Dirichlet series
| L(s) = 1 | + 40·5-s − 34·7-s + 64·11-s − 48·13-s − 212·17-s + 456·19-s + 166·23-s + 700·25-s + 110·29-s − 160·31-s − 1.36e3·35-s + 104·37-s + 280·41-s + 136·43-s + 594·47-s + 1.40e3·49-s − 752·53-s + 2.56e3·55-s + 24·59-s − 950·61-s − 1.92e3·65-s − 426·67-s + 136·71-s + 308·73-s − 2.17e3·77-s − 764·79-s + 1.36e3·83-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 1.83·7-s + 1.75·11-s − 1.02·13-s − 3.02·17-s + 5.50·19-s + 1.50·23-s + 28/5·25-s + 0.704·29-s − 0.926·31-s − 6.56·35-s + 0.462·37-s + 1.06·41-s + 0.482·43-s + 1.84·47-s + 4.09·49-s − 1.94·53-s + 6.27·55-s + 0.0529·59-s − 1.99·61-s − 3.66·65-s − 0.776·67-s + 0.227·71-s + 0.493·73-s − 3.22·77-s − 1.08·79-s + 1.80·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{48} \cdot 3^{48} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.38987\times 10^{28}\) |
| Root analytic conductor: | \(7.98261\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{48} \cdot 3^{48} \cdot 5^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(187.9679581\) |
| \(L(\frac12)\) | \(\approx\) | \(187.9679581\) |
| \(L(\frac{5}{2})\) | 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 | \( 1 \) | |
| 5 | \( ( 1 - p T + p^{2} T^{2} )^{8} \) | |
| good | 7 | \( 1 + 34 T - 247 T^{2} - 21134 T^{3} + 48576 T^{4} + 10359770 T^{5} + 90547945 T^{6} - 1690005466 T^{7} - 57651574555 T^{8} - 848926387420 T^{9} + 17202085010120 T^{10} + 757104043681060 T^{11} + 369908587147986 T^{12} - 276782630996841932 T^{13} - 2528111715924452682 T^{14} + 45100990194032721336 T^{15} + \)\(14\!\cdots\!16\)\( T^{16} + 45100990194032721336 p^{3} T^{17} - 2528111715924452682 p^{6} T^{18} - 276782630996841932 p^{9} T^{19} + 369908587147986 p^{12} T^{20} + 757104043681060 p^{15} T^{21} + 17202085010120 p^{18} T^{22} - 848926387420 p^{21} T^{23} - 57651574555 p^{24} T^{24} - 1690005466 p^{27} T^{25} + 90547945 p^{30} T^{26} + 10359770 p^{33} T^{27} + 48576 p^{36} T^{28} - 21134 p^{39} T^{29} - 247 p^{42} T^{30} + 34 p^{45} T^{31} + p^{48} T^{32} \) |
| 11 | \( 1 - 64 T - 3375 T^{2} + 15640 p T^{3} + 12542819 T^{4} - 288580560 T^{5} - 27115333536 T^{6} - 60493229664 T^{7} + 43588260389419 T^{8} + 1037408985847920 T^{9} - 3476229367145225 p T^{10} - 2492189862796374616 T^{11} + 1814006788540943266 T^{12} + \)\(31\!\cdots\!40\)\( T^{13} + \)\(62\!\cdots\!13\)\( T^{14} - \)\(16\!\cdots\!12\)\( T^{15} - \)\(11\!\cdots\!72\)\( T^{16} - \)\(16\!\cdots\!12\)\( p^{3} T^{17} + \)\(62\!\cdots\!13\)\( p^{6} T^{18} + \)\(31\!