Dirichlet series
| L(s) = 1 | + 2·5-s − 12·7-s − 63·9-s − 64·11-s + 32·13-s − 84·19-s − 384·23-s + 2·25-s + 32·29-s − 140·31-s − 24·35-s − 466·37-s − 98·41-s − 104·43-s − 126·45-s + 688·47-s + 72·49-s − 1.61e3·53-s − 128·55-s − 1.75e3·59-s + 1.28e3·61-s + 756·63-s + 64·65-s + 1.26e3·67-s + 968·71-s + 838·73-s + 768·77-s + ⋯ |
| L(s) = 1 | + 0.178·5-s − 0.647·7-s − 7/3·9-s − 1.75·11-s + 0.682·13-s − 1.01·19-s − 3.48·23-s + 0.0159·25-s + 0.204·29-s − 0.811·31-s − 0.115·35-s − 2.07·37-s − 0.373·41-s − 0.368·43-s − 0.417·45-s + 2.13·47-s + 0.209·49-s − 4.17·53-s − 0.313·55-s − 3.86·59-s + 2.69·61-s + 1.51·63-s + 0.122·65-s + 2.29·67-s + 1.61·71-s + 1.34·73-s + 1.13·77-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{14} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{14} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{56} \cdot 3^{14} \cdot 13^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.40832\times 10^{21}\) |
| Root analytic conductor: | \(6.06771\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{56} \cdot 3^{14} \cdot 13^{14} ,\ ( \ : [3/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.02538899096\) |
| \(L(\frac12)\) | \(\approx\) | \(0.02538899096\) |
| \(L(\frac{5}{2})\) | 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 | \( ( 1 + p^{2} T^{2} )^{7} \) | |
| 13 | \( 1 - 32 T + 135 T^{2} + 20560 T^{3} + 20165 p T^{4} + 614752 p^{2} T^{5} - 290753 p^{4} T^{6} + 10348256 p^{4} T^{7} - 290753 p^{7} T^{8} + 614752 p^{8} T^{9} + 20165 p^{10} T^{10} + 20560 p^{12} T^{11} + 135 p^{15} T^{12} - 32 p^{18} T^{13} + p^{21} T^{14} \) | |
| good | 5 | \( 1 - 2 T + 2 T^{2} + 1134 T^{3} - 10001 T^{4} + 244892 T^{5} + 173196 T^{6} + 823148 p^{2} T^{7} + 303323349 T^{8} - 1326893078 T^{9} + 57101564286 T^{10} - 153657591734 T^{11} + 1449893797451 T^{12} + 84434060546728 T^{13} + 59318683143592 T^{14} + 84434060546728 p^{3} T^{15} + 1449893797451 p^{6} T^{16} - 153657591734 p^{9} T^{17} + 57101564286 p^{12} T^{18} - 1326893078 p^{15} T^{19} + 303323349 p^{18} T^{20} + 823148 p^{23} T^{21} + 173196 p^{24} T^{22} + 244892 p^{27} T^{23} - 10001 p^{30} T^{24} + 1134 p^{33} T^{25} + 2 p^{36} T^{26} - 2 p^{39} T^{27} + p^{42} T^{28} \) |
| 7 | \( 1 + 12 T + 72 T^{2} + 4324 T^{3} - 210321 T^{4} - 4298184 T^{5} - 27086608 T^{6} - 1472789496 T^{7} + 10450981405 T^{8} + 575859013924 T^{9} + 475686584712 p T^{10} + 164621482384588 T^{11} + 82957566586003 T^{12} - 46213953561480048 T^{13} - 255900864025929184 T^{14} - 46213953561480048 p^{3} T^{15} + 82957566586003 p^{6} T^{16} + 164621482384588 p^{9} T^{17} + 475686584712 p^{13} T^{18} + 575859013924 p^{15} T^{19} + 10450981405 p^{18} T^{20} - 1472789496 p^{21} T^{21} - 27086608 p^{24} T^{22} - 4298184 p^{27} T^{23} - 210321 p^{30} T^{24} + 4324 p^{33} T^{25} + 72 p^{36} T^{26} + 12 p^{39} T^{27} + p^{42} T^{28} \) | |
