Dirichlet series
| L(s) = 1 | − 23·2-s + 378·3-s − 209·4-s + 69·5-s − 8.69e3·6-s − 3.27e3·7-s + 8.31e3·8-s + 7.65e4·9-s − 1.58e3·10-s − 7.90e4·12-s − 3.21e4·13-s + 7.53e4·14-s + 2.60e4·15-s + 5.72e3·16-s − 4.29e4·17-s − 1.76e6·18-s − 5.04e4·19-s − 1.44e4·20-s − 1.23e6·21-s − 8.34e4·23-s + 3.14e6·24-s − 3.75e5·25-s + 7.40e5·26-s + 1.10e7·27-s + 6.85e5·28-s − 2.27e5·29-s − 5.99e5·30-s + ⋯ |
| L(s) = 1 | − 2.03·2-s + 8.08·3-s − 1.63·4-s + 0.246·5-s − 16.4·6-s − 3.61·7-s + 5.73·8-s + 35·9-s − 0.501·10-s − 13.1·12-s − 4.06·13-s + 7.34·14-s + 1.99·15-s + 0.349·16-s − 2.11·17-s − 71.1·18-s − 1.68·19-s − 0.403·20-s − 29.1·21-s − 1.43·23-s + 46.3·24-s − 4.81·25-s + 8.26·26-s + 107.·27-s + 5.89·28-s − 1.72·29-s − 4.05·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 11^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 11^{28}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(3^{14} \cdot 11^{28}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.81240\times 10^{28}\) |
| Root analytic conductor: | \(10.6487\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(14\) |
| Selberg data: | \((28,\ 3^{14} \cdot 11^{28} ,\ ( \ : [7/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - p^{3} T )^{14} \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + 23 T + 369 p T^{2} + 13471 T^{3} + 267219 T^{4} + 1009795 p^{2} T^{5} + 7899241 p^{3} T^{6} + 51844779 p^{4} T^{7} + 176461619 p^{6} T^{8} + 2062953329 p^{6} T^{9} + 12813720683 p^{7} T^{10} + 34449610115 p^{9} T^{11} + 815222718557 p^{8} T^{12} + 532090898643 p^{12} T^{13} + 6359614883467 p^{12} T^{14} + 532090898643 p^{19} T^{15} + 815222718557 p^{22} T^{16} + 34449610115 p^{30} T^{17} + 12813720683 p^{35} T^{18} + 2062953329 p^{41} T^{19} + 176461619 p^{48} T^{20} + 51844779 p^{53} T^{21} + 7899241 p^{59} T^{22} + 1009795 p^{65} T^{23} + 267219 p^{70} T^{24} + 13471 p^{77} T^{25} + 369 p^{85} T^{26} + 23 p^{91} T^{27} + p^{98} T^{28} \) |