\cdots\!40\)\( p^{9} T^{19} + 1814006788540943266 p^{12} T^{20} - 2492189862796374616 p^{15} T^{21} - 3476229367145225 p^{19} T^{22} + 1037408985847920 p^{21} T^{23} + 43588260389419 p^{24} T^{24} - 60493229664 p^{27} T^{25} - 27115333536 p^{30} T^{26} - 288580560 p^{33} T^{27} + 12542819 p^{36} T^{28} + 15640 p^{40} T^{29} - 3375 p^{42} T^{30} - 64 p^{45} T^{31} + p^{48} T^{32} \) | |
| 13 | \( 1 + 48 T - 8912 T^{2} - 562608 T^{3} + 41733732 T^{4} + 271894392 p T^{5} - 108854198656 T^{6} - 14870531020512 T^{7} + 76843432937738 T^{8} + 3603669749525952 p T^{9} + 793796077535506752 T^{10} - \)\(11\!\cdots\!00\)\( T^{11} - \)\(47\!\cdots\!84\)\( T^{12} + \)\(19\!\cdots\!24\)\( T^{13} + \)\(16\!\cdots\!68\)\( T^{14} - \)\(16\!\cdots\!08\)\( T^{15} - \)\(41\!\cdots\!97\)\( T^{16} - \)\(16\!\cdots\!08\)\( p^{3} T^{17} + \)\(16\!\cdots\!68\)\( p^{6} T^{18} + \)\(19\!\cdots\!24\)\( p^{9} T^{19} - \)\(47\!\cdots\!84\)\( p^{12} T^{20} - \)\(11\!\cdots\!00\)\( p^{15} T^{21} + 793796077535506752 p^{18} T^{22} + 3603669749525952 p^{22} T^{23} + 76843432937738 p^{24} T^{24} - 14870531020512 p^{27} T^{25} - 108854198656 p^{30} T^{26} + 271894392 p^{34} T^{27} + 41733732 p^{36} T^{28} - 562608 p^{39} T^{29} - 8912 p^{42} T^{30} + 48 p^{45} T^{31} + p^{48} T^{32} \) | |
| 17 | \( ( 1 + 106 T + 31045 T^{2} + 2335986 T^{3} + 411874574 T^{4} + 24427914838 T^{5} + 3380410797435 T^{6} + 167720274601742 T^{7} + 19528516701345202 T^{8} + 167720274601742 p^{3} T^{9} + 3380410797435 p^{6} T^{10} + 24427914838 p^{9} T^{11} + 411874574 p^{12} T^{12} + 2335986 p^{15} T^{13} + 31045 p^{18} T^{14} + 106 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 19 | \( ( 1 - 12 p T + 58963 T^{2} - 9081276 T^{3} + 1417239154 T^{4} - 165377934468 T^{5} + 19163197645333 T^{6} - 93695886721476 p T^{7} + 163162294625990314 T^{8} - 93695886721476 p^{4} T^{9} + 19163197645333 p^{6} T^{10} - 165377934468 p^{9} T^{11} + 1417239154 p^{12} T^{12} - 9081276 p^{15} T^{13} + 58963 p^{18} T^{14} - 12 p^{22} T^{15} + p^{24} T^{16} )^{2} \) | |
| 23 | \( 1 - 166 T - 51771 T^{2} + 372718 p T^{3} + 1699180148 T^{4} - 262513793694 T^{5} - 38701416393735 T^{6} + 5634488619363702 T^{7} + 652010168572890505 T^{8} - 91452659189257903260 T^{9} - \)\(83\!\cdots\!80\)\( T^{10} + \)\(11\!\cdots\!44\)\( T^{11} + \)\(79\!\cdots\!10\)\( T^{12} - \)\(10\!\cdots\!60\)\( T^{13} - \)\(60\!\cdots\!98\)\( T^{14} + \)\(47\!\cdots\!24\)\( T^{15} + \)\(53\!\cdots\!68\)\( T^{16} + \)\(47\!\cdots\!24\)\( p^{3} T^{17} - \)\(60\!