Dirichlet series
| L(s) = 1 | − 2·2-s + 7.90e3·3-s + 2·4-s + 1.92e5·5-s − 1.58e4·6-s − 3.86e5·7-s − 3.54e6·8-s + 3.12e7·9-s − 3.85e5·10-s − 5.39e8·11-s + 1.58e4·12-s − 6.81e8·13-s + 7.72e5·14-s + 1.52e9·15-s + 2.45e9·16-s + 1.31e10·17-s − 6.25e7·18-s + 3.85e5·20-s − 3.05e9·21-s + 1.07e9·22-s + 2.36e11·23-s − 2.80e10·24-s + 2.51e10·25-s + 1.36e9·26-s + 3.98e11·27-s − 7.72e5·28-s − 3.05e9·30-s + ⋯ |
| L(s) = 1 | − 0.00781·2-s + 1.20·3-s + 3.05e−5·4-s + 0.493·5-s − 0.00941·6-s − 0.0670·7-s − 0.211·8-s + 0.726·9-s − 0.00385·10-s − 2.51·11-s + 3.67e−5·12-s − 0.835·13-s + 0.000523·14-s + 0.595·15-s + 0.570·16-s + 1.88·17-s − 0.00567·18-s + 1.50e−5·20-s − 0.0807·21-s + 0.0196·22-s + 3.01·23-s − 0.254·24-s + 0.164·25-s + 0.00652·26-s + 1.41·27-s − 2.04e − 6·28-s − 0.00464·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(17-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{14}\right)^{s/2} \, \Gamma_{\C}(s+8)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(5^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.38218\times 10^{12}\) |
| Root analytic conductor: | \(2.84889\) |
| Motivic weight: | \(16\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 5^{14} ,\ ( \ : [8]^{14} ),\ 1 )\) |
Particular Values
| \(L(\frac{17}{2})\) | \(\approx\) | \(7.249008410\) |
| \(L(\frac12)\) | \(\approx\) | \(7.249008410\) |
| \(L(9)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 - 38576 p T + 96234443 p^{3} T^{2} - 241205908224 p^{7} T^{3} - 338717849967 p^{15} T^{4} + 9165234043951536 p^{17} T^{5} - 67068806418101712 p^{23} T^{6} + 333917496793142272 p^{29} T^{7} - 67068806418101712 p^{39} T^{8} + 9165234043951536 p^{49} T^{9} - 338717849967 p^{63} T^{10} - 241205908224 p^{71} T^{11} + 96234443 p^{83} T^{12} - 38576 p^{97} T^{13} + p^{112} T^{14} \) |
| good | 2 | \( 1 + p T + p T^{2} + 110671 p^{5} T^{3} - 38064095 p^{6} T^{4} + 2050699491 p^{7} T^{5} + 53131744041 p^{7} T^{6} + 9002806430343 p^{12} T^{7} + 295440976561329 p^{16} T^{8} - 9838399607681565 p^{19} T^{9} + 40830725384592945 p^{22} T^{10} - 4620674410741138905 p^{26} T^{11} - 35725531530859667495 p^{30} T^{12} + \)\(18\!\cdots\!55\)\( p^{33} T^{13} - \)\(42\!\cdots\!55\)\( p^{35} T^{14} + \)\(18\!\cdots\!55\)\( p^{49} T^{15} - 35725531530859667495 p^{62} T^{16} - 4620674410741138905 p^{74} T^{17} + 40830725384592945 p^{86} T^{18} - 9838399607681565 p^{99} T^{19} + 295440976561329 p^{112} T^{20} + 9002806430343 p^{124} T^{21} + 53131744041 p^{135} T^{22} + 2050699491 p^{151} T^{23} - 38064095 p^{166} T^{24} + 110671 p^{181} T^{25} + p^{193} T^{26} + p^{209} T^{27} + p^{224} T^{28} \) |