| 11 | \( 1 + 64 T + 2048 T^{2} + 93432 T^{3} + 7363915 T^{4} + 282419672 T^{5} + 7358330400 T^{6} + 23353565560 p T^{7} + 6606900422049 T^{8} - 32556130742840 T^{9} - 5918534761699200 T^{10} - 47978839309457968 p T^{11} - 40280708786654326141 T^{12} - \)\(15\!\cdots\!84\)\( T^{13} - \)\(44\!\cdots\!24\)\( T^{14} - \)\(15\!\cdots\!84\)\( p^{3} T^{15} - 40280708786654326141 p^{6} T^{16} - 47978839309457968 p^{10} T^{17} - 5918534761699200 p^{12} T^{18} - 32556130742840 p^{15} T^{19} + 6606900422049 p^{18} T^{20} + 23353565560 p^{22} T^{21} + 7358330400 p^{24} T^{22} + 282419672 p^{27} T^{23} + 7363915 p^{30} T^{24} + 93432 p^{33} T^{25} + 2048 p^{36} T^{26} + 64 p^{39} T^{27} + p^{42} T^{28} \) | |
| 17 | \( 1 - 32082 T^{2} + 512266171 T^{4} - 5588594067796 T^{6} + 47814278890775529 T^{8} - \)\(34\!\cdots\!82\)\( T^{10} + \)\(20\!\cdots\!15\)\( T^{12} - \)\(11\!\cdots\!76\)\( T^{14} + \)\(20\!\cdots\!15\)\( p^{6} T^{16} - \)\(34\!\cdots\!82\)\( p^{12} T^{18} + 47814278890775529 p^{18} T^{20} - 5588594067796 p^{24} T^{22} + 512266171 p^{30} T^{24} - 32082 p^{36} T^{26} + p^{42} T^{28} \) | |
| 19 | \( 1 + 84 T + 3528 T^{2} + 1043020 T^{3} - 128816793 T^{4} - 17899199288 T^{5} - 505121734288 T^{6} - 165720796412264 T^{7} + 4397346637229005 T^{8} + 1562972956221072892 T^{9} + 38048290002698151096 T^{10} + \)\(11\!\cdots\!96\)\( T^{11} - \)\(25\!\cdots\!77\)\( T^{12} - \)\(84\!\cdots\!24\)\( T^{13} - \)\(20\!\cdots\!20\)\( T^{14} - \)\(84\!\cdots\!24\)\( p^{3} T^{15} - \)\(25\!\cdots\!77\)\( p^{6} T^{16} + \)\(11\!\cdots\!96\)\( p^{9} T^{17} + 38048290002698151096 p^{12} T^{18} + 1562972956221072892 p^{15} T^{19} + 4397346637229005 p^{18} T^{20} - 165720796412264 p^{21} T^{21} - 505121734288 p^{24} T^{22} - 17899199288 p^{27} T^{23} - 128816793 p^{30} T^{24} + 1043020 p^{33} T^{25} + 3528 p^{36} T^{26} + 84 p^{39} T^{27} + p^{42} T^{28} \) | |
