75\)\( T^{14} + \)\(80\!\cdots\!01\)\( T^{16} + \)\(30\!\cdots\!18\)\( T^{18} + \)\(10\!\cdots\!24\)\( T^{20} + \)\(30\!\cdots\!18\)\( p^{18} T^{22} + \)\(80\!\cdots\!01\)\( p^{36} T^{24} + \)\(18\!\cdots\!75\)\( p^{54} T^{26} + \)\(35\!\cdots\!28\)\( p^{72} T^{28} + \)\(59\!\cdots\!43\)\( p^{90} T^{30} + \)\(81\!\cdots\!38\)\( p^{108} T^{32} + \)\(87\!\cdots\!91\)\( p^{126} T^{34} + \)\(70\!\cdots\!10\)\( p^{144} T^{36} + 548575019 p^{165} T^{38} + p^{180} T^{40} \) | |
| 23 | \( ( 1 - 3126189 T + 11973013625246 T^{2} - 28963662194694384585 T^{3} + \)\(71\!\cdots\!66\)\( T^{4} - \)\(14\!\cdots\!83\)\( T^{5} + \)\(28\!\cdots\!24\)\( T^{6} - \)\(48\!\cdots\!13\)\( T^{7} + \)\(78\!\cdots\!57\)\( T^{8} - \)\(11\!\cdots\!22\)\( T^{9} + \)\(16\!\cdots\!00\)\( T^{10} - \)\(11\!\cdots\!22\)\( p^{9} T^{11} + \)\(78\!\cdots\!57\)\( p^{18} T^{12} - \)\(48\!\cdots\!13\)\( p^{27} T^{13} + \)\(28\!\cdots\!24\)\( p^{36} T^{14} - \)\(14\!\cdots\!83\)\( p^{45} T^{15} + \)\(71\!\cdots\!66\)\( p^{54} T^{16} - 28963662194694384585 p^{63} T^{17} + 11973013625246 p^{72} T^{18} - 3126189 p^{81} T^{19} + p^{90} T^{20} )^{2} \) | |
| 29 | \( ( 1 - 2712414 T + 101714019641951 T^{2} - \)\(22\!\cdots\!46\)\( T^{3} + \)\(49\!\cdots\!71\)\( T^{4} - \)\(93\!\cdots\!52\)\( T^{5} + \)\(15\!\cdots\!34\)\( T^{6} - \)\(25\!\cdots\!16\)\( T^{7} + \)\(34\!\cdots\!25\)\( T^{8} - \)\(49\!\cdots\!86\)\( T^{9} + \)\(58\!\cdots\!29\)\( T^{10} - \)\(49\!\cdots\!86\)\( p^{9} T^{11} + \)\(34\!\cdots\!25\)\( p^{18} T^{12} - \)\(25\!\cdots\!16\)\( p^{27} T^{13} + \)\(15\!\cdots\!34\)\( p^{36} T^{14} - \)\(93\!\cdots\!52\)\( p^{45} T^{15} + \)\(49\!\cdots\!71\)\( p^{54} T^{16} - \)\(22\!\cdots\!46\)\( p^{63} T^{17} + 101714019641951 p^{72} T^{18} - 2712414 p^{81} T^{19} + p^{90} T^{20} )^{2} \) | |
| 31 | \( 1 + 324518493957860 T^{2} + \)\(52\!\cdots\!38\)\( T^{4} + \)\(18\!\cdots\!44\)\( p T^{6} + \)\(44\!\cdots\!17\)\( T^{8} + \)\(28\!\cdots\!72\)\( T^{10} + \)\(14\!\cdots\!92\)\( T^{12} + \)\(63\!\cdots\!92\)\( T^{14} + \)\(23\!\cdots\!58\)\( T^{16} + \)\(76\!\cdots\!24\)\( T^{18} + \)\(21\!\cdots\!32\)\( T^{20} + \)\(76\!\cdots\!24\)\( p^{18} T^{22} + \)\(23\!\cdots\!58\)\( p^{36} T^{24} + \)\(63\!\cdots\!92\)\( p^{54} T^{26} + \)\(14\!\cdots\!92\)\( p^{72} T^{28} + \)\(28\!\cdots\!72\)\( p^{90} T^{30} + \)\(44\!\cdots\!17\)\( p^{108} T^{32} + \)\(18\!\cdots\!44\)\( p^{127} T^{34} + \)\(52\!\cdots\!38\)\( p^{144} T^{36} + 324518493957860 p^{162} T^{38} + p^{180} T^{40} \) | |