\cdots\!98\)\( p^{6} T^{18} - \)\(10\!\cdots\!60\)\( p^{9} T^{19} + \)\(79\!\cdots\!10\)\( p^{12} T^{20} + \)\(11\!\cdots\!44\)\( p^{15} T^{21} - \)\(83\!\cdots\!80\)\( p^{18} T^{22} - 91452659189257903260 p^{21} T^{23} + 652010168572890505 p^{24} T^{24} + 5634488619363702 p^{27} T^{25} - 38701416393735 p^{30} T^{26} - 262513793694 p^{33} T^{27} + 1699180148 p^{36} T^{28} + 372718 p^{40} T^{29} - 51771 p^{42} T^{30} - 166 p^{45} T^{31} + p^{48} T^{32} \) | |
| 29 | \( 1 - 110 T - 114183 T^{2} + 14921770 T^{3} + 6319555484 T^{4} - 889633556526 T^{5} - 240038497236243 T^{6} + 29575366178209986 T^{7} + 8073314784857263021 T^{8} - 21871795763629730760 p T^{9} - \)\(26\!\cdots\!24\)\( T^{10} + \)\(11\!\cdots\!20\)\( T^{11} + \)\(81\!\cdots\!38\)\( T^{12} - \)\(22\!\cdots\!64\)\( T^{13} - \)\(20\!\cdots\!14\)\( T^{14} + \)\(23\!\cdots\!84\)\( T^{15} + \)\(50\!\cdots\!40\)\( T^{16} + \)\(23\!\cdots\!84\)\( p^{3} T^{17} - \)\(20\!\cdots\!14\)\( p^{6} T^{18} - \)\(22\!\cdots\!64\)\( p^{9} T^{19} + \)\(81\!\cdots\!38\)\( p^{12} T^{20} + \)\(11\!\cdots\!20\)\( p^{15} T^{21} - \)\(26\!\cdots\!24\)\( p^{18} T^{22} - 21871795763629730760 p^{22} T^{23} + 8073314784857263021 p^{24} T^{24} + 29575366178209986 p^{27} T^{25} - 240038497236243 p^{30} T^{26} - 889633556526 p^{33} T^{27} + 6319555484 p^{36} T^{28} + 14921770 p^{39} T^{29} - 114183 p^{42} T^{30} - 110 p^{45} T^{31} + p^{48} T^{32} \) | |
| 31 | \( 1 + 160 T - 131956 T^{2} - 19682000 T^{3} + 9661184460 T^{4} + 1372562682008 T^{5} - 478307569546712 T^{6} - 71298900351812272 T^{7} + 16533677888868068618 T^{8} + \)\(29\!\cdots\!84\)\( T^{9} - \)\(36\!\cdots\!28\)\( T^{10} - \)\(94\!\cdots\!00\)\( T^{11} + \)\(26\!\cdots\!76\)\( T^{12} + \)\(22\!\cdots\!20\)\( T^{13} + \)\(18\!\cdots\!24\)\( T^{14} - \)\(25\!\cdots\!08\)\( T^{15} - \)\(86\!\cdots\!81\)\( T^{16} - \)\(25\!\cdots\!08\)\( p^{3} T^{17} + \)\(18\!\cdots\!24\)\( p^{6} T^{18} + \)\(22\!\cdots\!20\)\( p^{9} T^{19} + \)\(26\!\cdots\!76\)\( p^{12} T^{20} - \)\(94\!\cdots\!00\)\( p^{15} T^{21} - \)\(36\!\cdots\!28\)\( p^{18} T^{22} + \)\(29\!\cdots\!84\)\( p^{21} T^{23} + 16533677888868068618 p^{24} T^{24} - 71298900351812272 p^{27} T^{25} - 478307569546712 p^{30} T^{26} + 1372562682008 p^{33} T^{27} + 9661184460 p^{36} T^{28} - 19682000 p^{39} T^{29} - 131956 p^{42} T^{30} + 160 p^{45} T^{31} + p^{48} T^{32} \) | |
