Dirichlet series
| L(s) = 1 | + 2·2-s − 2·4-s + 2·5-s − 4·7-s + 24·8-s + 4·10-s − 94·11-s − 2·13-s − 8·14-s + 20·16-s + 4·17-s − 706·19-s − 4·20-s − 188·22-s − 1.14e3·23-s + 2·25-s − 4·26-s + 8·28-s − 862·29-s − 776·32-s + 8·34-s − 8·35-s − 1.82e3·37-s − 1.41e3·38-s + 48·40-s + 1.69e3·43-s + 188·44-s + ⋯ |
| L(s) = 1 | + 1/2·2-s − 1/8·4-s + 2/25·5-s − 0.0816·7-s + 3/8·8-s + 1/25·10-s − 0.776·11-s − 0.0118·13-s − 0.0408·14-s + 5/64·16-s + 0.0138·17-s − 1.95·19-s − 0.00999·20-s − 0.388·22-s − 2.17·23-s + 0.00319·25-s − 0.00591·26-s + 1/98·28-s − 1.02·29-s − 0.757·32-s + 0.00692·34-s − 0.00653·35-s − 1.33·37-s − 0.977·38-s + 0.0299·40-s + 0.916·43-s + 0.0971·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{28}\right)^{s/2} \, \Gamma_{\C}(s+2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{56} \cdot 3^{28}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.62177\times 10^{16}\) |
| Root analytic conductor: | \(3.85814\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{56} \cdot 3^{28} ,\ ( \ : [2]^{14} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(0.0004788098104\) |
| \(L(\frac12)\) | \(\approx\) | \(0.0004788098104\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p T + 3 p T^{2} - 5 p^{3} T^{3} + 15 p^{3} T^{4} + 11 p^{5} T^{5} - 45 p^{6} T^{6} - 5 p^{9} T^{7} - 45 p^{10} T^{8} + 11 p^{13} T^{9} + 15 p^{15} T^{10} - 5 p^{19} T^{11} + 3 p^{21} T^{12} - p^{25} T^{13} + p^{28} T^{14} \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + 2 T^{2} - 3938 T^{3} - 18913 p T^{4} + 9371916 T^{5} - 2160156 p T^{6} + 6597908556 T^{7} - 151886931511 T^{8} - 563896366366 T^{9} + 18344031697054 T^{10} + 1149733079121218 T^{11} + 112600777623819 p^{4} T^{12} - 53569344141254552 p^{2} T^{13} + 14585682673141755608 T^{14} - 53569344141254552 p^{6} T^{15} + 112600777623819 p^{12} T^{16} + 1149733079121218 p^{12} T^{17} + 18344031697054 p^{16} T^{18} - 563896366366 p^{20} T^{19} - 151886931511 p^{24} T^{20} + 6597908556 p^{28} T^{21} - 2160156 p^{33} T^{22} + 9371916 p^{36} T^{23} - 18913 p^{41} T^{24} - 3938 p^{44} T^{25} + 2 p^{48} T^{26} - 2 p^{52} T^{27} + p^{56} T^{28} \) |
| 7 | \( ( 1 + 2 T + 8235 T^{2} + 64404 T^{3} + 38860249 T^{4} + 447351454 T^{5} + 125587217723 T^{6} + 196872839848 p T^{7} + 125587217723 p^{4} T^{8} + 447351454 p^{8} T^{9} + 38860249 p^{12} T^{10} + 64404 p^{16} T^{11} + 8235 p^{20} T^{12} + 2 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 11 | \( 1 + 94 T + 4418 T^{2} - 965570 T^{3} - 470088133 T^{4} - 283723804 p T^{5} + 2249641670708 T^{6} + 796542815073292 T^{7} + 97369777194709097 T^{8} - 5317394377195027646 T^{9} - \)\(63\!