| 3 | \( 1 - 2636 p T + 3474248 p^{2} T^{2} - 14762823484 p^{3} T^{3} - 89143322705 p^{4} T^{4} + 1501393328708792 p^{6} T^{5} + 10823018464118113648 p^{8} T^{6} - \)\(14\!\cdots\!84\)\( p^{9} T^{7} + \)\(81\!\cdots\!21\)\( p^{10} T^{8} - \)\(42\!\cdots\!20\)\( p^{12} T^{9} + \)\(29\!\cdots\!20\)\( p^{14} T^{10} - \)\(28\!\cdots\!40\)\( p^{17} T^{11} + \)\(18\!\cdots\!95\)\( p^{20} T^{12} - \)\(30\!\cdots\!40\)\( p^{24} T^{13} + \)\(13\!\cdots\!60\)\( p^{28} T^{14} - \)\(30\!\cdots\!40\)\( p^{40} T^{15} + \)\(18\!\cdots\!95\)\( p^{52} T^{16} - \)\(28\!\cdots\!40\)\( p^{65} T^{17} + \)\(29\!\cdots\!20\)\( p^{78} T^{18} - \)\(42\!\cdots\!20\)\( p^{92} T^{19} + \)\(81\!\cdots\!21\)\( p^{106} T^{20} - \)\(14\!\cdots\!84\)\( p^{121} T^{21} + 10823018464118113648 p^{136} T^{22} + 1501393328708792 p^{150} T^{23} - 89143322705 p^{164} T^{24} - 14762823484 p^{179} T^{25} + 3474248 p^{194} T^{26} - 2636 p^{209} T^{27} + p^{224} T^{28} \) | |
| 7 | \( 1 + 386452 T + 74672574152 T^{2} + 13670552451431962236 p T^{3} + \)\(39\!\cdots\!55\)\( p^{2} T^{4} - \)\(98\!\cdots\!84\)\( p^{3} T^{5} + \)\(13\!\cdots\!88\)\( p^{4} T^{6} + \)\(88\!\cdots\!84\)\( p^{5} T^{7} + \)\(41\!\cdots\!41\)\( p^{6} T^{8} - \)\(25\!\cdots\!60\)\( p^{9} T^{9} + \)\(50\!\cdots\!60\)\( p^{11} T^{10} - \)\(48\!\cdots\!20\)\( p^{12} T^{11} - \)\(72\!\cdots\!45\)\( p^{14} T^{12} - \)\(31\!\cdots\!40\)\( p^{16} T^{13} + \)\(88\!\cdots\!40\)\( p^{18} T^{14} - \)\(31\!\cdots\!40\)\( p^{32} T^{15} - \)\(72\!\cdots\!45\)\( p^{46} T^{16} - \)\(48\!\cdots\!20\)\( p^{60} T^{17} + \)\(50\!\cdots\!60\)\( p^{75} T^{18} - \)\(25\!\cdots\!60\)\( p^{89} T^{19} + \)\(41\!\cdots\!41\)\( p^{102} T^{20} + \)\(88\!\cdots\!84\)\( p^{117} T^{21} + \)\(13\!\cdots\!88\)\( p^{132} T^{22} - \)\(98\!\cdots\!84\)\( p^{147} T^{23} + \)\(39\!\cdots\!55\)\( p^{162} T^{24} + 13670552451431962236 p^{177} T^{25} + 74672574152 p^{192} T^{26} + 386452 p^{208} T^{27} + p^{224} T^{28} \) | |
| 11 | \( ( 1 + 269570816 T + 16428642964398941 p T^{2} + \)\(23\!\cdots\!16\)\( p T^{3} + \)\(90\!\cdots\!91\)\( p^{3} T^{4} + \)\(49\!\cdots\!08\)\( p^{3} T^{5} + \)\(34\!\cdots\!23\)\( p^{4} T^{6} + \)\(38\!\cdots\!48\)\( p^{5} T^{7} + \)\(34\!\cdots\!23\)\( p^{20} T^{8} + \)\(49\!\cdots\!08\)\( p^{35} T^{9} + \)\(90\!\cdots\!91\)\( p^{51} T^{10} + \)\(23\!\cdots\!16\)\( p^{65} T^{11} + 16428642964398941 p^{81} T^{12} + 269570816 p^{96} T^{13} + p^{112} T^{14} )^{2} \) | |
| 13 | \( 1 + 681358582 T + 232124758632525362 T^{2} + \)\(58\!\cdots\!74\)\( p T^{3} + \)\(46\!\cdots\!55\)\( p^{2} T^{4} + \)\(66\!\cdots\!84\)\( p^{3} T^{5} + \)\(71\!\cdots\!88\)\( p^{4} T^{6} + \)\(14\!\cdots\!76\)\( p^{5} T^{7} + \)\(19\!\cdots\!61\)\( p^{6} T^{8} - \)\(24\!\cdots\!70\)\( p^{8} T^{9} + \)\(11\!\cdots\!30\)\( p^{8} T^{10} + \)\(64\!\cdots\!10\)\( p^{9} T^{11} - \)\(15\!\cdots\!45\)\( p^{10} T^{12} - \)\(71\!\cdots\!20\)\( p^{11} T^{13} + \)\(15\!