| 5 | \( 1 - 69 T + 76152 p T^{2} - 28960416 T^{3} + 74730951182 T^{4} - 5384138101377 T^{5} + 10009616315318179 T^{6} - 124550469557986608 p T^{7} + 1683645545879603583 p^{4} T^{8} - \)\(38\!\cdots\!97\)\( p^{3} T^{9} + \)\(15\!\cdots\!19\)\( p^{4} T^{10} - \)\(82\!\cdots\!89\)\( p^{5} T^{11} + \)\(10\!\cdots\!72\)\( p^{7} T^{12} - \)\(13\!\cdots\!69\)\( p^{7} T^{13} + \)\(16\!\cdots\!39\)\( p^{8} T^{14} - \)\(13\!\cdots\!69\)\( p^{14} T^{15} + \)\(10\!\cdots\!72\)\( p^{21} T^{16} - \)\(82\!\cdots\!89\)\( p^{26} T^{17} + \)\(15\!\cdots\!19\)\( p^{32} T^{18} - \)\(38\!\cdots\!97\)\( p^{38} T^{19} + 1683645545879603583 p^{46} T^{20} - 124550469557986608 p^{50} T^{21} + 10009616315318179 p^{56} T^{22} - 5384138101377 p^{63} T^{23} + 74730951182 p^{70} T^{24} - 28960416 p^{77} T^{25} + 76152 p^{85} T^{26} - 69 p^{91} T^{27} + p^{98} T^{28} \) | |
| 7 | \( 1 + 3278 T + 10716428 T^{2} + 22993401552 T^{3} + 46293354697739 T^{4} + 76779511172751200 T^{5} + \)\(11\!\cdots\!05\)\( T^{6} + \)\(16\!\cdots\!44\)\( T^{7} + \)\(21\!\cdots\!66\)\( T^{8} + \)\(26\!\cdots\!22\)\( T^{9} + \)\(29\!\cdots\!74\)\( T^{10} + \)\(32\!\cdots\!46\)\( T^{11} + \)\(33\!\cdots\!68\)\( T^{12} + \)\(32\!\cdots\!48\)\( T^{13} + \)\(30\!\cdots\!43\)\( T^{14} + \)\(32\!\cdots\!48\)\( p^{7} T^{15} + \)\(33\!\cdots\!68\)\( p^{14} T^{16} + \)\(32\!\cdots\!46\)\( p^{21} T^{17} + \)\(29\!\cdots\!74\)\( p^{28} T^{18} + \)\(26\!\cdots\!22\)\( p^{35} T^{19} + \)\(21\!\cdots\!66\)\( p^{42} T^{20} + \)\(16\!\cdots\!44\)\( p^{49} T^{21} + \)\(11\!\cdots\!05\)\( p^{56} T^{22} + 76779511172751200 p^{63} T^{23} + 46293354697739 p^{70} T^{24} + 22993401552 p^{77} T^{25} + 10716428 p^{84} T^{26} + 3278 p^{91} T^{27} + p^{98} T^{28} \) | |
| 13 | \( 1 + 2476 p T + 847194133 T^{2} + 15983663325780 T^{3} + 268225155774630669 T^{4} + \)\(38\!\cdots\!64\)\( T^{5} + \)\(49\!\cdots\!60\)\( T^{6} + \)\(44\!\cdots\!80\)\( p T^{7} + \)\(63\!\cdots\!76\)\( T^{8} + \)\(64\!\cdots\!56\)\( T^{9} + \)\(62\!\cdots\!04\)\( T^{10} + \)\(56\!\cdots\!88\)\( T^{11} + \)\(49\!\cdots\!98\)\( T^{12} + \)\(40\!\cdots\!44\)\( T^{13} + \)\(33\!\cdots\!58\)\( T^{14} + \)\(40\!\cdots\!44\)\( p^{7} T^{15} + \)\(49\!\cdots\!98\)\( p^{14} T^{16} + \)\(56\!\cdots\!88\)\( p^{21} T^{17} + \)\(62\!\cdots\!04\)\( p^{28} T^{18} + \)\(64\!\cdots\!56\)\( p^{35} T^{19} + \)\(63\!\cdots\!76\)\( p^{42} T^{20} + \)\(44\!\cdots\!80\)\( p^{50} T^{21} + \)\(49\!\cdots\!60\)\( p^{56} T^{22} + \)\(38\!\cdots\!64\)\( p^{63} T^{23} + 268225155774630669 p^{70} T^{24} + 15983663325780 p^{77} T^{25} + 847194133 p^{84} T^{26} + 2476 p^{92} T^{27} + p^{98} T^{28} \) | |