\cdots\!78\)\( T^{10} - \)\(14\!\cdots\!86\)\( T^{11} - \)\(19\!\cdots\!01\)\( p^{2} T^{12} + \)\(20\!\cdots\!32\)\( p T^{13} + \)\(18\!\cdots\!92\)\( T^{14} + \)\(20\!\cdots\!32\)\( p^{5} T^{15} - \)\(19\!\cdots\!01\)\( p^{10} T^{16} - \)\(14\!\cdots\!86\)\( p^{12} T^{17} - \)\(63\!\cdots\!78\)\( p^{16} T^{18} - 5317394377195027646 p^{20} T^{19} + 97369777194709097 p^{24} T^{20} + 796542815073292 p^{28} T^{21} + 2249641670708 p^{32} T^{22} - 283723804 p^{37} T^{23} - 470088133 p^{40} T^{24} - 965570 p^{44} T^{25} + 4418 p^{48} T^{26} + 94 p^{52} T^{27} + p^{56} T^{28} \) | |
| 13 | \( 1 + 2 T + 2 T^{2} + 6883234 T^{3} + 1464853339 T^{4} + 65707775476 T^{5} + 1832149307204 p T^{6} + 16199073445624116 T^{7} + 1142495439970904649 T^{8} + \)\(10\!\cdots\!70\)\( T^{9} + \)\(78\!\cdots\!46\)\( T^{10} + \)\(14\!\cdots\!90\)\( T^{11} + \)\(74\!\cdots\!35\)\( T^{12} + \)\(22\!\cdots\!32\)\( T^{13} + \)\(94\!\cdots\!60\)\( T^{14} + \)\(22\!\cdots\!32\)\( p^{4} T^{15} + \)\(74\!\cdots\!35\)\( p^{8} T^{16} + \)\(14\!\cdots\!90\)\( p^{12} T^{17} + \)\(78\!\cdots\!46\)\( p^{16} T^{18} + \)\(10\!\cdots\!70\)\( p^{20} T^{19} + 1142495439970904649 p^{24} T^{20} + 16199073445624116 p^{28} T^{21} + 1832149307204 p^{33} T^{22} + 65707775476 p^{36} T^{23} + 1464853339 p^{40} T^{24} + 6883234 p^{44} T^{25} + 2 p^{48} T^{26} + 2 p^{52} T^{27} + p^{56} T^{28} \) | |
| 17 | \( ( 1 - 2 T + 333755 T^{2} + 12776716 T^{3} + 3318766585 p T^{4} + 174198961346 p T^{5} + 6454907058757691 T^{6} + 326099715157486120 T^{7} + 6454907058757691 p^{4} T^{8} + 174198961346 p^{9} T^{9} + 3318766585 p^{13} T^{10} + 12776716 p^{16} T^{11} + 333755 p^{20} T^{12} - 2 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 19 | \( 1 + 706 T + 249218 T^{2} + 101381538 T^{3} + 32599242619 T^{4} + 3617133340788 T^{5} - 431513784802892 T^{6} - 960761925731564364 T^{7} - \)\(71\!\cdots\!35\)\( T^{8} - \)\(22\!\cdots\!14\)\( T^{9} - \)\(57\!\cdots\!66\)\( T^{10} - \)\(18\!\cdots\!22\)\( T^{11} - \)\(19\!\cdots\!61\)\( T^{12} + \)\(38\!\cdots\!56\)\( T^{13} + \)\(11\!\cdots\!48\)\( T^{14} + \)\(38\!\cdots\!56\)\( p^{4} T^{15} - \)\(19\!\cdots\!61\)\( p^{8} T^{16} - \)\(18\!\cdots\!22\)\( p^{12} T^{17} - \)\(57\!\cdots\!66\)\( p^{16} T^{18} - \)\(22\!\cdots\!14\)\( p^{20} T^{19} - \)\(71\!\cdots\!35\)\( p^{24} T^{20} - 960761925731564364 p^{28} T^{21} - 431513784802892 p^{32} T^{22} + 3617133340788 p^{36} T^{23} + 32599242619 p^{40} T^{24} + 101381538 p^{44} T^{25} + 249218 p^{48} T^{26} + 706 p^{52} T^{27} + p^{56} T^{28} \) | |