| 23 | \( ( 1 + 192 T + 42885 T^{2} + 5118192 T^{3} + 756751489 T^{4} + 84821599872 T^{5} + 10588624986797 T^{6} + 1133963364063648 T^{7} + 10588624986797 p^{3} T^{8} + 84821599872 p^{6} T^{9} + 756751489 p^{9} T^{10} + 5118192 p^{12} T^{11} + 42885 p^{15} T^{12} + 192 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 29 | \( ( 1 - 16 T + 84663 T^{2} - 1908992 T^{3} + 3695648849 T^{4} - 35040068464 T^{5} + 114383563883663 T^{6} - 188326090459648 T^{7} + 114383563883663 p^{3} T^{8} - 35040068464 p^{6} T^{9} + 3695648849 p^{9} T^{10} - 1908992 p^{12} T^{11} + 84663 p^{15} T^{12} - 16 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 31 | \( 1 + 140 T + 9800 T^{2} + 4533844 T^{3} + 1443444255 T^{4} - 10911446648 T^{5} - 5395485521552 T^{6} - 3476994262353800 T^{7} - 1426342382235877667 T^{8} - \)\(11\!\cdots\!84\)\( T^{9} - \)\(94\!\cdots\!48\)\( T^{10} - \)\(52\!\cdots\!60\)\( T^{11} - \)\(32\!\cdots\!69\)\( T^{12} + \)\(44\!\cdots\!08\)\( p T^{13} + \)\(12\!\cdots\!88\)\( p^{2} T^{14} + \)\(44\!\cdots\!08\)\( p^{4} T^{15} - \)\(32\!\cdots\!69\)\( p^{6} T^{16} - \)\(52\!\cdots\!60\)\( p^{9} T^{17} - \)\(94\!\cdots\!48\)\( p^{12} T^{18} - \)\(11\!\cdots\!84\)\( p^{15} T^{19} - 1426342382235877667 p^{18} T^{20} - 3476994262353800 p^{21} T^{21} - 5395485521552 p^{24} T^{22} - 10911446648 p^{27} T^{23} + 1443444255 p^{30} T^{24} + 4533844 p^{33} T^{25} + 9800 p^{36} T^{26} + 140 p^{39} T^{27} + p^{42} T^{28} \) | |
| 37 | \( 1 + 466 T + 108578 T^{2} + 24427594 T^{3} + 968546771 T^{4} - 1120016712908 T^{5} - 328736985202348 T^{6} - 99157555778271260 T^{7} - 21369678360840577767 T^{8} - 45233581998687625834 p T^{9} + \)\(63\!\cdots\!98\)\( T^{10} + \)\(10\!\cdots\!30\)\( T^{11} + \)\(33\!\cdots\!47\)\( T^{12} + \)\(42\!\cdots\!84\)\( T^{13} + \)\(57\!\cdots\!88\)\( T^{14} + \)\(42\!\cdots\!84\)\( p^{3} T^{15} + \)\(33\!\cdots\!47\)\( p^{6} T^{16} + \)\(10\!\cdots\!30\)\( p^{9} T^{17} + \)\(63\!\cdots\!98\)\( p^{12} T^{18} - 45233581998687625834 p^{16} T^{19} - 21369678360840577767 p^{18} T^{20} - 99157555778271260 p^{21} T^{21} - 328736985202348 p^{24} T^{22} - 1120016712908 p^{27} T^{23} + 968546771 p^{30} T^{24} + 24427594 p^{33} T^{25} + 108578 p^{36} T^{26} + 466 p^{39} T^{27} + p^{42} T^{28} \) | |
| 41 | \( 1 + 98 T + 4802 T^{2} - 11585430 T^{3} - 2869423865 T^{4} - 108110097692 T^{5} + 70295277968364 T^{6} - 71023953638171420 T^{7} + 3521616073827310629 T^{8} + \)\(40\!\cdots\!66\)\( T^{9} + \)\(14\!\cdots\!02\)\( T^{10} - \)\(11\!\cdots\!82\)\( T^{11} - \)\(46\!\cdots\!01\)\( T^{12} - \)\(95\!\cdots\!96\)\( T^{13} + \)\(53\!\cdots\!56\)\( T^{14} - \)\(95\!\cdots\!96\)\( p^{3} T^{15} - \)\(46\!\cdots\!01\)\( p^{6} T^{16} - \)\(11\!\cdots\!82\)\( p^{9} T^{17} + \)\(14\!\cdots\!02\)\( p^{12} T^{18} + \)\(40\!\cdots\!66\)\( p^{15} T^{19} + 3521616073827310629 p^{18} T^{20} - 71023953638171420 p^{21} T^{21} + 70295277968364 p^{24} T^{22} - 108110097692 p^{27} T^{23} - 2869423865 p^{30} T^{24} - 11585430 p^{33} T^{25} + 4802 p^{36} T^{26} + 98 p^{39} T^{27} + p^{42} T^{28} \) | |