| 37 | \( ( 1 - 52 T + 222212 T^{2} - 19043060 T^{3} + 23921733572 T^{4} - 2894521636028 T^{5} + 1714674068827500 T^{6} - 245529880580365212 T^{7} + 95601077014450447414 T^{8} - 245529880580365212 p^{3} T^{9} + 1714674068827500 p^{6} T^{10} - 2894521636028 p^{9} T^{11} + 23921733572 p^{12} T^{12} - 19043060 p^{15} T^{13} + 222212 p^{18} T^{14} - 52 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 41 | \( 1 - 280 T - 103236 T^{2} + 19375568 T^{3} + 8301492446 T^{4} - 1595379050976 T^{5} + 170809264120344 T^{6} + 21390503852909280 T^{7} - 38963103211351884887 T^{8} - \)\(64\!\cdots\!04\)\( T^{9} + \)\(50\!\cdots\!24\)\( T^{10} + \)\(38\!\cdots\!56\)\( T^{11} - \)\(15\!\cdots\!14\)\( T^{12} - \)\(68\!\cdots\!72\)\( T^{13} + \)\(39\!\cdots\!44\)\( T^{14} + \)\(95\!\cdots\!40\)\( T^{15} + \)\(76\!\cdots\!76\)\( T^{16} + \)\(95\!\cdots\!40\)\( p^{3} T^{17} + \)\(39\!\cdots\!44\)\( p^{6} T^{18} - \)\(68\!\cdots\!72\)\( p^{9} T^{19} - \)\(15\!\cdots\!14\)\( p^{12} T^{20} + \)\(38\!\cdots\!56\)\( p^{15} T^{21} + \)\(50\!\cdots\!24\)\( p^{18} T^{22} - \)\(64\!\cdots\!04\)\( p^{21} T^{23} - 38963103211351884887 p^{24} T^{24} + 21390503852909280 p^{27} T^{25} + 170809264120344 p^{30} T^{26} - 1595379050976 p^{33} T^{27} + 8301492446 p^{36} T^{28} + 19375568 p^{39} T^{29} - 103236 p^{42} T^{30} - 280 p^{45} T^{31} + p^{48} T^{32} \) | |
| 43 | \( 1 - 136 T - 180175 T^{2} + 34451312 T^{3} + 14958572235 T^{4} - 2779537534976 T^{5} - 561319680677288 T^{6} - 2754366420294656 T^{7} - 7527751933340363125 T^{8} + \)\(27\!\cdots\!24\)\( T^{9} - \)\(34\!\cdots\!51\)\( T^{10} - \)\(29\!\cdots\!64\)\( T^{11} + \)\(69\!\cdots\!34\)\( T^{12} + \)\(16\!\cdots\!44\)\( T^{13} + \)\(13\!\cdots\!05\)\( T^{14} - \)\(47\!\cdots\!12\)\( T^{15} - \)\(18\!\cdots\!44\)\( T^{16} - \)\(47\!\cdots\!12\)\( p^{3} T^{17} + \)\(13\!\cdots\!05\)\( p^{6} T^{18} + \)\(16\!\cdots\!44\)\( p^{9} T^{19} + \)\(69\!\cdots\!34\)\( p^{12} T^{20} - \)\(29\!\cdots\!64\)\( p^{15} T^{21} - \)\(34\!\cdots\!51\)\( p^{18} T^{22} + \)\(27\!\cdots\!24\)\( p^{21} T^{23} - 7527751933340363125 p^{24} T^{24} - 2754366420294656 p^{27} T^{25} - 561319680677288 p^{30} T^{26} - 2779537534976 p^{33} T^{27} + 14958572235 p^{36} T^{28} + 34451312 p^{39} T^{29} - 180175 p^{42} T^{30} - 136 p^{45} T^{31} + p^{48} T^{32} \) | |