\cdots\!60\)\( p^{12} T^{14} - \)\(71\!\cdots\!20\)\( p^{27} T^{15} - \)\(15\!\cdots\!45\)\( p^{42} T^{16} + \)\(64\!\cdots\!10\)\( p^{57} T^{17} + \)\(11\!\cdots\!30\)\( p^{72} T^{18} - \)\(24\!\cdots\!70\)\( p^{88} T^{19} + \)\(19\!\cdots\!61\)\( p^{102} T^{20} + \)\(14\!\cdots\!76\)\( p^{117} T^{21} + \)\(71\!\cdots\!88\)\( p^{132} T^{22} + \)\(66\!\cdots\!84\)\( p^{147} T^{23} + \)\(46\!\cdots\!55\)\( p^{162} T^{24} + \)\(58\!\cdots\!74\)\( p^{177} T^{25} + 232124758632525362 p^{192} T^{26} + 681358582 p^{208} T^{27} + p^{224} T^{28} \) | |
| 17 | \( 1 - 13124182498 T + 86122083120404760002 T^{2} - \)\(94\!\cdots\!38\)\( T^{3} + \)\(72\!\cdots\!35\)\( p T^{4} - \)\(79\!\cdots\!32\)\( T^{5} + \)\(42\!\cdots\!68\)\( T^{6} - \)\(38\!\cdots\!92\)\( T^{7} + \)\(17\!\cdots\!89\)\( T^{8} + \)\(42\!\cdots\!30\)\( T^{9} - \)\(48\!\cdots\!70\)\( T^{10} + \)\(67\!\cdots\!30\)\( T^{11} - \)\(12\!\cdots\!05\)\( T^{12} + \)\(88\!\cdots\!60\)\( T^{13} - \)\(47\!\cdots\!40\)\( T^{14} + \)\(88\!\cdots\!60\)\( p^{16} T^{15} - \)\(12\!\cdots\!05\)\( p^{32} T^{16} + \)\(67\!\cdots\!30\)\( p^{48} T^{17} - \)\(48\!\cdots\!70\)\( p^{64} T^{18} + \)\(42\!\cdots\!30\)\( p^{80} T^{19} + \)\(17\!\cdots\!89\)\( p^{96} T^{20} - \)\(38\!\cdots\!92\)\( p^{112} T^{21} + \)\(42\!\cdots\!68\)\( p^{128} T^{22} - \)\(79\!\cdots\!32\)\( p^{144} T^{23} + \)\(72\!\cdots\!35\)\( p^{161} T^{24} - \)\(94\!\cdots\!38\)\( p^{176} T^{25} + 86122083120404760002 p^{192} T^{26} - 13124182498 p^{208} T^{27} + p^{224} T^{28} \) | |
| 19 | \( 1 - \)\(23\!\cdots\!34\)\( T^{2} + \)\(26\!\cdots\!51\)\( T^{4} - \)\(17\!\cdots\!24\)\( T^{6} + \)\(80\!\cdots\!21\)\( T^{8} - \)\(27\!\cdots\!02\)\( T^{10} + \)\(78\!\cdots\!43\)\( T^{12} - \)\(21\!\cdots\!52\)\( T^{14} + \)\(78\!\cdots\!43\)\( p^{32} T^{16} - \)\(27\!\cdots\!02\)\( p^{64} T^{18} + \)\(80\!\cdots\!21\)\( p^{96} T^{20} - \)\(17\!\cdots\!24\)\( p^{128} T^{22} + \)\(26\!\cdots\!51\)\( p^{160} T^{24} - \)\(23\!\cdots\!34\)\( p^{192} T^{26} + p^{224} T^{28} \) | |
| 23 | \( 1 - 236373052228 T + \)\(27\!\cdots\!92\)\( T^{2} - \)\(35\!\cdots\!08\)\( T^{3} + \)\(47\!\cdots\!95\)\( T^{4} - \)\(43\!\cdots\!72\)\( T^{5} + \)\(33\!\cdots\!08\)\( T^{6} - \)\(30\!\cdots\!92\)\( T^{7} + \)\(18\!\cdots\!69\)\( T^{8} - \)\(23\!\cdots\!20\)\( T^{9} - \)\(38\!\cdots\!20\)\( T^{10} + \)\(83\!\cdots\!80\)\( T^{11} - \)\(14\!\cdots\!05\)\( T^{12} + \)\(13\!\cdots\!60\)\( T^{13} - \)\(96\!\cdots\!40\)\( T^{14} + \)\(13\!\cdots\!60\)\( p^{16} T^{15} - \)\(14\!\cdots\!05\)\( p^{32} T^{16} + \)\(83\!\cdots\!80\)\( p^{48} T^{17} - \)\(38\!\cdots\!20\)\( p^{64} T^{18} - \)\(23\!\cdots\!20\)\( p^{80} T^{19} + \)\(18\!\cdots\!69\)\( p^{96} T^{20} - \)\(30\!\cdots\!92\)\( p^{112} T^{21} + \)\(33\!\cdots\!08\)\( p^{128} T^{22} - \)\(43\!\cdots\!72\)\( p^{144} T^{23} + \)\(47\!\cdots\!95\)\( p^{160} T^{24} - \)\(35\!\cdots\!08\)\( p^{176} T^{25} + \)\(27\!\cdots\!92\)\( p^{192} T^{26} - 236373052228 p^{208} T^{27} + p^{224} T^{28} \) | |