| 43 | \( ( 1 + 52 T + 365109 T^{2} + 4603560 T^{3} + 64105379245 T^{4} - 13723066684 p T^{5} + 7251798669740641 T^{6} - 102377883080916048 T^{7} + 7251798669740641 p^{3} T^{8} - 13723066684 p^{7} T^{9} + 64105379245 p^{9} T^{10} + 4603560 p^{12} T^{11} + 365109 p^{15} T^{12} + 52 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 47 | \( 1 - 688 T + 236672 T^{2} - 100584376 T^{3} + 67421638707 T^{4} - 25188356243656 T^{5} + 6431383367226912 T^{6} - 2509352731944689944 T^{7} + \)\(13\!\cdots\!97\)\( T^{8} - \)\(41\!\cdots\!08\)\( T^{9} + \)\(10\!\cdots\!48\)\( T^{10} - \)\(45\!\cdots\!36\)\( T^{11} + \)\(21\!\cdots\!27\)\( T^{12} - \)\(59\!\cdots\!16\)\( T^{13} + \)\(15\!\cdots\!16\)\( T^{14} - \)\(59\!\cdots\!16\)\( p^{3} T^{15} + \)\(21\!\cdots\!27\)\( p^{6} T^{16} - \)\(45\!\cdots\!36\)\( p^{9} T^{17} + \)\(10\!\cdots\!48\)\( p^{12} T^{18} - \)\(41\!\cdots\!08\)\( p^{15} T^{19} + \)\(13\!\cdots\!97\)\( p^{18} T^{20} - 2509352731944689944 p^{21} T^{21} + 6431383367226912 p^{24} T^{22} - 25188356243656 p^{27} T^{23} + 67421638707 p^{30} T^{24} - 100584376 p^{33} T^{25} + 236672 p^{36} T^{26} - 688 p^{39} T^{27} + p^{42} T^{28} \) | |
| 53 | \( ( 1 + 806 T + 885023 T^{2} + 476354188 T^{3} + 323134745393 T^{4} + 138760496866442 T^{5} + 72791256902468983 T^{6} + 25702187260664874728 T^{7} + 72791256902468983 p^{3} T^{8} + 138760496866442 p^{6} T^{9} + 323134745393 p^{9} T^{10} + 476354188 p^{12} T^{11} + 885023 p^{15} T^{12} + 806 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 59 | \( 1 + 1752 T + 1534752 T^{2} + 929069952 T^{3} + 394101494475 T^{4} + 111453691795656 T^{5} + 22004299032135264 T^{6} + 5631240979066561656 T^{7} + \)\(48\!\cdots\!13\)\( T^{8} + \)\(34\!\cdots\!04\)\( T^{9} + \)\(12\!\cdots\!12\)\( T^{10} - \)\(22\!\cdots\!16\)\( T^{11} - \)\(43\!\cdots\!29\)\( T^{12} - \)\(36\!\cdots\!80\)\( T^{13} - \)\(19\!\cdots\!24\)\( T^{14} - \)\(36\!\cdots\!80\)\( p^{3} T^{15} - \)\(43\!\cdots\!29\)\( p^{6} T^{16} - \)\(22\!\cdots\!16\)\( p^{9} T^{17} + \)\(12\!\cdots\!12\)\( p^{12} T^{18} + \)\(34\!\cdots\!04\)\( p^{15} T^{19} + \)\(48\!\cdots\!13\)\( p^{18} T^{20} + 5631240979066561656 p^{21} T^{21} + 22004299032135264 p^{24} T^{22} + 111453691795656 p^{27} T^{23} + 394101494475 p^{30} T^{24} + 929069952 p^{33} T^{25} + 1534752 p^{36} T^{26} + 1752 p^{39} T^{27} + p^{42} T^{28} \) | |