| 23 | \( ( 1 + 574 T + 1024043 T^{2} + 635922028 T^{3} + 551773439769 T^{4} + 314763003369506 T^{5} + 213700110666561659 T^{6} + \)\(10\!\cdots\!08\)\( T^{7} + 213700110666561659 p^{4} T^{8} + 314763003369506 p^{8} T^{9} + 551773439769 p^{12} T^{10} + 635922028 p^{16} T^{11} + 1024043 p^{20} T^{12} + 574 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 29 | \( 1 + 862 T + 371522 T^{2} + 1045006654 T^{3} + 1721779716827 T^{4} + 691721668187596 T^{5} + 502604487474844916 T^{6} + \)\(98\!\cdots\!28\)\( T^{7} + \)\(75\!\cdots\!49\)\( T^{8} + \)\(31\!\cdots\!74\)\( T^{9} + \)\(39\!\cdots\!98\)\( T^{10} + \)\(15\!\cdots\!86\)\( p T^{11} + \)\(34\!\cdots\!87\)\( T^{12} + \)\(29\!\cdots\!64\)\( T^{13} + \)\(24\!\cdots\!96\)\( T^{14} + \)\(29\!\cdots\!64\)\( p^{4} T^{15} + \)\(34\!\cdots\!87\)\( p^{8} T^{16} + \)\(15\!\cdots\!86\)\( p^{13} T^{17} + \)\(39\!\cdots\!98\)\( p^{16} T^{18} + \)\(31\!\cdots\!74\)\( p^{20} T^{19} + \)\(75\!\cdots\!49\)\( p^{24} T^{20} + \)\(98\!\cdots\!28\)\( p^{28} T^{21} + 502604487474844916 p^{32} T^{22} + 691721668187596 p^{36} T^{23} + 1721779716827 p^{40} T^{24} + 1045006654 p^{44} T^{25} + 371522 p^{48} T^{26} + 862 p^{52} T^{27} + p^{56} T^{28} \) | |
| 31 | \( 1 - 6904334 T^{2} + 24182883262811 T^{4} - 57459081771770667372 T^{6} + \)\(10\!\cdots\!69\)\( T^{8} - \)\(14\!\cdots\!50\)\( T^{10} + \)\(57\!\cdots\!93\)\( p T^{12} - \)\(17\!\cdots\!20\)\( T^{14} + \)\(57\!\cdots\!93\)\( p^{9} T^{16} - \)\(14\!\cdots\!50\)\( p^{16} T^{18} + \)\(10\!\cdots\!69\)\( p^{24} T^{20} - 57459081771770667372 p^{32} T^{22} + 24182883262811 p^{40} T^{24} - 6904334 p^{48} T^{26} + p^{56} T^{28} \) | |
| 37 | \( 1 + 1826 T + 1667138 T^{2} + 4976934274 T^{3} + 7030206539163 T^{4} + 1716272691299380 T^{5} + 102662887807662052 p T^{6} + \)\(11\!\cdots\!60\)\( T^{7} - \)\(10\!\cdots\!63\)\( T^{8} - \)\(16\!\cdots\!22\)\( T^{9} + \)\(49\!\cdots\!50\)\( p T^{10} - \)\(21\!\cdots\!18\)\( T^{11} - \)\(10\!\cdots\!97\)\( T^{12} - \)\(14\!\cdots\!60\)\( T^{13} - \)\(67\!\cdots\!04\)\( T^{14} - \)\(14\!\cdots\!60\)\( p^{4} T^{15} - \)\(10\!\cdots\!97\)\( p^{8} T^{16} - \)\(21\!\cdots\!18\)\( p^{12} T^{17} + \)\(49\!\cdots\!50\)\( p^{17} T^{18} - \)\(16\!\cdots\!22\)\( p^{20} T^{19} - \)\(10\!\cdots\!63\)\( p^{24} T^{20} + \)\(11\!\cdots\!60\)\( p^{28} T^{21} + 102662887807662052 p^{33} T^{22} + 1716272691299380 p^{36} T^{23} + 7030206539163 p^{40} T^{24} + 4976934274 p^{44} T^{25} + 1667138 p^{48} T^{26} + 1826 p^{52} T^{27} + p^{56} T^{28} \) | |