| 29 | \( 1 - \)\(18\!\cdots\!94\)\( T^{2} + \)\(16\!\cdots\!31\)\( T^{4} - \)\(10\!\cdots\!04\)\( T^{6} + \)\(51\!\cdots\!81\)\( T^{8} - \)\(20\!\cdots\!02\)\( T^{10} + \)\(64\!\cdots\!63\)\( T^{12} - \)\(17\!\cdots\!12\)\( T^{14} + \)\(64\!\cdots\!63\)\( p^{32} T^{16} - \)\(20\!\cdots\!02\)\( p^{64} T^{18} + \)\(51\!\cdots\!81\)\( p^{96} T^{20} - \)\(10\!\cdots\!04\)\( p^{128} T^{22} + \)\(16\!\cdots\!31\)\( p^{160} T^{24} - \)\(18\!\cdots\!94\)\( p^{192} T^{26} + p^{224} T^{28} \) | |
| 31 | \( ( 1 + 552611537476 T + \)\(26\!\cdots\!71\)\( T^{2} + \)\(52\!\cdots\!56\)\( T^{3} + \)\(27\!\cdots\!61\)\( T^{4} - \)\(13\!\cdots\!92\)\( p T^{5} + \)\(18\!\cdots\!43\)\( p^{2} T^{6} - \)\(24\!\cdots\!12\)\( p^{3} T^{7} + \)\(18\!\cdots\!43\)\( p^{18} T^{8} - \)\(13\!\cdots\!92\)\( p^{33} T^{9} + \)\(27\!\cdots\!61\)\( p^{48} T^{10} + \)\(52\!\cdots\!56\)\( p^{64} T^{11} + \)\(26\!\cdots\!71\)\( p^{80} T^{12} + 552611537476 p^{96} T^{13} + p^{112} T^{14} )^{2} \) | |
| 37 | \( 1 + 9027827087002 T + \)\(40\!\cdots\!02\)\( T^{2} + \)\(98\!\cdots\!82\)\( T^{3} + \)\(83\!\cdots\!95\)\( T^{4} + \)\(51\!\cdots\!28\)\( T^{5} + \)\(60\!\cdots\!28\)\( T^{6} + \)\(34\!\cdots\!48\)\( T^{7} + \)\(13\!\cdots\!49\)\( T^{8} + \)\(33\!\cdots\!30\)\( T^{9} + \)\(70\!\cdots\!30\)\( T^{10} + \)\(14\!\cdots\!30\)\( T^{11} + \)\(98\!\cdots\!95\)\( T^{12} + \)\(73\!\cdots\!60\)\( T^{13} + \)\(32\!\cdots\!60\)\( T^{14} + \)\(73\!\cdots\!60\)\( p^{16} T^{15} + \)\(98\!\cdots\!95\)\( p^{32} T^{16} + \)\(14\!\cdots\!30\)\( p^{48} T^{17} + \)\(70\!\cdots\!30\)\( p^{64} T^{18} + \)\(33\!\cdots\!30\)\( p^{80} T^{19} + \)\(13\!\cdots\!49\)\( p^{96} T^{20} + \)\(34\!\cdots\!48\)\( p^{112} T^{21} + \)\(60\!\cdots\!28\)\( p^{128} T^{22} + \)\(51\!\cdots\!28\)\( p^{144} T^{23} + \)\(83\!\cdots\!95\)\( p^{160} T^{24} + \)\(98\!\cdots\!82\)\( p^{176} T^{25} + \)\(40\!\cdots\!02\)\( p^{192} T^{26} + 9027827087002 p^{208} T^{27} + p^{224} T^{28} \) | |
| 41 | \( ( 1 - 19771950742144 T + \)\(44\!\cdots\!31\)\( T^{2} - \)\(58\!\cdots\!04\)\( T^{3} + \)\(78\!\cdots\!81\)\( T^{4} - \)\(78\!\cdots\!52\)\( T^{5} + \)\(79\!\cdots\!63\)\( T^{6} - \)\(63\!\cdots\!12\)\( T^{7} + \)\(79\!\cdots\!63\)\( p^{16} T^{8} - \)\(78\!\cdots\!52\)\( p^{32} T^{9} + \)\(78\!\cdots\!81\)\( p^{48} T^{10} - \)\(58\!\cdots\!04\)\( p^{64} T^{11} + \)\(44\!\cdots\!31\)\( p^{80} T^{12} - 19771950742144 p^{96} T^{13} + p^{112} T^{14} )^{2} \) | |