| 61 | \( ( 1 - 642 T + 1241139 T^{2} - 551402300 T^{3} + 646018139709 T^{4} - 216142085939822 T^{5} + 205020993475509463 T^{6} - 56472587302484824200 T^{7} + 205020993475509463 p^{3} T^{8} - 216142085939822 p^{6} T^{9} + 646018139709 p^{9} T^{10} - 551402300 p^{12} T^{11} + 1241139 p^{15} T^{12} - 642 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 67 | \( 1 - 1260 T + 793800 T^{2} - 632962916 T^{3} + 470798697255 T^{4} - 218020032390120 T^{5} + 101306261446143728 T^{6} - 58343394290690111352 T^{7} + \)\(11\!\cdots\!49\)\( T^{8} + \)\(47\!\cdots\!32\)\( T^{9} - \)\(19\!\cdots\!28\)\( T^{10} + \)\(19\!\cdots\!60\)\( T^{11} - \)\(22\!\cdots\!21\)\( T^{12} + \)\(12\!\cdots\!20\)\( T^{13} - \)\(47\!\cdots\!80\)\( T^{14} + \)\(12\!\cdots\!20\)\( p^{3} T^{15} - \)\(22\!\cdots\!21\)\( p^{6} T^{16} + \)\(19\!\cdots\!60\)\( p^{9} T^{17} - \)\(19\!\cdots\!28\)\( p^{12} T^{18} + \)\(47\!\cdots\!32\)\( p^{15} T^{19} + \)\(11\!\cdots\!49\)\( p^{18} T^{20} - 58343394290690111352 p^{21} T^{21} + 101306261446143728 p^{24} T^{22} - 218020032390120 p^{27} T^{23} + 470798697255 p^{30} T^{24} - 632962916 p^{33} T^{25} + 793800 p^{36} T^{26} - 1260 p^{39} T^{27} + p^{42} T^{28} \) | |
| 71 | \( 1 - 968 T + 468512 T^{2} - 589990416 T^{3} + 344954291491 T^{4} - 8564754923944 T^{5} + 20719803237272928 T^{6} + 349726326561831368 T^{7} - \)\(54\!\cdots\!15\)\( T^{8} + \)\(20\!\cdots\!16\)\( T^{9} - \)\(15\!\cdots\!76\)\( T^{10} + \)\(35\!\cdots\!12\)\( T^{11} + \)\(38\!\cdots\!87\)\( T^{12} - \)\(45\!\cdots\!52\)\( T^{13} + \)\(19\!\cdots\!44\)\( T^{14} - \)\(45\!\cdots\!52\)\( p^{3} T^{15} + \)\(38\!\cdots\!87\)\( p^{6} T^{16} + \)\(35\!\cdots\!12\)\( p^{9} T^{17} - \)\(15\!\cdots\!76\)\( p^{12} T^{18} + \)\(20\!\cdots\!16\)\( p^{15} T^{19} - \)\(54\!\cdots\!15\)\( p^{18} T^{20} + 349726326561831368 p^{21} T^{21} + 20719803237272928 p^{24} T^{22} - 8564754923944 p^{27} T^{23} + 344954291491 p^{30} T^{24} - 589990416 p^{33} T^{25} + 468512 p^{36} T^{26} - 968 p^{39} T^{27} + p^{42} T^{28} \) | |
| 73 | \( 1 - 838 T + 351122 T^{2} - 370158166 T^{3} + 193339764427 T^{4} - 8458628172060 T^{5} + 7711019571290964 T^{6} - 11214646953986889660 T^{7} + \)\(30\!\cdots\!53\)\( T^{8} + \)\(21\!\cdots\!34\)\( T^{9} - \)\(17\!\cdots\!26\)\( T^{10} + \)\(42\!\cdots\!54\)\( T^{11} - \)\(71\!\cdots\!29\)\( T^{12} + \)\(51\!\cdots\!56\)\( T^{13} - \)\(19\!\cdots\!84\)\( T^{14} + \)\(51\!\cdots\!56\)\( p^{3} T^{15} - \)\(71\!\cdots\!29\)\( p^{6} T^{16} + \)\(42\!\cdots\!54\)\( p^{9} T^{17} - \)\(17\!\cdots\!26\)\( p^{12} T^{18} + \)\(21\!\cdots\!34\)\( p^{15} T^{19} + \)\(30\!\cdots\!53\)\( p^{18} T^{20} - 11214646953986889660 p^{21} T^{21} + 7711019571290964 p^{24} T^{22} - 8458628172060 p^{27} T^{23} + 193339764427 p^{30} T^{24} - 370158166 p^{33} T^{25} + 351122 p^{36} T^{26} - 838 p^{39} T^{27} + p^{42} T^{28} \) | |