| 37 | \( 1 + 1419832968167534 T^{2} + \)\(10\!\cdots\!39\)\( T^{4} + \)\(47\!\cdots\!34\)\( T^{6} + \)\(16\!\cdots\!83\)\( T^{8} + \)\(46\!\cdots\!00\)\( T^{10} + \)\(10\!\cdots\!02\)\( T^{12} + \)\(20\!\cdots\!56\)\( T^{14} + \)\(35\!\cdots\!25\)\( T^{16} + \)\(54\!\cdots\!46\)\( T^{18} + \)\(74\!\cdots\!21\)\( T^{20} + \)\(54\!\cdots\!46\)\( p^{18} T^{22} + \)\(35\!\cdots\!25\)\( p^{36} T^{24} + \)\(20\!\cdots\!56\)\( p^{54} T^{26} + \)\(10\!\cdots\!02\)\( p^{72} T^{28} + \)\(46\!\cdots\!00\)\( p^{90} T^{30} + \)\(16\!\cdots\!83\)\( p^{108} T^{32} + \)\(47\!\cdots\!34\)\( p^{126} T^{34} + \)\(10\!\cdots\!39\)\( p^{144} T^{36} + 1419832968167534 p^{162} T^{38} + p^{180} T^{40} \) | |
| 41 | \( 1 + 1965380062206758 T^{2} + \)\(17\!\cdots\!55\)\( T^{4} + \)\(98\!\cdots\!94\)\( T^{6} + \)\(49\!\cdots\!23\)\( T^{8} + \)\(23\!\cdots\!44\)\( T^{10} + \)\(98\!\cdots\!62\)\( T^{12} + \)\(38\!\cdots\!04\)\( T^{14} + \)\(14\!\cdots\!01\)\( T^{16} + \)\(50\!\cdots\!38\)\( T^{18} + \)\(16\!\cdots\!81\)\( T^{20} + \)\(50\!\cdots\!38\)\( p^{18} T^{22} + \)\(14\!\cdots\!01\)\( p^{36} T^{24} + \)\(38\!\cdots\!04\)\( p^{54} T^{26} + \)\(98\!\cdots\!62\)\( p^{72} T^{28} + \)\(23\!\cdots\!44\)\( p^{90} T^{30} + \)\(49\!\cdots\!23\)\( p^{108} T^{32} + \)\(98\!\cdots\!94\)\( p^{126} T^{34} + \)\(17\!\cdots\!55\)\( p^{144} T^{36} + 1965380062206758 p^{162} T^{38} + p^{180} T^{40} \) | |
| 43 | \( ( 1 - 26621029 T + 3731482435669156 T^{2} - \)\(77\!\cdots\!13\)\( T^{3} + \)\(64\!\cdots\!94\)\( T^{4} - \)\(11\!\cdots\!03\)\( T^{5} + \)\(71\!\cdots\!42\)\( T^{6} - \)\(10\!\cdots\!45\)\( T^{7} + \)\(55\!\cdots\!97\)\( T^{8} - \)\(70\!\cdots\!38\)\( T^{9} + \)\(32\!\cdots\!16\)\( T^{10} - \)\(70\!\cdots\!38\)\( p^{9} T^{11} + \)\(55\!\cdots\!97\)\( p^{18} T^{12} - \)\(10\!\cdots\!45\)\( p^{27} T^{13} + \)\(71\!\cdots\!42\)\( p^{36} T^{14} - \)\(11\!\cdots\!03\)\( p^{45} T^{15} + \)\(64\!\cdots\!94\)\( p^{54} T^{16} - \)\(77\!\cdots\!13\)\( p^{63} T^{17} + 3731482435669156 p^{72} T^{18} - 26621029 p^{81} T^{19} + p^{90} T^{20} )^{2} \) | |