| 17 | \( 1 + 42917 T + 3435503581 T^{2} + 118711004929155 T^{3} + 5785619978011667516 T^{4} + \)\(17\!\cdots\!07\)\( T^{5} + \)\(65\!\cdots\!25\)\( T^{6} + \)\(17\!\cdots\!09\)\( T^{7} + \)\(55\!\cdots\!21\)\( T^{8} + \)\(13\!\cdots\!84\)\( T^{9} + \)\(37\!\cdots\!82\)\( T^{10} + \)\(83\!\cdots\!34\)\( T^{11} + \)\(20\!\cdots\!30\)\( T^{12} + \)\(41\!\cdots\!66\)\( T^{13} + \)\(91\!\cdots\!84\)\( T^{14} + \)\(41\!\cdots\!66\)\( p^{7} T^{15} + \)\(20\!\cdots\!30\)\( p^{14} T^{16} + \)\(83\!\cdots\!34\)\( p^{21} T^{17} + \)\(37\!\cdots\!82\)\( p^{28} T^{18} + \)\(13\!\cdots\!84\)\( p^{35} T^{19} + \)\(55\!\cdots\!21\)\( p^{42} T^{20} + \)\(17\!\cdots\!09\)\( p^{49} T^{21} + \)\(65\!\cdots\!25\)\( p^{56} T^{22} + \)\(17\!\cdots\!07\)\( p^{63} T^{23} + 5785619978011667516 p^{70} T^{24} + 118711004929155 p^{77} T^{25} + 3435503581 p^{84} T^{26} + 42917 p^{91} T^{27} + p^{98} T^{28} \) | |
| 19 | \( 1 + 50419 T + 6021753547 T^{2} + 265837040767095 T^{3} + 18985420174984005684 T^{4} + \)\(74\!\cdots\!17\)\( T^{5} + \)\(41\!\cdots\!47\)\( T^{6} + \)\(14\!\cdots\!33\)\( T^{7} + \)\(68\!\cdots\!09\)\( T^{8} + \)\(22\!\cdots\!04\)\( T^{9} + \)\(93\!\cdots\!06\)\( T^{10} + \)\(28\!\cdots\!14\)\( T^{11} + \)\(10\!\cdots\!62\)\( T^{12} + \)\(15\!\cdots\!38\)\( p T^{13} + \)\(10\!\cdots\!28\)\( T^{14} + \)\(15\!\cdots\!38\)\( p^{8} T^{15} + \)\(10\!\cdots\!62\)\( p^{14} T^{16} + \)\(28\!\cdots\!14\)\( p^{21} T^{17} + \)\(93\!\cdots\!06\)\( p^{28} T^{18} + \)\(22\!\cdots\!04\)\( p^{35} T^{19} + \)\(68\!\cdots\!09\)\( p^{42} T^{20} + \)\(14\!\cdots\!33\)\( p^{49} T^{21} + \)\(41\!\cdots\!47\)\( p^{56} T^{22} + \)\(74\!\cdots\!17\)\( p^{63} T^{23} + 18985420174984005684 p^{70} T^{24} + 265837040767095 p^{77} T^{25} + 6021753547 p^{84} T^{26} + 50419 p^{91} T^{27} + p^{98} T^{28} \) | |