| 79 | \( 1 - 2783938 T^{2} + 3808691851643 T^{4} - 3370884845396762548 T^{6} + \)\(21\!\cdots\!17\)\( T^{8} - \)\(99\!\cdots\!46\)\( T^{10} + \)\(38\!\cdots\!63\)\( T^{12} - \)\(16\!\cdots\!32\)\( T^{14} + \)\(38\!\cdots\!63\)\( p^{6} T^{16} - \)\(99\!\cdots\!46\)\( p^{12} T^{18} + \)\(21\!\cdots\!17\)\( p^{18} T^{20} - 3370884845396762548 p^{24} T^{22} + 3808691851643 p^{30} T^{24} - 2783938 p^{36} T^{26} + p^{42} T^{28} \) | |
| 83 | \( 1 + 1688 T + 1424672 T^{2} + 528589728 T^{3} + 400858733275 T^{4} + 674087847373672 T^{5} + 706471623387954528 T^{6} + \)\(31\!\cdots\!08\)\( T^{7} - \)\(70\!\cdots\!11\)\( T^{8} - \)\(17\!\cdots\!64\)\( T^{9} - \)\(76\!\cdots\!24\)\( T^{10} + \)\(23\!\cdots\!72\)\( T^{11} - \)\(58\!\cdots\!69\)\( T^{12} - \)\(81\!\cdots\!32\)\( T^{13} - \)\(10\!\cdots\!00\)\( T^{14} - \)\(81\!\cdots\!32\)\( p^{3} T^{15} - \)\(58\!\cdots\!69\)\( p^{6} T^{16} + \)\(23\!\cdots\!72\)\( p^{9} T^{17} - \)\(76\!\cdots\!24\)\( p^{12} T^{18} - \)\(17\!\cdots\!64\)\( p^{15} T^{19} - \)\(70\!\cdots\!11\)\( p^{18} T^{20} + \)\(31\!\cdots\!08\)\( p^{21} T^{21} + 706471623387954528 p^{24} T^{22} + 674087847373672 p^{27} T^{23} + 400858733275 p^{30} T^{24} + 528589728 p^{33} T^{25} + 1424672 p^{36} T^{26} + 1688 p^{39} T^{27} + p^{42} T^{28} \) | |
| 89 | \( 1 + 2666 T + 3553778 T^{2} + 4198526306 T^{3} + 6104662161159 T^{4} + 7824296595289124 T^{5} + 7978772208368498700 T^{6} + \)\(78\!\cdots\!32\)\( T^{7} + \)\(81\!\cdots\!05\)\( T^{8} + \)\(80\!\cdots\!02\)\( T^{9} + \)\(71\!\cdots\!22\)\( T^{10} + \)\(63\!\cdots\!66\)\( T^{11} + \)\(55\!\cdots\!31\)\( T^{12} + \)\(48\!\cdots\!48\)\( T^{13} + \)\(41\!\cdots\!08\)\( T^{14} + \)\(48\!\cdots\!48\)\( p^{3} T^{15} + \)\(55\!\cdots\!31\)\( p^{6} T^{16} + \)\(63\!\cdots\!66\)\( p^{9} T^{17} + \)\(71\!\cdots\!22\)\( p^{12} T^{18} + \)\(80\!\cdots\!02\)\( p^{15} T^{19} + \)\(81\!\cdots\!05\)\( p^{18} T^{20} + \)\(78\!\cdots\!32\)\( p^{21} T^{21} + 7978772208368498700 p^{24} T^{22} + 7824296595289124 p^{27} T^{23} + 6104662161159 p^{30} T^{24} + 4198526306 p^{33} T^{25} + 3553778 p^{36} T^{26} + 2666 p^{39} T^{27} + p^{42} T^{28} \) | |