| 43 | \( 1 + 63017457929452 T + \)\(19\!\cdots\!52\)\( T^{2} + \)\(45\!\cdots\!52\)\( T^{3} + \)\(90\!\cdots\!95\)\( T^{4} + \)\(16\!\cdots\!88\)\( T^{5} + \)\(27\!\cdots\!88\)\( T^{6} + \)\(42\!\cdots\!88\)\( T^{7} + \)\(59\!\cdots\!09\)\( T^{8} + \)\(80\!\cdots\!80\)\( T^{9} + \)\(10\!\cdots\!80\)\( T^{10} + \)\(12\!\cdots\!80\)\( T^{11} + \)\(15\!\cdots\!95\)\( T^{12} + \)\(17\!\cdots\!60\)\( T^{13} + \)\(20\!\cdots\!60\)\( T^{14} + \)\(17\!\cdots\!60\)\( p^{16} T^{15} + \)\(15\!\cdots\!95\)\( p^{32} T^{16} + \)\(12\!\cdots\!80\)\( p^{48} T^{17} + \)\(10\!\cdots\!80\)\( p^{64} T^{18} + \)\(80\!\cdots\!80\)\( p^{80} T^{19} + \)\(59\!\cdots\!09\)\( p^{96} T^{20} + \)\(42\!\cdots\!88\)\( p^{112} T^{21} + \)\(27\!\cdots\!88\)\( p^{128} T^{22} + \)\(16\!\cdots\!88\)\( p^{144} T^{23} + \)\(90\!\cdots\!95\)\( p^{160} T^{24} + \)\(45\!\cdots\!52\)\( p^{176} T^{25} + \)\(19\!\cdots\!52\)\( p^{192} T^{26} + 63017457929452 p^{208} T^{27} + p^{224} T^{28} \) | |
| 47 | \( 1 + 2826187575452 T + \)\(39\!\cdots\!52\)\( T^{2} - \)\(92\!\cdots\!08\)\( T^{3} + \)\(76\!\cdots\!95\)\( T^{4} + \)\(22\!\cdots\!08\)\( T^{5} + \)\(63\!\cdots\!08\)\( T^{6} + \)\(69\!\cdots\!68\)\( T^{7} - \)\(18\!\cdots\!71\)\( T^{8} + \)\(10\!\cdots\!80\)\( T^{9} + \)\(48\!\cdots\!80\)\( T^{10} + \)\(13\!\cdots\!40\)\( p T^{11} + \)\(13\!\cdots\!55\)\( p^{2} T^{12} - \)\(90\!\cdots\!80\)\( p^{3} T^{13} + \)\(21\!\cdots\!60\)\( p^{4} T^{14} - \)\(90\!\cdots\!80\)\( p^{19} T^{15} + \)\(13\!\cdots\!55\)\( p^{34} T^{16} + \)\(13\!\cdots\!40\)\( p^{49} T^{17} + \)\(48\!\cdots\!80\)\( p^{64} T^{18} + \)\(10\!\cdots\!80\)\( p^{80} T^{19} - \)\(18\!\cdots\!71\)\( p^{96} T^{20} + \)\(69\!\cdots\!68\)\( p^{112} T^{21} + \)\(63\!\cdots\!08\)\( p^{128} T^{22} + \)\(22\!\cdots\!08\)\( p^{144} T^{23} + \)\(76\!\cdots\!95\)\( p^{160} T^{24} - \)\(92\!\cdots\!08\)\( p^{176} T^{25} + \)\(39\!\cdots\!52\)\( p^{192} T^{26} + 2826187575452 p^{208} T^{27} + p^{224} T^{28} \) | |
| 53 | \( 1 + 275142037498442 T + \)\(37\!\cdots\!82\)\( T^{2} + \)\(41\!\cdots\!82\)\( T^{3} + \)\(38\!\cdots\!95\)\( T^{4} + \)\(28\!\cdots\!68\)\( T^{5} + \)\(36\!\cdots\!76\)\( p T^{6} + \)\(11\!\cdots\!28\)\( T^{7} + \)\(64\!\cdots\!29\)\( T^{8} + \)\(74\!\cdots\!10\)\( p T^{9} + \)\(28\!\cdots\!30\)\( T^{10} + \)\(22\!\cdots\!30\)\( T^{11} + \)\(17\!\cdots\!95\)\( T^{12} + \)\(13\!\cdots\!60\)\( T^{13} + \)\(85\!\cdots\!60\)\( T^{14} + \)\(13\!\cdots\!60\)\( p^{16} T^{15} + \)\(17\!\cdots\!95\)\( p^{32} T^{16} + \)\(22\!\cdots\!30\)\( p^{48} T^{17} + \)\(28\!\cdots\!30\)\( p^{64} T^{18} + \)\(74\!\cdots\!10\)\( p^{81} T^{19} + \)\(64\!\cdots\!29\)\( p^{96} T^{20} + \)\(11\!\cdots\!28\)\( p^{112} T^{21} + \)\(36\!\cdots\!76\)\( p^{129} T^{22} + \)\(28\!\cdots\!68\)\( p^{144} T^{23} + \)\(38\!\cdots\!95\)\( p^{160} T^{24} + \)\(41\!\cdots\!82\)\( p^{176} T^{25} + \)\(37\!\cdots\!82\)\( p^{192} T^{26} + 275142037498442 p^{208} T^{27} + p^{224} T^{28} \) | |