| 23 | \( 1 + 83492 T + 26109466605 T^{2} + 1966526486062264 T^{3} + \)\(33\!\cdots\!13\)\( T^{4} + \)\(22\!\cdots\!88\)\( T^{5} + \)\(28\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!76\)\( T^{7} + \)\(18\!\cdots\!76\)\( T^{8} + \)\(10\!\cdots\!20\)\( T^{9} + \)\(93\!\cdots\!60\)\( T^{10} + \)\(50\!\cdots\!40\)\( T^{11} + \)\(17\!\cdots\!30\)\( p T^{12} + \)\(20\!\cdots\!00\)\( T^{13} + \)\(14\!\cdots\!10\)\( T^{14} + \)\(20\!\cdots\!00\)\( p^{7} T^{15} + \)\(17\!\cdots\!30\)\( p^{15} T^{16} + \)\(50\!\cdots\!40\)\( p^{21} T^{17} + \)\(93\!\cdots\!60\)\( p^{28} T^{18} + \)\(10\!\cdots\!20\)\( p^{35} T^{19} + \)\(18\!\cdots\!76\)\( p^{42} T^{20} + \)\(17\!\cdots\!76\)\( p^{49} T^{21} + \)\(28\!\cdots\!80\)\( p^{56} T^{22} + \)\(22\!\cdots\!88\)\( p^{63} T^{23} + \)\(33\!\cdots\!13\)\( p^{70} T^{24} + 1966526486062264 p^{77} T^{25} + 26109466605 p^{84} T^{26} + 83492 p^{91} T^{27} + p^{98} T^{28} \) | |
| 29 | \( 1 + 227028 T + 166410964107 T^{2} + 26368811431571736 T^{3} + \)\(11\!\cdots\!36\)\( T^{4} + \)\(12\!\cdots\!36\)\( T^{5} + \)\(47\!\cdots\!52\)\( T^{6} + \)\(35\!\cdots\!60\)\( T^{7} + \)\(14\!\cdots\!82\)\( T^{8} + \)\(69\!\cdots\!44\)\( T^{9} + \)\(34\!\cdots\!25\)\( T^{10} + \)\(46\!\cdots\!00\)\( p T^{11} + \)\(77\!\cdots\!53\)\( T^{12} + \)\(26\!\cdots\!16\)\( T^{13} + \)\(14\!\cdots\!88\)\( T^{14} + \)\(26\!\cdots\!16\)\( p^{7} T^{15} + \)\(77\!\cdots\!53\)\( p^{14} T^{16} + \)\(46\!\cdots\!00\)\( p^{22} T^{17} + \)\(34\!\cdots\!25\)\( p^{28} T^{18} + \)\(69\!\cdots\!44\)\( p^{35} T^{19} + \)\(14\!\cdots\!82\)\( p^{42} T^{20} + \)\(35\!\cdots\!60\)\( p^{49} T^{21} + \)\(47\!\cdots\!52\)\( p^{56} T^{22} + \)\(12\!\cdots\!36\)\( p^{63} T^{23} + \)\(11\!\cdots\!36\)\( p^{70} T^{24} + 26368811431571736 p^{77} T^{25} + 166410964107 p^{84} T^{26} + 227028 p^{91} T^{27} + p^{98} T^{28} \) | |
| 31 | \( 1 + 36481 T + 197597198442 T^{2} + 7179229922465742 T^{3} + \)\(18\!\cdots\!32\)\( T^{4} + \)\(76\!\cdots\!91\)\( T^{5} + \)\(11\!\cdots\!71\)\( T^{6} + \)\(57\!\cdots\!60\)\( T^{7} + \)\(52\!\cdots\!31\)\( T^{8} + \)\(32\!\cdots\!41\)\( T^{9} + \)\(18\!\cdots\!41\)\( T^{10} + \)\(14\!\cdots\!31\)\( T^{11} + \)\(58\!\cdots\!20\)\( T^{12} + \)\(51\!\cdots\!69\)\( T^{13} + \)\(16\!\cdots\!49\)\( T^{14} + \)\(51\!\cdots\!69\)\( p^{7} T^{15} + \)\(58\!\cdots\!20\)\( p^{14} T^{16} + \)\(14\!\cdots\!31\)\( p^{21} T^{17} + \)\(18\!\cdots\!41\)\( p^{28} T^{18} + \)\(32\!\cdots\!41\)\( p^{35} T^{19} + \)\(52\!\cdots\!31\)\( p^{42} T^{20} + \)\(57\!\cdots\!60\)\( p^{49} T^{21} + \)\(11\!\cdots\!71\)\( p^{56} T^{22} + \)\(76\!\cdots\!91\)\( p^{63} T^{23} + \)\(18\!\cdots\!32\)\( p^{70} T^{24} + 7179229922465742 p^{77} T^{25} + 197597198442 p^{84} T^{26} + 36481 p^{91} T^{27} + p^{98} T^{28} \) | |