| 47 | \( 1 + 14897963086138820 T^{2} + \)\(10\!\cdots\!98\)\( T^{4} + \)\(50\!\cdots\!12\)\( T^{6} + \)\(17\!\cdots\!97\)\( T^{8} + \)\(44\!\cdots\!12\)\( T^{10} + \)\(94\!\cdots\!68\)\( T^{12} + \)\(16\!\cdots\!32\)\( T^{14} + \)\(25\!\cdots\!70\)\( T^{16} + \)\(33\!\cdots\!20\)\( T^{18} + \)\(40\!\cdots\!52\)\( T^{20} + \)\(33\!\cdots\!20\)\( p^{18} T^{22} + \)\(25\!\cdots\!70\)\( p^{36} T^{24} + \)\(16\!\cdots\!32\)\( p^{54} T^{26} + \)\(94\!\cdots\!68\)\( p^{72} T^{28} + \)\(44\!\cdots\!12\)\( p^{90} T^{30} + \)\(17\!\cdots\!97\)\( p^{108} T^{32} + \)\(50\!\cdots\!12\)\( p^{126} T^{34} + \)\(10\!\cdots\!98\)\( p^{144} T^{36} + 14897963086138820 p^{162} T^{38} + p^{180} T^{40} \) | |
| 53 | \( ( 1 + 18214893 T + 22487841037440566 T^{2} + \)\(15\!\cdots\!39\)\( T^{3} + \)\(24\!\cdots\!46\)\( T^{4} - \)\(15\!\cdots\!77\)\( T^{5} + \)\(16\!\cdots\!52\)\( T^{6} - \)\(83\!\cdots\!01\)\( T^{7} + \)\(84\!\cdots\!53\)\( T^{8} - \)\(55\!\cdots\!50\)\( T^{9} + \)\(31\!\cdots\!56\)\( T^{10} - \)\(55\!\cdots\!50\)\( p^{9} T^{11} + \)\(84\!\cdots\!53\)\( p^{18} T^{12} - \)\(83\!\cdots\!01\)\( p^{27} T^{13} + \)\(16\!\cdots\!52\)\( p^{36} T^{14} - \)\(15\!\cdots\!77\)\( p^{45} T^{15} + \)\(24\!\cdots\!46\)\( p^{54} T^{16} + \)\(15\!\cdots\!39\)\( p^{63} T^{17} + 22487841037440566 p^{72} T^{18} + 18214893 p^{81} T^{19} + p^{90} T^{20} )^{2} \) | |
| 59 | \( 1 + 92619383932328741 T^{2} + \)\(41\!\cdots\!54\)\( T^{4} + \)\(11\!\cdots\!31\)\( T^{6} + \)\(25\!\cdots\!26\)\( T^{8} + \)\(42\!\cdots\!39\)\( T^{10} + \)\(59\!\cdots\!80\)\( T^{12} + \)\(73\!\cdots\!71\)\( T^{14} + \)\(82\!\cdots\!05\)\( T^{16} + \)\(82\!\cdots\!26\)\( T^{18} + \)\(75\!\cdots\!76\)\( T^{20} + \)\(82\!\cdots\!26\)\( p^{18} T^{22} + \)\(82\!\cdots\!05\)\( p^{36} T^{24} + \)\(73\!\cdots\!71\)\( p^{54} T^{26} + \)\(59\!\cdots\!80\)\( p^{72} T^{28} + \)\(42\!\cdots\!39\)\( p^{90} T^{30} + \)\(25\!\cdots\!26\)\( p^{108} T^{32} + \)\(11\!\cdots\!31\)\( p^{126} T^{34} + \)\(41\!\cdots\!54\)\( p^{144} T^{36} + 92619383932328741 p^{162} T^{38} + p^{180} T^{40} \) | |
| 61 | \( ( 1 - 268212896 T + 103138971494572405 T^{2} - \)\(19\!\cdots\!20\)\( T^{3} + \)\(45\!\cdots\!75\)\( T^{4} - \)\(68\!\cdots\!96\)\( T^{5} + \)\(12\!\cdots\!26\)\( T^{6} - \)\(15\!\cdots\!72\)\( T^{7} + \)\(22\!\cdots\!17\)\( T^{8} - \)\(24\!\cdots\!20\)\( T^{9} + \)\(30\!\cdots\!59\)\( T^{10} - \)\(24\!\cdots\!20\)\( p^{9} T^{11} + \)\(22\!\cdots\!17\)\( p^{18} T^{12} - \)\(15\!\cdots\!72\)\( p^{27} T^{13} + \)\(12\!\cdots\!26\)\( p^{36} T^{14} - \)\(68\!\cdots\!96\)\( p^{45} T^{15} + \)\(45\!\cdots\!75\)\( p^{54} T^{16} - \)\(19\!\cdots\!20\)\( p^{63} T^{17} + 103138971494572405 p^{72} T^{18} - 268212896 p^{81} T^{19} + p^{90} T^{20} )^{2} \) | |