| 47 | \( 1 - 594 T - 310011 T^{2} + 304407366 T^{3} + 20774197748 T^{4} - 72479330517210 T^{5} + 9579853530361713 T^{6} + 9306120146230275786 T^{7} - \)\(28\!\cdots\!15\)\( T^{8} - \)\(39\!\cdots\!60\)\( T^{9} + \)\(33\!\cdots\!60\)\( T^{10} - \)\(71\!\cdots\!56\)\( T^{11} - \)\(21\!\cdots\!42\)\( T^{12} + \)\(13\!\cdots\!00\)\( T^{13} - \)\(50\!\cdots\!22\)\( T^{14} - \)\(15\!\cdots\!80\)\( p T^{15} + \)\(36\!\cdots\!20\)\( p^{2} T^{16} - \)\(15\!\cdots\!80\)\( p^{4} T^{17} - \)\(50\!\cdots\!22\)\( p^{6} T^{18} + \)\(13\!\cdots\!00\)\( p^{9} T^{19} - \)\(21\!\cdots\!42\)\( p^{12} T^{20} - \)\(71\!\cdots\!56\)\( p^{15} T^{21} + \)\(33\!\cdots\!60\)\( p^{18} T^{22} - \)\(39\!\cdots\!60\)\( p^{21} T^{23} - \)\(28\!\cdots\!15\)\( p^{24} T^{24} + 9306120146230275786 p^{27} T^{25} + 9579853530361713 p^{30} T^{26} - 72479330517210 p^{33} T^{27} + 20774197748 p^{36} T^{28} + 304407366 p^{39} T^{29} - 310011 p^{42} T^{30} - 594 p^{45} T^{31} + p^{48} T^{32} \) | |
| 53 | \( ( 1 + 376 T + 817240 T^{2} + 341940264 T^{3} + 325408964924 T^{4} + 136823541940888 T^{5} + 83073726681403560 T^{6} + 31907086786968433928 T^{7} + \)\(14\!\cdots\!02\)\( T^{8} + 31907086786968433928 p^{3} T^{9} + 83073726681403560 p^{6} T^{10} + 136823541940888 p^{9} T^{11} + 325408964924 p^{12} T^{12} + 341940264 p^{15} T^{13} + 817240 p^{18} T^{14} + 376 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 59 | \( 1 - 24 T - 360487 T^{2} + 324780624 T^{3} + 4070478771 T^{4} - 107701398520056 T^{5} + 59557369091373760 T^{6} + 3647129114664346296 T^{7} - \)\(16\!\cdots\!13\)\( T^{8} + \)\(67\!\cdots\!24\)\( T^{9} + \)\(75\!\cdots\!81\)\( T^{10} - \)\(16\!\cdots\!40\)\( T^{11} + \)\(41\!\cdots\!10\)\( T^{12} + \)\(10\!\cdots\!80\)\( T^{13} - \)\(98\!\cdots\!71\)\( T^{14} - \)\(13\!\cdots\!48\)\( T^{15} + \)\(11\!\cdots\!20\)\( T^{16} - \)\(13\!\cdots\!48\)\( p^{3} T^{17} - \)\(98\!\cdots\!71\)\( p^{6} T^{18} + \)\(10\!\cdots\!80\)\( p^{9} T^{19} + \)\(41\!\cdots\!10\)\( p^{12} T^{20} - \)\(16\!\cdots\!40\)\( p^{15} T^{21} + \)\(75\!\cdots\!81\)\( p^{18} T^{22} + \)\(67\!\cdots\!24\)\( p^{21} T^{23} - \)\(16\!\cdots\!13\)\( p^{24} T^{24} + 3647129114664346296 p^{27} T^{25} + 59557369091373760 p^{30} T^{26} - 107701398520056 p^{33} T^{27} + 4070478771 p^{36} T^{28} + 324780624 p^{39} T^{29} - 360487 p^{42} T^{30} - 24 p^{45} T^{31} + p^{48} T^{32} \) | |
| 61 | \( 1 + 950 T - 305263 T^{2} - 363109666 T^{3} + 135233907396 T^{4} + 60599113270798 T^{5} - 46775165658112883 T^{6} + 10324883264962546822 T^{7} + \)\(13\!\cdots\!65\)\( T^{8} - \)\(58\!\cdots\!72\)\( T^{9} - \)\(23\!\cdots\!80\)\( T^{10} + \)\(83\!\cdots\!64\)\( T^{11} + \)\(94\!\cdots\!86\)\( T^{12} - \)\(31\!\cdots\!72\)\( T^{13} + \)\(14\!\cdots\!94\)\( T^{14} - \)\(11\!\cdots\!64\)\( T^{15} - \)\(94\!\cdots\!88\)\( T^{16} - \)\(11\!\cdots\!64\)\( p^{3} T^{17} + \)\(14\!\cdots\!94\)\( p^{6} T^{18} - \)\(31\!