| 37 | \( 1 - 634524 T + 721886106797 T^{2} - 395888004071656040 T^{3} + \)\(76\!\cdots\!53\)\( p T^{4} - \)\(13\!\cdots\!72\)\( T^{5} + \)\(74\!\cdots\!00\)\( T^{6} - \)\(31\!\cdots\!92\)\( T^{7} + \)\(14\!\cdots\!72\)\( T^{8} - \)\(56\!\cdots\!80\)\( T^{9} + \)\(62\!\cdots\!96\)\( p T^{10} - \)\(80\!\cdots\!84\)\( T^{11} + \)\(29\!\cdots\!22\)\( T^{12} - \)\(92\!\cdots\!72\)\( T^{13} + \)\(30\!\cdots\!42\)\( T^{14} - \)\(92\!\cdots\!72\)\( p^{7} T^{15} + \)\(29\!\cdots\!22\)\( p^{14} T^{16} - \)\(80\!\cdots\!84\)\( p^{21} T^{17} + \)\(62\!\cdots\!96\)\( p^{29} T^{18} - \)\(56\!\cdots\!80\)\( p^{35} T^{19} + \)\(14\!\cdots\!72\)\( p^{42} T^{20} - \)\(31\!\cdots\!92\)\( p^{49} T^{21} + \)\(74\!\cdots\!00\)\( p^{56} T^{22} - \)\(13\!\cdots\!72\)\( p^{63} T^{23} + \)\(76\!\cdots\!53\)\( p^{71} T^{24} - 395888004071656040 p^{77} T^{25} + 721886106797 p^{84} T^{26} - 634524 p^{91} T^{27} + p^{98} T^{28} \) | |
| 41 | \( 1 + 1286544 T + 2271948865281 T^{2} + 2270251348934307624 T^{3} + \)\(24\!\cdots\!73\)\( T^{4} + \)\(19\!\cdots\!16\)\( T^{5} + \)\(15\!\cdots\!40\)\( T^{6} + \)\(11\!\cdots\!08\)\( T^{7} + \)\(74\!\cdots\!00\)\( T^{8} + \)\(44\!\cdots\!40\)\( T^{9} + \)\(26\!\cdots\!20\)\( T^{10} + \)\(13\!\cdots\!88\)\( T^{11} + \)\(71\!\cdots\!10\)\( T^{12} + \)\(33\!\cdots\!20\)\( T^{13} + \)\(15\!\cdots\!70\)\( T^{14} + \)\(33\!\cdots\!20\)\( p^{7} T^{15} + \)\(71\!\cdots\!10\)\( p^{14} T^{16} + \)\(13\!\cdots\!88\)\( p^{21} T^{17} + \)\(26\!\cdots\!20\)\( p^{28} T^{18} + \)\(44\!\cdots\!40\)\( p^{35} T^{19} + \)\(74\!\cdots\!00\)\( p^{42} T^{20} + \)\(11\!\cdots\!08\)\( p^{49} T^{21} + \)\(15\!\cdots\!40\)\( p^{56} T^{22} + \)\(19\!\cdots\!16\)\( p^{63} T^{23} + \)\(24\!\cdots\!73\)\( p^{70} T^{24} + 2270251348934307624 p^{77} T^{25} + 2271948865281 p^{84} T^{26} + 1286544 p^{91} T^{27} + p^{98} T^{28} \) | |