| 67 | \( 1 + 246887561783242241 T^{2} + \)\(30\!\cdots\!18\)\( T^{4} + \)\(25\!\cdots\!47\)\( T^{6} + \)\(16\!\cdots\!42\)\( T^{8} + \)\(85\!\cdots\!75\)\( T^{10} + \)\(37\!\cdots\!64\)\( T^{12} + \)\(14\!\cdots\!99\)\( T^{14} + \)\(50\!\cdots\!05\)\( T^{16} + \)\(16\!\cdots\!18\)\( T^{18} + \)\(45\!\cdots\!24\)\( T^{20} + \)\(16\!\cdots\!18\)\( p^{18} T^{22} + \)\(50\!\cdots\!05\)\( p^{36} T^{24} + \)\(14\!\cdots\!99\)\( p^{54} T^{26} + \)\(37\!\cdots\!64\)\( p^{72} T^{28} + \)\(85\!\cdots\!75\)\( p^{90} T^{30} + \)\(16\!\cdots\!42\)\( p^{108} T^{32} + \)\(25\!\cdots\!47\)\( p^{126} T^{34} + \)\(30\!\cdots\!18\)\( p^{144} T^{36} + 246887561783242241 p^{162} T^{38} + p^{180} T^{40} \) | |
| 71 | \( 1 + 538838541563825777 T^{2} + \)\(14\!\cdots\!58\)\( T^{4} + \)\(24\!\cdots\!27\)\( T^{6} + \)\(32\!\cdots\!66\)\( T^{8} + \)\(33\!\cdots\!03\)\( T^{10} + \)\(28\!\cdots\!24\)\( T^{12} + \)\(20\!\cdots\!79\)\( T^{14} + \)\(13\!\cdots\!45\)\( T^{16} + \)\(73\!\cdots\!38\)\( T^{18} + \)\(35\!\cdots\!20\)\( T^{20} + \)\(73\!\cdots\!38\)\( p^{18} T^{22} + \)\(13\!\cdots\!45\)\( p^{36} T^{24} + \)\(20\!\cdots\!79\)\( p^{54} T^{26} + \)\(28\!\cdots\!24\)\( p^{72} T^{28} + \)\(33\!\cdots\!03\)\( p^{90} T^{30} + \)\(32\!\cdots\!66\)\( p^{108} T^{32} + \)\(24\!\cdots\!27\)\( p^{126} T^{34} + \)\(14\!\cdots\!58\)\( p^{144} T^{36} + 538838541563825777 p^{162} T^{38} + p^{180} T^{40} \) | |
| 73 | \( 1 + 948791083991149685 T^{2} + \)\(43\!\cdots\!42\)\( T^{4} + \)\(13\!\cdots\!83\)\( T^{6} + \)\(28\!\cdots\!62\)\( T^{8} + \)\(46\!\cdots\!91\)\( T^{10} + \)\(62\!\cdots\!64\)\( T^{12} + \)\(67\!\cdots\!51\)\( T^{14} + \)\(61\!\cdots\!85\)\( T^{16} + \)\(46\!\cdots\!86\)\( T^{18} + \)\(29\!\cdots\!72\)\( T^{20} + \)\(46\!\cdots\!86\)\( p^{18} T^{22} + \)\(61\!\cdots\!85\)\( p^{36} T^{24} + \)\(67\!\cdots\!51\)\( p^{54} T^{26} + \)\(62\!\cdots\!64\)\( p^{72} T^{28} + \)\(46\!\cdots\!91\)\( p^{90} T^{30} + \)\(28\!\cdots\!62\)\( p^{108} T^{32} + \)\(13\!\cdots\!83\)\( p^{126} T^{34} + \)\(43\!\cdots\!42\)\( p^{144} T^{36} + 948791083991149685 p^{162} T^{38} + p^{180} T^{40} \) | |