| 59 | \( 1 - \)\(10\!\cdots\!74\)\( T^{2} + \)\(64\!\cdots\!71\)\( T^{4} - \)\(29\!\cdots\!44\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} - \)\(33\!\cdots\!02\)\( T^{10} + \)\(88\!\cdots\!23\)\( T^{12} - \)\(20\!\cdots\!92\)\( T^{14} + \)\(88\!\cdots\!23\)\( p^{32} T^{16} - \)\(33\!\cdots\!02\)\( p^{64} T^{18} + \)\(10\!\cdots\!61\)\( p^{96} T^{20} - \)\(29\!\cdots\!44\)\( p^{128} T^{22} + \)\(64\!\cdots\!71\)\( p^{160} T^{24} - \)\(10\!\cdots\!74\)\( p^{192} T^{26} + p^{224} T^{28} \) | |
| 61 | \( ( 1 - 341817264197984 T + \)\(20\!\cdots\!51\)\( T^{2} - \)\(55\!\cdots\!24\)\( T^{3} + \)\(19\!\cdots\!21\)\( T^{4} - \)\(42\!\cdots\!52\)\( T^{5} + \)\(11\!\cdots\!43\)\( T^{6} - \)\(19\!\cdots\!52\)\( T^{7} + \)\(11\!\cdots\!43\)\( p^{16} T^{8} - \)\(42\!\cdots\!52\)\( p^{32} T^{9} + \)\(19\!\cdots\!21\)\( p^{48} T^{10} - \)\(55\!\cdots\!24\)\( p^{64} T^{11} + \)\(20\!\cdots\!51\)\( p^{80} T^{12} - 341817264197984 p^{96} T^{13} + p^{112} T^{14} )^{2} \) | |
| 67 | \( 1 + 181547742064252 T + \)\(16\!\cdots\!52\)\( T^{2} - \)\(12\!\cdots\!88\)\( T^{3} - \)\(36\!\cdots\!05\)\( T^{4} + \)\(21\!\cdots\!68\)\( T^{5} + \)\(12\!\cdots\!68\)\( T^{6} + \)\(41\!\cdots\!08\)\( T^{7} - \)\(32\!\cdots\!11\)\( T^{8} - \)\(11\!\cdots\!20\)\( T^{9} + \)\(31\!\cdots\!80\)\( T^{10} + \)\(40\!\cdots\!40\)\( p T^{11} + \)\(10\!\cdots\!95\)\( T^{12} - \)\(28\!\cdots\!40\)\( T^{13} - \)\(16\!\cdots\!40\)\( T^{14} - \)\(28\!\cdots\!40\)\( p^{16} T^{15} + \)\(10\!\cdots\!95\)\( p^{32} T^{16} + \)\(40\!\cdots\!40\)\( p^{49} T^{17} + \)\(31\!\cdots\!80\)\( p^{64} T^{18} - \)\(11\!\cdots\!20\)\( p^{80} T^{19} - \)\(32\!\cdots\!11\)\( p^{96} T^{20} + \)\(41\!\cdots\!08\)\( p^{112} T^{21} + \)\(12\!\cdots\!68\)\( p^{128} T^{22} + \)\(21\!\cdots\!68\)\( p^{144} T^{23} - \)\(36\!\cdots\!05\)\( p^{160} T^{24} - \)\(12\!\cdots\!88\)\( p^{176} T^{25} + \)\(16\!\cdots\!52\)\( p^{192} T^{26} + 181547742064252 p^{208} T^{27} + p^{224} T^{28} \) | |
| 71 | \( ( 1 + 668772005549396 T + \)\(23\!\cdots\!11\)\( T^{2} + \)\(14\!\cdots\!16\)\( T^{3} + \)\(25\!\cdots\!41\)\( T^{4} + \)\(13\!\cdots\!48\)\( T^{5} + \)\(16\!\cdots\!83\)\( T^{6} + \)\(74\!\cdots\!28\)\( T^{7} + \)\(16\!\cdots\!83\)\( p^{16} T^{8} + \)\(13\!\cdots\!48\)\( p^{32} T^{9} + \)\(25\!\cdots\!41\)\( p^{48} T^{10} + \)\(14\!\cdots\!16\)\( p^{64} T^{11} + \)\(23\!\cdots\!11\)\( p^{80} T^{12} + 668772005549396 p^{96} T^{13} + p^{112} T^{14} )^{2} \) | |