| 97 | \( 1 - 422 T + 89042 T^{2} - 829693862 T^{3} - 2537662262981 T^{4} + 1383048580696356 T^{5} - 13492025513270508 T^{6} + \)\(17\!\cdots\!96\)\( T^{7} + \)\(24\!\cdots\!01\)\( T^{8} - \)\(18\!\cdots\!10\)\( T^{9} + \)\(17\!\cdots\!46\)\( T^{10} - \)\(16\!\cdots\!66\)\( T^{11} - \)\(82\!\cdots\!93\)\( T^{12} + \)\(15\!\cdots\!88\)\( T^{13} - \)\(88\!\cdots\!56\)\( T^{14} + \)\(15\!\cdots\!88\)\( p^{3} T^{15} - \)\(82\!\cdots\!93\)\( p^{6} T^{16} - \)\(16\!\cdots\!66\)\( p^{9} T^{17} + \)\(17\!\cdots\!46\)\( p^{12} T^{18} - \)\(18\!\cdots\!10\)\( p^{15} T^{19} + \)\(24\!\cdots\!01\)\( p^{18} T^{20} + \)\(17\!\cdots\!96\)\( p^{21} T^{21} - 13492025513270508 p^{24} T^{22} + 1383048580696356 p^{27} T^{23} - 2537662262981 p^{30} T^{24} - 829693862 p^{33} T^{25} + 89042 p^{36} T^{26} - 422 p^{39} T^{27} + p^{42} T^{28} \) | |
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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
−2.66905423506545591947240128897, −2.45890993820501713937861051789, −2.43118948798730806224713856167, −2.36998686689187596597977626815, −2.33859982047193005290307648919, −2.23061633828203988047603946776, −2.11406535913623345910337907484, −1.95464912685730309604969058327, −1.90425420273852264438603228693, −1.70033517794047250907376750996, −1.63072890309804207226950167795, −1.60933332826798657579025514959, −1.56852098806055637258776876117, −1.44850036597546235010416674447, −1.37613199248275229985001730932, −1.13356449016387120454976156793, −0.943696653316710370201525849280, −0.827413758147348270092259177185, −0.71713859550421417579982959567, −0.60840921672726683110401495527, −0.41082435127325292781907834831, −0.34545284598404102526047970210, −0.24919973644096252361852252561, −0.06892834099267237386994502347, −0.04537173566823505055614158910, 0.04537173566823505055614158910, 0.06892834099267237386994502347, 0.24919973644096252361852252561, 0.34545284598404102526047970210, 0.41082435127325292781907834831, 0.60840921672726683110401495527, 0.71713859550421417579982959567, 0.827413758147348270092259177185, 0.943696653316710370201525849280, 1.13356449016387120454976156793, 1.37613199248275229985001730932, 1.44850036597546235010416674447, 1.56852098806055637258776876117, 1.60933332826798657579025514959, 1.63072890309804207226950167795, 1.70033517794047250907376750996, 1.90425420273852264438603228693, 1.95464912685730309604969058327, 2.11406535913623345910337907484, 2.23061633828203988047603946776, 2.33859982047193005290307648919, 2.36998686689187596597977626815, 2.43118948798730806224713856167, 2.45890993820501713937861051789, 2.66905423506545591947240128897
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.