\cdots\!72\)\( p^{9} T^{19} + \)\(94\!\cdots\!86\)\( p^{12} T^{20} + \)\(83\!\cdots\!64\)\( p^{15} T^{21} - \)\(23\!\cdots\!80\)\( p^{18} T^{22} - \)\(58\!\cdots\!72\)\( p^{21} T^{23} + \)\(13\!\cdots\!65\)\( p^{24} T^{24} + 10324883264962546822 p^{27} T^{25} - 46775165658112883 p^{30} T^{26} + 60599113270798 p^{33} T^{27} + 135233907396 p^{36} T^{28} - 363109666 p^{39} T^{29} - 305263 p^{42} T^{30} + 950 p^{45} T^{31} + p^{48} T^{32} \) | |
| 67 | \( 1 + 426 T - 1027390 T^{2} - 86373564 T^{3} + 543051301344 T^{4} - 138749580868254 T^{5} - 184501395331331228 T^{6} + 50416767182111770926 T^{7} + \)\(45\!\cdots\!91\)\( T^{8} + \)\(33\!\cdots\!94\)\( T^{9} - \)\(20\!\cdots\!36\)\( T^{10} - \)\(15\!\cdots\!70\)\( T^{11} + \)\(10\!\cdots\!72\)\( T^{12} - \)\(90\!\cdots\!36\)\( T^{13} - \)\(29\!\cdots\!66\)\( T^{14} + \)\(22\!\cdots\!58\)\( T^{15} + \)\(63\!\cdots\!68\)\( T^{16} + \)\(22\!\cdots\!58\)\( p^{3} T^{17} - \)\(29\!\cdots\!66\)\( p^{6} T^{18} - \)\(90\!\cdots\!36\)\( p^{9} T^{19} + \)\(10\!\cdots\!72\)\( p^{12} T^{20} - \)\(15\!\cdots\!70\)\( p^{15} T^{21} - \)\(20\!\cdots\!36\)\( p^{18} T^{22} + \)\(33\!\cdots\!94\)\( p^{21} T^{23} + \)\(45\!\cdots\!91\)\( p^{24} T^{24} + 50416767182111770926 p^{27} T^{25} - 184501395331331228 p^{30} T^{26} - 138749580868254 p^{33} T^{27} + 543051301344 p^{36} T^{28} - 86373564 p^{39} T^{29} - 1027390 p^{42} T^{30} + 426 p^{45} T^{31} + p^{48} T^{32} \) | |
| 71 | \( ( 1 - 68 T + 1927240 T^{2} - 195755628 T^{3} + 1736810112812 T^{4} - 199064965710284 T^{5} + 997860891273011592 T^{6} - \)\(11\!\cdots\!80\)\( T^{7} + \)\(41\!\cdots\!82\)\( T^{8} - \)\(11\!\cdots\!80\)\( p^{3} T^{9} + 997860891273011592 p^{6} T^{10} - 199064965710284 p^{9} T^{11} + 1736810112812 p^{12} T^{12} - 195755628 p^{15} T^{13} + 1927240 p^{18} T^{14} - 68 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 154 T + 1392497 T^{2} - 202741334 T^{3} + 1131066261986 T^{4} - 169870001525330 T^{5} + 658070598272208879 T^{6} - 95024107331567434782 T^{7} + \)\(28\!\cdots\!66\)\( T^{8} - 95024107331567434782 p^{3} T^{9} + 658070598272208879 p^{6} T^{10} - 169870001525330 p^{9} T^{11} + 1131066261986 p^{12} T^{12} - 202741334 p^{15} T^{13} + 1392497 p^{18} T^{14} - 154 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 79 | \( 1 + 764 T - 3047264 T^{2} - 1993391576 T^{3} + 68938798172 p T^{4} + 2913454312151068 T^{5} - 7040014760926552608 T^{6} - \)\(29\!\cdots\!12\)\( T^{7} + \)\(72\!\cdots\!94\)\( T^{8} + \)\(21\!\cdots\!20\)\( T^{9} - \)\(61\!