| 79 | \( ( 1 - 778351808 T + 1162270557708283606 T^{2} - \)\(70\!\cdots\!64\)\( T^{3} + \)\(59\!\cdots\!05\)\( T^{4} - \)\(29\!\cdots\!48\)\( T^{5} + \)\(18\!\cdots\!60\)\( T^{6} - \)\(76\!\cdots\!96\)\( T^{7} + \)\(37\!\cdots\!86\)\( T^{8} - \)\(13\!\cdots\!92\)\( T^{9} + \)\(53\!\cdots\!84\)\( T^{10} - \)\(13\!\cdots\!92\)\( p^{9} T^{11} + \)\(37\!\cdots\!86\)\( p^{18} T^{12} - \)\(76\!\cdots\!96\)\( p^{27} T^{13} + \)\(18\!\cdots\!60\)\( p^{36} T^{14} - \)\(29\!\cdots\!48\)\( p^{45} T^{15} + \)\(59\!\cdots\!05\)\( p^{54} T^{16} - \)\(70\!\cdots\!64\)\( p^{63} T^{17} + 1162270557708283606 p^{72} T^{18} - 778351808 p^{81} T^{19} + p^{90} T^{20} )^{2} \) | |
| 83 | \( 1 + 1541985118648764608 T^{2} + \)\(12\!\cdots\!42\)\( T^{4} + \)\(72\!\cdots\!88\)\( T^{6} + \)\(32\!\cdots\!73\)\( T^{8} + \)\(11\!\cdots\!56\)\( T^{10} + \)\(36\!\cdots\!48\)\( T^{12} + \)\(99\!\cdots\!60\)\( T^{14} + \)\(24\!\cdots\!38\)\( T^{16} + \)\(52\!\cdots\!12\)\( T^{18} + \)\(10\!\cdots\!16\)\( T^{20} + \)\(52\!\cdots\!12\)\( p^{18} T^{22} + \)\(24\!\cdots\!38\)\( p^{36} T^{24} + \)\(99\!\cdots\!60\)\( p^{54} T^{26} + \)\(36\!\cdots\!48\)\( p^{72} T^{28} + \)\(11\!\cdots\!56\)\( p^{90} T^{30} + \)\(32\!\cdots\!73\)\( p^{108} T^{32} + \)\(72\!\cdots\!88\)\( p^{126} T^{34} + \)\(12\!\cdots\!42\)\( p^{144} T^{36} + 1541985118648764608 p^{162} T^{38} + p^{180} T^{40} \) | |
| 89 | \( 1 + 3681988986653886869 T^{2} + \)\(66\!\cdots\!94\)\( T^{4} + \)\(78\!\cdots\!47\)\( T^{6} + \)\(66\!\cdots\!50\)\( T^{8} + \)\(42\!\cdots\!03\)\( T^{10} + \)\(21\!\cdots\!92\)\( T^{12} + \)\(88\!\cdots\!75\)\( T^{14} + \)\(29\!\cdots\!77\)\( T^{16} + \)\(88\!\cdots\!58\)\( T^{18} + \)\(28\!\cdots\!04\)\( T^{20} + \)\(88\!\cdots\!58\)\( p^{18} T^{22} + \)\(29\!\cdots\!77\)\( p^{36} T^{24} + \)\(88\!\cdots\!75\)\( p^{54} T^{26} + \)\(21\!\cdots\!92\)\( p^{72} T^{28} + \)\(42\!\cdots\!03\)\( p^{90} T^{30} + \)\(66\!\cdots\!50\)\( p^{108} T^{32} + \)\(78\!\cdots\!47\)\( p^{126} T^{34} + \)\(66\!\cdots\!94\)\( p^{144} T^{36} + 3681988986653886869 p^{162} T^{38} + p^{180} T^{40} \) | |