| 41 | \( 1 - 24523982 T^{2} + 302442312166171 T^{4} - \)\(24\!\cdots\!76\)\( T^{6} + \)\(14\!\cdots\!01\)\( T^{8} - \)\(70\!\cdots\!70\)\( T^{10} + \)\(27\!\cdots\!51\)\( T^{12} - \)\(84\!\cdots\!88\)\( T^{14} + \)\(27\!\cdots\!51\)\( p^{8} T^{16} - \)\(70\!\cdots\!70\)\( p^{16} T^{18} + \)\(14\!\cdots\!01\)\( p^{24} T^{20} - \)\(24\!\cdots\!76\)\( p^{32} T^{22} + 302442312166171 p^{40} T^{24} - 24523982 p^{48} T^{26} + p^{56} T^{28} \) | |
| 43 | \( 1 - 1694 T + 1434818 T^{2} - 14278395262 T^{3} + 44454402050619 T^{4} - 13474016894363980 T^{5} + 60977294006539554100 T^{6} - \)\(38\!\cdots\!44\)\( T^{7} + \)\(23\!\cdots\!81\)\( T^{8} + \)\(78\!\cdots\!82\)\( T^{9} + \)\(12\!\cdots\!62\)\( T^{10} - \)\(18\!\cdots\!22\)\( T^{11} - \)\(91\!\cdots\!45\)\( T^{12} + \)\(10\!\cdots\!08\)\( T^{13} + \)\(16\!\cdots\!52\)\( T^{14} + \)\(10\!\cdots\!08\)\( p^{4} T^{15} - \)\(91\!\cdots\!45\)\( p^{8} T^{16} - \)\(18\!\cdots\!22\)\( p^{12} T^{17} + \)\(12\!\cdots\!62\)\( p^{16} T^{18} + \)\(78\!\cdots\!82\)\( p^{20} T^{19} + \)\(23\!\cdots\!81\)\( p^{24} T^{20} - \)\(38\!\cdots\!44\)\( p^{28} T^{21} + 60977294006539554100 p^{32} T^{22} - 13474016894363980 p^{36} T^{23} + 44454402050619 p^{40} T^{24} - 14278395262 p^{44} T^{25} + 1434818 p^{48} T^{26} - 1694 p^{52} T^{27} + p^{56} T^{28} \) | |
| 47 | \( 1 - 51887758 T^{2} + 1309844227745755 T^{4} - \)\(45\!\cdots\!48\)\( p T^{6} + \)\(24\!\cdots\!09\)\( T^{8} - \)\(21\!\cdots\!30\)\( T^{10} + \)\(15\!\cdots\!39\)\( T^{12} - \)\(82\!\cdots\!04\)\( T^{14} + \)\(15\!\cdots\!39\)\( p^{8} T^{16} - \)\(21\!\cdots\!30\)\( p^{16} T^{18} + \)\(24\!\cdots\!09\)\( p^{24} T^{20} - \)\(45\!\cdots\!48\)\( p^{33} T^{22} + 1309844227745755 p^{40} T^{24} - 51887758 p^{48} T^{26} + p^{56} T^{28} \) | |
| 53 | \( 1 - 482 T + 116162 T^{2} + 5558326078 T^{3} + 43583027341595 T^{4} - 155411473123116980 T^{5} + 85293132817975457012 T^{6} - \)\(11\!\cdots\!76\)\( T^{7} + \)\(35\!\cdots\!13\)\( T^{8} + \)\(92\!\cdots\!94\)\( T^{9} + \)\(12\!\cdots\!18\)\( T^{10} - \)\(13\!\cdots\!10\)\( T^{11} + \)\(28\!\cdots\!31\)\( T^{12} - \)\(18\!\cdots\!24\)\( T^{13} + \)\(12\!\cdots\!68\)\( T^{14} - \)\(18\!\cdots\!24\)\( p^{4} T^{15} + \)\(28\!\cdots\!31\)\( p^{8} T^{16} - \)\(13\!\cdots\!10\)\( p^{12} T^{17} + \)\(12\!\cdots\!18\)\( p^{16} T^{18} + \)\(92\!\cdots\!94\)\( p^{20} T^{19} + \)\(35\!\cdots\!13\)\( p^{24} T^{20} - \)\(11\!\cdots\!76\)\( p^{28} T^{21} + 85293132817975457012 p^{32} T^{22} - 155411473123116980 p^{36} T^{23} + 43583027341595 p^{40} T^{24} + 5558326078 p^{44} T^{25} + 116162 p^{48} T^{26} - 482 p^{52} T^{27} + p^{56} T^{28} \) | |