\cdots\!28\)\( T^{10} - \)\(12\!\cdots\!20\)\( T^{11} + \)\(44\!\cdots\!00\)\( T^{12} + \)\(49\!\cdots\!96\)\( T^{13} - \)\(27\!\cdots\!52\)\( T^{14} - \)\(94\!\cdots\!24\)\( T^{15} + \)\(14\!\cdots\!91\)\( T^{16} - \)\(94\!\cdots\!24\)\( p^{3} T^{17} - \)\(27\!\cdots\!52\)\( p^{6} T^{18} + \)\(49\!\cdots\!96\)\( p^{9} T^{19} + \)\(44\!\cdots\!00\)\( p^{12} T^{20} - \)\(12\!\cdots\!20\)\( p^{15} T^{21} - \)\(61\!\cdots\!28\)\( p^{18} T^{22} + \)\(21\!\cdots\!20\)\( p^{21} T^{23} + \)\(72\!\cdots\!94\)\( p^{24} T^{24} - \)\(29\!\cdots\!12\)\( p^{27} T^{25} - 7040014760926552608 p^{30} T^{26} + 2913454312151068 p^{33} T^{27} + 68938798172 p^{37} T^{28} - 1993391576 p^{39} T^{29} - 3047264 p^{42} T^{30} + 764 p^{45} T^{31} + p^{48} T^{32} \) | |
| 83 | \( 1 - 1362 T - 501535 T^{2} + 1578572910 T^{3} - 839637144504 T^{4} + 73779217952670 T^{5} + 371756007556129993 T^{6} - \)\(41\!\cdots\!90\)\( T^{7} + \)\(26\!\cdots\!89\)\( T^{8} - \)\(66\!\cdots\!48\)\( T^{9} - \)\(21\!\cdots\!20\)\( T^{10} + \)\(11\!\cdots\!76\)\( T^{11} + \)\(13\!\cdots\!50\)\( T^{12} - \)\(88\!\cdots\!00\)\( T^{13} - \)\(37\!\cdots\!94\)\( T^{14} + \)\(39\!\cdots\!68\)\( T^{15} - \)\(49\!\cdots\!20\)\( T^{16} + \)\(39\!\cdots\!68\)\( p^{3} T^{17} - \)\(37\!\cdots\!94\)\( p^{6} T^{18} - \)\(88\!\cdots\!00\)\( p^{9} T^{19} + \)\(13\!\cdots\!50\)\( p^{12} T^{20} + \)\(11\!\cdots\!76\)\( p^{15} T^{21} - \)\(21\!\cdots\!20\)\( p^{18} T^{22} - \)\(66\!\cdots\!48\)\( p^{21} T^{23} + \)\(26\!\cdots\!89\)\( p^{24} T^{24} - \)\(41\!\cdots\!90\)\( p^{27} T^{25} + 371756007556129993 p^{30} T^{26} + 73779217952670 p^{33} T^{27} - 839637144504 p^{36} T^{28} + 1578572910 p^{39} T^{29} - 501535 p^{42} T^{30} - 1362 p^{45} T^{31} + p^{48} T^{32} \) | |
| 89 | \( ( 1 - 86 T + 3710795 T^{2} - 32267226 T^{3} + 6867350553353 T^{4} + 187052548442468 T^{5} + 8210089612772983054 T^{6} + \)\(29\!\cdots\!96\)\( T^{7} + \)\(68\!\cdots\!02\)\( T^{8} + \)\(29\!\cdots\!96\)\( p^{3} T^{9} + 8210089612772983054 p^{6} T^{10} + 187052548442468 p^{9} T^{11} + 6867350553353 p^{12} T^{12} - 32267226 p^{15} T^{13} + 3710795 p^{18} T^{14} - 86 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 97 | \( 1 - 182 T - 4934009 T^{2} + 1331500142 T^{3} + 12475138995551 T^{4} - 4130371292501872 T^{5} - 21906218554765302816 T^{6} + \)\(81\!\cdots\!80\)\( T^{7} + \)\(30\!\cdots\!55\)\( T^{8} - \)\(11\!\cdots\!54\)\( T^{9} - \)\(36\!\cdots\!45\)\( T^{10} + \)\(11\!\cdots\!22\)\( T^{11} + \)\(38\!\cdots\!06\)\( T^{12} - \)\(81\!