| 43 | \( 1 + 2632644 T + 5058448201357 T^{2} + 7053104639566519144 T^{3} + \)\(84\!\cdots\!97\)\( T^{4} + \)\(85\!\cdots\!44\)\( T^{5} + \)\(78\!\cdots\!68\)\( T^{6} + \)\(64\!\cdots\!92\)\( T^{7} + \)\(48\!\cdots\!12\)\( T^{8} + \)\(34\!\cdots\!64\)\( T^{9} + \)\(22\!\cdots\!48\)\( T^{10} + \)\(14\!\cdots\!72\)\( T^{11} + \)\(82\!\cdots\!78\)\( T^{12} + \)\(46\!\cdots\!08\)\( T^{13} + \)\(24\!\cdots\!22\)\( T^{14} + \)\(46\!\cdots\!08\)\( p^{7} T^{15} + \)\(82\!\cdots\!78\)\( p^{14} T^{16} + \)\(14\!\cdots\!72\)\( p^{21} T^{17} + \)\(22\!\cdots\!48\)\( p^{28} T^{18} + \)\(34\!\cdots\!64\)\( p^{35} T^{19} + \)\(48\!\cdots\!12\)\( p^{42} T^{20} + \)\(64\!\cdots\!92\)\( p^{49} T^{21} + \)\(78\!\cdots\!68\)\( p^{56} T^{22} + \)\(85\!\cdots\!44\)\( p^{63} T^{23} + \)\(84\!\cdots\!97\)\( p^{70} T^{24} + 7053104639566519144 p^{77} T^{25} + 5058448201357 p^{84} T^{26} + 2632644 p^{91} T^{27} + p^{98} T^{28} \) | |
| 47 | \( 1 + 1692245 T + 5659098036117 T^{2} + 7887231301354550905 T^{3} + \)\(15\!\cdots\!24\)\( T^{4} + \)\(18\!\cdots\!15\)\( T^{5} + \)\(25\!\cdots\!81\)\( T^{6} + \)\(26\!\cdots\!35\)\( T^{7} + \)\(31\!\cdots\!53\)\( T^{8} + \)\(28\!\cdots\!80\)\( T^{9} + \)\(28\!\cdots\!62\)\( T^{10} + \)\(23\!\cdots\!50\)\( T^{11} + \)\(20\!\cdots\!94\)\( T^{12} + \)\(15\!\cdots\!30\)\( T^{13} + \)\(11\!\cdots\!80\)\( T^{14} + \)\(15\!\cdots\!30\)\( p^{7} T^{15} + \)\(20\!\cdots\!94\)\( p^{14} T^{16} + \)\(23\!\cdots\!50\)\( p^{21} T^{17} + \)\(28\!\cdots\!62\)\( p^{28} T^{18} + \)\(28\!\cdots\!80\)\( p^{35} T^{19} + \)\(31\!\cdots\!53\)\( p^{42} T^{20} + \)\(26\!\cdots\!35\)\( p^{49} T^{21} + \)\(25\!\cdots\!81\)\( p^{56} T^{22} + \)\(18\!\cdots\!15\)\( p^{63} T^{23} + \)\(15\!\cdots\!24\)\( p^{70} T^{24} + 7887231301354550905 p^{77} T^{25} + 5659098036117 p^{84} T^{26} + 1692245 p^{91} T^{27} + p^{98} T^{28} \) | |
| 53 | \( 1 - 1211317 T + 11121286239186 T^{2} - 12619915266155463944 T^{3} + \)\(61\!\cdots\!00\)\( T^{4} - \)\(64\!\cdots\!09\)\( T^{5} + \)\(21\!\cdots\!27\)\( T^{6} - \)\(21\!\cdots\!56\)\( T^{7} + \)\(57\!\cdots\!59\)\( T^{8} - \)\(50\!\cdots\!81\)\( T^{9} + \)\(11\!\cdots\!49\)\( T^{10} - \)\(93\!\cdots\!69\)\( T^{11} + \)\(18\!\cdots\!84\)\( T^{12} - \)\(13\!\cdots\!85\)\( T^{13} + \)\(24\!\cdots\!73\)\( T^{14} - \)\(13\!