| 59 | \( 1 - 2786 T + 3880898 T^{2} - 95235375746 T^{3} + 282835430943931 T^{4} + 951817240129082700 T^{5} + \)\(78\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!24\)\( T^{7} - \)\(10\!\cdots\!59\)\( T^{8} + \)\(16\!\cdots\!78\)\( T^{9} + \)\(95\!\cdots\!50\)\( T^{10} + \)\(27\!\cdots\!94\)\( T^{11} + \)\(66\!\cdots\!23\)\( T^{12} - \)\(40\!\cdots\!76\)\( T^{13} + \)\(34\!\cdots\!76\)\( T^{14} - \)\(40\!\cdots\!76\)\( p^{4} T^{15} + \)\(66\!\cdots\!23\)\( p^{8} T^{16} + \)\(27\!\cdots\!94\)\( p^{12} T^{17} + \)\(95\!\cdots\!50\)\( p^{16} T^{18} + \)\(16\!\cdots\!78\)\( p^{20} T^{19} - \)\(10\!\cdots\!59\)\( p^{24} T^{20} + \)\(10\!\cdots\!24\)\( p^{28} T^{21} + \)\(78\!\cdots\!20\)\( p^{32} T^{22} + 951817240129082700 p^{36} T^{23} + 282835430943931 p^{40} T^{24} - 95235375746 p^{44} T^{25} + 3880898 p^{48} T^{26} - 2786 p^{52} T^{27} + p^{56} T^{28} \) | |
| 61 | \( 1 + 3778 T + 7136642 T^{2} + 13584988130 T^{3} + 15885658886619 T^{4} - 322904339201283724 T^{5} - \)\(12\!\cdots\!20\)\( T^{6} - \)\(11\!\cdots\!80\)\( T^{7} - \)\(34\!\cdots\!59\)\( T^{8} - \)\(57\!\cdots\!82\)\( T^{9} - \)\(99\!\cdots\!50\)\( p T^{10} - \)\(13\!\cdots\!70\)\( T^{11} + \)\(15\!\cdots\!55\)\( T^{12} + \)\(19\!\cdots\!60\)\( T^{13} + \)\(92\!\cdots\!08\)\( T^{14} + \)\(19\!\cdots\!60\)\( p^{4} T^{15} + \)\(15\!\cdots\!55\)\( p^{8} T^{16} - \)\(13\!\cdots\!70\)\( p^{12} T^{17} - \)\(99\!\cdots\!50\)\( p^{17} T^{18} - \)\(57\!\cdots\!82\)\( p^{20} T^{19} - \)\(34\!\cdots\!59\)\( p^{24} T^{20} - \)\(11\!\cdots\!80\)\( p^{28} T^{21} - \)\(12\!\cdots\!20\)\( p^{32} T^{22} - 322904339201283724 p^{36} T^{23} + 15885658886619 p^{40} T^{24} + 13584988130 p^{44} T^{25} + 7136642 p^{48} T^{26} + 3778 p^{52} T^{27} + p^{56} T^{28} \) | |
| 67 | \( 1 - 7998 T + 31984002 T^{2} - 47670849246 T^{3} - 439076236005637 T^{4} + 648765759780793716 T^{5} + \)\(99\!\cdots\!64\)\( T^{6} - \)\(93\!\cdots\!52\)\( T^{7} + \)\(28\!\cdots\!21\)\( T^{8} + \)\(69\!\cdots\!70\)\( T^{9} - \)\(55\!\cdots\!14\)\( T^{10} + \)\(21\!\cdots\!74\)\( T^{11} - \)\(39\!\cdots\!49\)\( T^{12} + \)\(17\!\cdots\!60\)\( T^{13} - \)\(39\!\cdots\!76\)\( T^{14} + \)\(17\!\cdots\!60\)\( p^{4} T^{15} - \)\(39\!\cdots\!49\)\( p^{8} T^{16} + \)\(21\!\cdots\!74\)\( p^{12} T^{17} - \)\(55\!\cdots\!14\)\( p^{16} T^{18} + \)\(69\!\cdots\!70\)\( p^{20} T^{19} + \)\(28\!\cdots\!21\)\( p^{24} T^{20} - \)\(93\!\cdots\!52\)\( p^{28} T^{21} + \)\(99\!\cdots\!64\)\( p^{32} T^{22} + 648765759780793716 p^{36} T^{23} - 439076236005637 p^{40} T^{24} - 47670849246 p^{44} T^{25} + 31984002 p^{48} T^{26} - 7998 p^{52} T^{27} + p^{56} T^{28} \) | |