| 97 | \( 1 + 7554282362075248997 T^{2} + \)\(27\!\cdots\!98\)\( T^{4} + \)\(68\!\cdots\!55\)\( T^{6} + \)\(13\!\cdots\!02\)\( p T^{8} + \)\(19\!\cdots\!31\)\( T^{10} + \)\(24\!\cdots\!68\)\( T^{12} + \)\(28\!\cdots\!07\)\( T^{14} + \)\(28\!\cdots\!89\)\( T^{16} + \)\(24\!\cdots\!86\)\( T^{18} + \)\(19\!\cdots\!08\)\( T^{20} + \)\(24\!\cdots\!86\)\( p^{18} T^{22} + \)\(28\!\cdots\!89\)\( p^{36} T^{24} + \)\(28\!\cdots\!07\)\( p^{54} T^{26} + \)\(24\!\cdots\!68\)\( p^{72} T^{28} + \)\(19\!\cdots\!31\)\( p^{90} T^{30} + \)\(13\!\cdots\!02\)\( p^{109} T^{32} + \)\(68\!\cdots\!55\)\( p^{126} T^{34} + \)\(27\!\cdots\!98\)\( p^{144} T^{36} + 7554282362075248997 p^{162} T^{38} + p^{180} T^{40} \) | |
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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
−1.74451733796302865746292711444, −1.73564013312765298345913558245, −1.70848428039180612854078971095, −1.60780227677873792212305543978, −1.51207203583332517303263685083, −1.48413834546891760170289977474, −1.38113466754478444774452007422, −1.10908296915474580129948565733, −1.01603238640048720536987308497, −0.932941644921500772518140071669, −0.791093685801411966944518458910, −0.78640675477444238228283901288, −0.72517276833201205949597909060, −0.70871005264827107480786044906, −0.67703404490834649031348740942, −0.62411101472687854066428048216, −0.54847556976169284744047146749, −0.49087772301207862546912254038, −0.38516062550432490548226981807, −0.34683510779733686527024076195, −0.34147922415108211296722420949, −0.32864148517694965286591539972, −0.17489838438410025236386069558, −0.04180032332132954381315805149, −0.03196910568340086720941339439, 0.03196910568340086720941339439, 0.04180032332132954381315805149, 0.17489838438410025236386069558, 0.32864148517694965286591539972, 0.34147922415108211296722420949, 0.34683510779733686527024076195, 0.38516062550432490548226981807, 0.49087772301207862546912254038, 0.54847556976169284744047146749, 0.62411101472687854066428048216, 0.67703404490834649031348740942, 0.70871005264827107480786044906, 0.72517276833201205949597909060, 0.78640675477444238228283901288, 0.791093685801411966944518458910, 0.932941644921500772518140071669, 1.01603238640048720536987308497, 1.10908296915474580129948565733, 1.38113466754478444774452007422, 1.48413834546891760170289977474, 1.51207203583332517303263685083, 1.60780227677873792212305543978, 1.70848428039180612854078971095, 1.73564013312765298345913558245, 1.74451733796302865746292711444
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.