\cdots\!85\)\( p^{7} T^{15} + \)\(18\!\cdots\!84\)\( p^{14} T^{16} - \)\(93\!\cdots\!69\)\( p^{21} T^{17} + \)\(11\!\cdots\!49\)\( p^{28} T^{18} - \)\(50\!\cdots\!81\)\( p^{35} T^{19} + \)\(57\!\cdots\!59\)\( p^{42} T^{20} - \)\(21\!\cdots\!56\)\( p^{49} T^{21} + \)\(21\!\cdots\!27\)\( p^{56} T^{22} - \)\(64\!\cdots\!09\)\( p^{63} T^{23} + \)\(61\!\cdots\!00\)\( p^{70} T^{24} - 12619915266155463944 p^{77} T^{25} + 11121286239186 p^{84} T^{26} - 1211317 p^{91} T^{27} + p^{98} T^{28} \) | |
| 59 | \( 1 - 2551943 T + 20648552417368 T^{2} - 56137769925668723334 T^{3} + \)\(22\!\cdots\!34\)\( T^{4} - \)\(57\!\cdots\!49\)\( T^{5} + \)\(16\!\cdots\!35\)\( T^{6} - \)\(38\!\cdots\!08\)\( T^{7} + \)\(88\!\cdots\!87\)\( T^{8} - \)\(18\!\cdots\!95\)\( T^{9} + \)\(37\!\cdots\!03\)\( T^{10} - \)\(70\!\cdots\!65\)\( T^{11} + \)\(12\!\cdots\!48\)\( T^{12} - \)\(21\!\cdots\!95\)\( T^{13} + \)\(34\!\cdots\!23\)\( T^{14} - \)\(21\!\cdots\!95\)\( p^{7} T^{15} + \)\(12\!\cdots\!48\)\( p^{14} T^{16} - \)\(70\!\cdots\!65\)\( p^{21} T^{17} + \)\(37\!\cdots\!03\)\( p^{28} T^{18} - \)\(18\!\cdots\!95\)\( p^{35} T^{19} + \)\(88\!\cdots\!87\)\( p^{42} T^{20} - \)\(38\!\cdots\!08\)\( p^{49} T^{21} + \)\(16\!\cdots\!35\)\( p^{56} T^{22} - \)\(57\!\cdots\!49\)\( p^{63} T^{23} + \)\(22\!\cdots\!34\)\( p^{70} T^{24} - 56137769925668723334 p^{77} T^{25} + 20648552417368 p^{84} T^{26} - 2551943 p^{91} T^{27} + p^{98} T^{28} \) | |
| 61 | \( 1 + 10221163 T + 69025952851969 T^{2} + \)\(35\!\cdots\!17\)\( T^{3} + \)\(15\!\cdots\!48\)\( T^{4} + \)\(55\!\cdots\!33\)\( T^{5} + \)\(18\!\cdots\!45\)\( T^{6} + \)\(55\!\cdots\!79\)\( T^{7} + \)\(15\!\cdots\!25\)\( T^{8} + \)\(38\!\cdots\!52\)\( T^{9} + \)\(91\!\cdots\!02\)\( T^{10} + \)\(20\!\cdots\!54\)\( T^{11} + \)\(41\!\cdots\!14\)\( T^{12} + \)\(13\!\cdots\!46\)\( p T^{13} + \)\(14\!\cdots\!44\)\( T^{14} + \)\(13\!\cdots\!46\)\( p^{8} T^{15} + \)\(41\!\cdots\!14\)\( p^{14} T^{16} + \)\(20\!\cdots\!54\)\( p^{21} T^{17} + \)\(91\!\cdots\!02\)\( p^{28} T^{18} + \)\(38\!\cdots\!52\)\( p^{35} T^{19} + \)\(15\!\cdots\!25\)\( p^{42} T^{20} + \)\(55\!\cdots\!79\)\( p^{49} T^{21} + \)\(18\!\cdots\!45\)\( p^{56} T^{22} + \)\(55\!\cdots\!33\)\( p^{63} T^{23} + \)\(15\!\cdots\!48\)\( p^{70} T^{24} + \)\(35\!\cdots\!17\)\( p^{77} T^{25} + 69025952851969 p^{84} T^{26} + 10221163 p^{91} T^{27} + p^{98} T^{28} \) | |