| 71 | \( ( 1 + 9982 T + 145017323 T^{2} + 1165310044396 T^{3} + 10014081489420185 T^{4} + 63641985337531169890 T^{5} + \)\(40\!\cdots\!59\)\( T^{6} + \)\(20\!\cdots\!72\)\( T^{7} + \)\(40\!\cdots\!59\)\( p^{4} T^{8} + 63641985337531169890 p^{8} T^{9} + 10014081489420185 p^{12} T^{10} + 1165310044396 p^{16} T^{11} + 145017323 p^{20} T^{12} + 9982 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 73 | \( 1 - 168573838 T^{2} + 13353714116727067 T^{4} - \)\(70\!\cdots\!96\)\( T^{6} + \)\(30\!\cdots\!57\)\( T^{8} - \)\(11\!\cdots\!98\)\( T^{10} + \)\(39\!\cdots\!07\)\( T^{12} - \)\(11\!\cdots\!44\)\( T^{14} + \)\(39\!\cdots\!07\)\( p^{8} T^{16} - \)\(11\!\cdots\!98\)\( p^{16} T^{18} + \)\(30\!\cdots\!57\)\( p^{24} T^{20} - \)\(70\!\cdots\!96\)\( p^{32} T^{22} + 13353714116727067 p^{40} T^{24} - 168573838 p^{48} T^{26} + p^{56} T^{28} \) | |
| 79 | \( 1 - 364033678 T^{2} + 65454647116587227 T^{4} - \)\(76\!\cdots\!56\)\( T^{6} + \)\(65\!\cdots\!81\)\( T^{8} - \)\(43\!\cdots\!10\)\( T^{10} + \)\(23\!\cdots\!67\)\( T^{12} - \)\(99\!\cdots\!88\)\( T^{14} + \)\(23\!\cdots\!67\)\( p^{8} T^{16} - \)\(43\!\cdots\!10\)\( p^{16} T^{18} + \)\(65\!\cdots\!81\)\( p^{24} T^{20} - \)\(76\!\cdots\!56\)\( p^{32} T^{22} + 65454647116587227 p^{40} T^{24} - 364033678 p^{48} T^{26} + p^{56} T^{28} \) | |
| 83 | \( 1 - 17282 T + 149333762 T^{2} - 1088719641698 T^{3} + 16964332297412731 T^{4} - \)\(21\!\cdots\!96\)\( T^{5} + \)\(17\!\cdots\!52\)\( T^{6} - \)\(12\!\cdots\!12\)\( T^{7} + \)\(11\!\cdots\!61\)\( T^{8} - \)\(11\!\cdots\!62\)\( T^{9} + \)\(90\!\cdots\!74\)\( T^{10} - \)\(61\!\cdots\!82\)\( T^{11} + \)\(44\!\cdots\!67\)\( T^{12} - \)\(35\!\cdots\!48\)\( T^{13} + \)\(26\!\cdots\!76\)\( T^{14} - \)\(35\!\cdots\!48\)\( p^{4} T^{15} + \)\(44\!\cdots\!67\)\( p^{8} T^{16} - \)\(61\!\cdots\!82\)\( p^{12} T^{17} + \)\(90\!\cdots\!74\)\( p^{16} T^{18} - \)\(11\!\cdots\!62\)\( p^{20} T^{19} + \)\(11\!\cdots\!61\)\( p^{24} T^{20} - \)\(12\!\cdots\!12\)\( p^{28} T^{21} + \)\(17\!\cdots\!52\)\( p^{32} T^{22} - \)\(21\!\cdots\!96\)\( p^{36} T^{23} + 16964332297412731 p^{40} T^{24} - 1088719641698 p^{44} T^{25} + 149333762 p^{48} T^{26} - 17282 p^{52} T^{27} + p^{56} T^{28} \) | |