| 73 | \( 1 - 778038215296478 T + \)\(30\!\cdots\!42\)\( T^{2} + \)\(44\!\cdots\!42\)\( T^{3} - \)\(80\!\cdots\!05\)\( T^{4} - \)\(60\!\cdots\!72\)\( T^{5} + \)\(34\!\cdots\!08\)\( T^{6} - \)\(38\!\cdots\!92\)\( T^{7} - \)\(97\!\cdots\!31\)\( T^{8} + \)\(26\!\cdots\!30\)\( T^{9} - \)\(73\!\cdots\!70\)\( T^{10} - \)\(12\!\cdots\!70\)\( T^{11} + \)\(11\!\cdots\!95\)\( T^{12} + \)\(78\!\cdots\!60\)\( T^{13} - \)\(10\!\cdots\!40\)\( T^{14} + \)\(78\!\cdots\!60\)\( p^{16} T^{15} + \)\(11\!\cdots\!95\)\( p^{32} T^{16} - \)\(12\!\cdots\!70\)\( p^{48} T^{17} - \)\(73\!\cdots\!70\)\( p^{64} T^{18} + \)\(26\!\cdots\!30\)\( p^{80} T^{19} - \)\(97\!\cdots\!31\)\( p^{96} T^{20} - \)\(38\!\cdots\!92\)\( p^{112} T^{21} + \)\(34\!\cdots\!08\)\( p^{128} T^{22} - \)\(60\!\cdots\!72\)\( p^{144} T^{23} - \)\(80\!\cdots\!05\)\( p^{160} T^{24} + \)\(44\!\cdots\!42\)\( p^{176} T^{25} + \)\(30\!\cdots\!42\)\( p^{192} T^{26} - 778038215296478 p^{208} T^{27} + p^{224} T^{28} \) | |
| 79 | \( 1 - \)\(21\!\cdots\!94\)\( T^{2} + \)\(22\!\cdots\!31\)\( T^{4} - \)\(15\!\cdots\!04\)\( T^{6} + \)\(79\!\cdots\!81\)\( T^{8} - \)\(31\!\cdots\!02\)\( T^{10} + \)\(98\!\cdots\!63\)\( T^{12} - \)\(25\!\cdots\!12\)\( T^{14} + \)\(98\!\cdots\!63\)\( p^{32} T^{16} - \)\(31\!\cdots\!02\)\( p^{64} T^{18} + \)\(79\!\cdots\!81\)\( p^{96} T^{20} - \)\(15\!\cdots\!04\)\( p^{128} T^{22} + \)\(22\!\cdots\!31\)\( p^{160} T^{24} - \)\(21\!\cdots\!94\)\( p^{192} T^{26} + p^{224} T^{28} \) | |
| 83 | \( 1 - 13357181209550188 T + \)\(89\!\cdots\!72\)\( T^{2} - \)\(42\!\cdots\!28\)\( T^{3} + \)\(17\!\cdots\!95\)\( T^{4} - \)\(65\!\cdots\!92\)\( T^{5} + \)\(22\!\cdots\!48\)\( T^{6} - \)\(71\!\cdots\!52\)\( T^{7} + \)\(21\!\cdots\!89\)\( T^{8} - \)\(61\!\cdots\!20\)\( T^{9} + \)\(20\!\cdots\!60\)\( p T^{10} - \)\(43\!\cdots\!20\)\( T^{11} + \)\(10\!\cdots\!95\)\( T^{12} - \)\(25\!\cdots\!40\)\( T^{13} + \)\(59\!\cdots\!60\)\( T^{14} - \)\(25\!\cdots\!40\)\( p^{16} T^{15} + \)\(10\!\cdots\!95\)\( p^{32} T^{16} - \)\(43\!\cdots\!20\)\( p^{48} T^{17} + \)\(20\!\cdots\!60\)\( p^{65} T^{18} - \)\(61\!\cdots\!20\)\( p^{80} T^{19} + \)\(21\!\cdots\!89\)\( p^{96} T^{20} - \)\(71\!\cdots\!52\)\( p^{112} T^{21} + \)\(22\!\cdots\!48\)\( p^{128} T^{22} - \)\(65\!\cdots\!92\)\( p^{144} T^{23} + \)\(17\!\cdots\!95\)\( p^{160} T^{24} - \)\(42\!\cdots\!28\)\( p^{176} T^{25} + \)\(89\!\cdots\!72\)\( p^{192} T^{26} - 13357181209550188 p^{208} T^{27} + p^{224} T^{28} \) | |
| 89 | \( 1 - \)\(11\!\cdots\!54\)\( T^{2} + \)\(65\!\cdots\!11\)\( T^{4} - \)\(26\!\cdots\!84\)\( T^{6} + \)\(78\!\cdots\!41\)\( T^{8} - \)\(19\!\cdots\!02\)\( T^{10} + \)\(38\!\cdots\!83\)\( T^{12} - \)\(65\!\cdots\!72\)\( T^{14} + \)\(38\!\cdots\!83\)\( p^{32} T^{16} - \)\(19\!\cdots\!02\)\( p^{64} T^{18} + \)\(78\!\cdots\!41\)\( p^{96} T^{20} - \)\(26\!\cdots\!84\)\( p^{128} T^{22} + \)\(65\!\cdots\!11\)\( p^{160} T^{24} - \)\(11\!\cdots\!54\)\( p^{192} T^{26} + p^{224} T^{28} \) | |