\cdots\!74\)\( T^{13} - \)\(37\!\cdots\!89\)\( T^{14} + \)\(27\!\cdots\!54\)\( T^{15} + \)\(35\!\cdots\!80\)\( T^{16} + \)\(27\!\cdots\!54\)\( p^{3} T^{17} - \)\(37\!\cdots\!89\)\( p^{6} T^{18} - \)\(81\!\cdots\!74\)\( p^{9} T^{19} + \)\(38\!\cdots\!06\)\( p^{12} T^{20} + \)\(11\!\cdots\!22\)\( p^{15} T^{21} - \)\(36\!\cdots\!45\)\( p^{18} T^{22} - \)\(11\!\cdots\!54\)\( p^{21} T^{23} + \)\(30\!\cdots\!55\)\( p^{24} T^{24} + \)\(81\!\cdots\!80\)\( p^{27} T^{25} - 21906218554765302816 p^{30} T^{26} - 4130371292501872 p^{33} T^{27} + 12475138995551 p^{36} T^{28} + 1331500142 p^{39} T^{29} - 4934009 p^{42} T^{30} - 182 p^{45} T^{31} + p^{48} T^{32} \) | |
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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
−2.11766907974849488703532136641, −2.01350430001043497577195155840, −2.00012971407846460153740006220, −1.98761251939656945097138807232, −1.70801675192981595812604754706, −1.68100138959522248630391765211, −1.64479209648235772809587875663, −1.61211397728884793589311818428, −1.55380600618910434879512103191, −1.47342946392375342878489548340, −1.22384999051852081406197189834, −1.20211403675780708226863211099, −1.06649980997233887573692557358, −1.02573074435147117870745669416, −1.01625550493065547390390530177, −0.909382887373357294965172197012, −0.879024014179753533100156401233, −0.74839865723743780885002828413, −0.70240786420118850706470098125, −0.56733308957320025138425989079, −0.48681599201180110676960817485, −0.34824048138637673589025697095, −0.23884986015233813257548457064, −0.19916558507979431833503798155, −0.18149272136969412997292230117, 0.18149272136969412997292230117, 0.19916558507979431833503798155, 0.23884986015233813257548457064, 0.34824048138637673589025697095, 0.48681599201180110676960817485, 0.56733308957320025138425989079, 0.70240786420118850706470098125, 0.74839865723743780885002828413, 0.879024014179753533100156401233, 0.909382887373357294965172197012, 1.01625550493065547390390530177, 1.02573074435147117870745669416, 1.06649980997233887573692557358, 1.20211403675780708226863211099, 1.22384999051852081406197189834, 1.47342946392375342878489548340, 1.55380600618910434879512103191, 1.61211397728884793589311818428, 1.64479209648235772809587875663, 1.68100138959522248630391765211, 1.70801675192981595812604754706, 1.98761251939656945097138807232, 2.00012971407846460153740006220, 2.01350430001043497577195155840, 2.11766907974849488703532136641
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.