| 89 | \( 1 - 548528910 T^{2} + 149200943223060123 T^{4} - \)\(26\!\cdots\!16\)\( T^{6} + \)\(35\!\cdots\!49\)\( T^{8} - \)\(37\!\cdots\!50\)\( T^{10} + \)\(31\!\cdots\!11\)\( T^{12} - \)\(21\!\cdots\!00\)\( T^{14} + \)\(31\!\cdots\!11\)\( p^{8} T^{16} - \)\(37\!\cdots\!50\)\( p^{16} T^{18} + \)\(35\!\cdots\!49\)\( p^{24} T^{20} - \)\(26\!\cdots\!16\)\( p^{32} T^{22} + 149200943223060123 p^{40} T^{24} - 548528910 p^{48} T^{26} + p^{56} T^{28} \) | |
| 97 | \( ( 1 + 2 T + 387850619 T^{2} + 251760181236 T^{3} + 75114732161345545 T^{4} + 73269666487293981214 T^{5} + \)\(94\!\cdots\!67\)\( T^{6} + \)\(89\!\cdots\!44\)\( T^{7} + \)\(94\!\cdots\!67\)\( p^{4} T^{8} + 73269666487293981214 p^{8} T^{9} + 75114732161345545 p^{12} T^{10} + 251760181236 p^{16} T^{11} + 387850619 p^{20} T^{12} + 2 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.44954593440434304466378904757, −3.22431547566672980856933053650, −3.20536668140822338701977826809, −3.18336168469215405016065746381, −2.93794956260707445751958242590, −2.82227905672313522555408653171, −2.72953045220007123843327154128, −2.42252473854482319401342887464, −2.31560487697406706104674077743, −2.22409339070852393593260279311, −2.09576362132065360838107033181, −2.07627814804223706611539547103, −2.07081313353225645032158736572, −1.83984781577586061316061931315, −1.73432062376155469798870707688, −1.49747303424628091448160508549, −1.32187892319035094599812979733, −1.24061782965770926847366593455, −1.18591679479907123520938094146, −0.961242237147534793497763818875, −0.66837895189519884016144280105, −0.55183708149761486415092245458, −0.13712263965820740763824016922, −0.04258773369302950163090173301, −0.01924817015758369746532817381, 0.01924817015758369746532817381, 0.04258773369302950163090173301, 0.13712263965820740763824016922, 0.55183708149761486415092245458, 0.66837895189519884016144280105, 0.961242237147534793497763818875, 1.18591679479907123520938094146, 1.24061782965770926847366593455, 1.32187892319035094599812979733, 1.49747303424628091448160508549, 1.73432062376155469798870707688, 1.83984781577586061316061931315, 2.07081313353225645032158736572, 2.07627814804223706611539547103, 2.09576362132065360838107033181, 2.22409339070852393593260279311, 2.31560487697406706104674077743, 2.42252473854482319401342887464, 2.72953045220007123843327154128, 2.82227905672313522555408653171, 2.93794956260707445751958242590, 3.18336168469215405016065746381, 3.20536668140822338701977826809, 3.22431547566672980856933053650, 3.44954593440434304466378904757
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.