| 97 | \( 1 + 3303551663290402 T + \)\(54\!\cdots\!02\)\( T^{2} + \)\(86\!\cdots\!42\)\( T^{3} - \)\(69\!\cdots\!05\)\( T^{4} - \)\(68\!\cdots\!92\)\( T^{5} + \)\(19\!\cdots\!08\)\( T^{6} - \)\(65\!\cdots\!32\)\( T^{7} + \)\(82\!\cdots\!29\)\( T^{8} + \)\(45\!\cdots\!30\)\( T^{9} - \)\(62\!\cdots\!70\)\( T^{10} + \)\(25\!\cdots\!30\)\( T^{11} + \)\(17\!\cdots\!95\)\( T^{12} - \)\(15\!\cdots\!40\)\( T^{13} + \)\(19\!\cdots\!60\)\( T^{14} - \)\(15\!\cdots\!40\)\( p^{16} T^{15} + \)\(17\!\cdots\!95\)\( p^{32} T^{16} + \)\(25\!\cdots\!30\)\( p^{48} T^{17} - \)\(62\!\cdots\!70\)\( p^{64} T^{18} + \)\(45\!\cdots\!30\)\( p^{80} T^{19} + \)\(82\!\cdots\!29\)\( p^{96} T^{20} - \)\(65\!\cdots\!32\)\( p^{112} T^{21} + \)\(19\!\cdots\!08\)\( p^{128} T^{22} - \)\(68\!\cdots\!92\)\( p^{144} T^{23} - \)\(69\!\cdots\!05\)\( p^{160} T^{24} + \)\(86\!\cdots\!42\)\( p^{176} T^{25} + \)\(54\!\cdots\!02\)\( p^{192} T^{26} + 3303551663290402 p^{208} T^{27} + p^{224} T^{28} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.06465533628580331732801507528, −5.02198372615212692129361508528, −4.94835694151311107726804212224, −4.29832200705151142134872233598, −4.17469482125694636681492872773, −4.02217683579096243043307409786, −3.66519258132799312901559417841, −3.47661038553769700523810931346, −3.25273147713355816423070396581, −3.20519801970275065866420826922, −3.05193745574313408609632618834, −2.88954408001752653467619848818, −2.63260052953291424265924496484, −2.62777263027109366004750586380, −2.26489030504391811823621079441, −2.03488538909236594379878829824, −1.76602115989685963906017089087, −1.66740665256179102188107741316, −1.58321768430100799345468796020, −0.996232177040116314764800138517, −0.993651271522980028498778061184, −0.960897831969735379012764956905, −0.41931919893275409725806967857, −0.39641366595429618069398260963, −0.16207162313868732527680386068, 0.16207162313868732527680386068, 0.39641366595429618069398260963, 0.41931919893275409725806967857, 0.960897831969735379012764956905, 0.993651271522980028498778061184, 0.996232177040116314764800138517, 1.58321768430100799345468796020, 1.66740665256179102188107741316, 1.76602115989685963906017089087, 2.03488538909236594379878829824, 2.26489030504391811823621079441, 2.62777263027109366004750586380, 2.63260052953291424265924496484, 2.88954408001752653467619848818, 3.05193745574313408609632618834, 3.20519801970275065866420826922, 3.25273147713355816423070396581, 3.47661038553769700523810931346, 3.66519258132799312901559417841, 4.02217683579096243043307409786, 4.17469482125694636681492872773, 4.29832200705151142134872233598, 4.94835694151311107726804212224, 5.02198372615212692129361508528, 5.06465533628580331732801507528
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.