| 67 | \( 1 + 53293 p T + 74739295526303 T^{2} + \)\(23\!\cdots\!47\)\( T^{3} + \)\(26\!\cdots\!68\)\( T^{4} + \)\(75\!\cdots\!21\)\( T^{5} + \)\(56\!\cdots\!27\)\( T^{6} + \)\(14\!\cdots\!77\)\( T^{7} + \)\(85\!\cdots\!53\)\( T^{8} + \)\(20\!\cdots\!08\)\( T^{9} + \)\(96\!\cdots\!46\)\( T^{10} + \)\(20\!\cdots\!86\)\( T^{11} + \)\(84\!\cdots\!22\)\( T^{12} + \)\(16\!\cdots\!02\)\( T^{13} + \)\(57\!\cdots\!20\)\( T^{14} + \)\(16\!\cdots\!02\)\( p^{7} T^{15} + \)\(84\!\cdots\!22\)\( p^{14} T^{16} + \)\(20\!\cdots\!86\)\( p^{21} T^{17} + \)\(96\!\cdots\!46\)\( p^{28} T^{18} + \)\(20\!\cdots\!08\)\( p^{35} T^{19} + \)\(85\!\cdots\!53\)\( p^{42} T^{20} + \)\(14\!\cdots\!77\)\( p^{49} T^{21} + \)\(56\!\cdots\!27\)\( p^{56} T^{22} + \)\(75\!\cdots\!21\)\( p^{63} T^{23} + \)\(26\!\cdots\!68\)\( p^{70} T^{24} + \)\(23\!\cdots\!47\)\( p^{77} T^{25} + 74739295526303 p^{84} T^{26} + 53293 p^{92} T^{27} + p^{98} T^{28} \) | |
| 71 | \( 1 - 4528437 T + 101013641343065 T^{2} - \)\(42\!\cdots\!97\)\( T^{3} + \)\(49\!\cdots\!40\)\( T^{4} - \)\(18\!\cdots\!07\)\( T^{5} + \)\(15\!\cdots\!61\)\( T^{6} - \)\(53\!\cdots\!23\)\( T^{7} + \)\(34\!\cdots\!25\)\( T^{8} - \)\(10\!\cdots\!68\)\( T^{9} + \)\(58\!\cdots\!74\)\( T^{10} - \)\(16\!\cdots\!26\)\( T^{11} + \)\(75\!\cdots\!14\)\( T^{12} - \)\(19\!\cdots\!02\)\( T^{13} + \)\(77\!\cdots\!00\)\( T^{14} - \)\(19\!\cdots\!02\)\( p^{7} T^{15} + \)\(75\!\cdots\!14\)\( p^{14} T^{16} - \)\(16\!\cdots\!26\)\( p^{21} T^{17} + \)\(58\!\cdots\!74\)\( p^{28} T^{18} - \)\(10\!\cdots\!68\)\( p^{35} T^{19} + \)\(34\!\cdots\!25\)\( p^{42} T^{20} - \)\(53\!\cdots\!23\)\( p^{49} T^{21} + \)\(15\!\cdots\!61\)\( p^{56} T^{22} - \)\(18\!\cdots\!07\)\( p^{63} T^{23} + \)\(49\!\cdots\!40\)\( p^{70} T^{24} - \)\(42\!\cdots\!97\)\( p^{77} T^{25} + 101013641343065 p^{84} T^{26} - 4528437 p^{91} T^{27} + p^{98} T^{28} \) | |
| 73 | \( 1 + 13052276 T + 169858754045429 T^{2} + \)\(13\!\cdots\!56\)\( T^{3} + \)\(10\!\cdots\!44\)\( T^{4} + \)\(66\!\cdots\!26\)\( T^{5} + \)\(40\!\cdots\!08\)\( T^{6} + \)\(20\!\cdots\!94\)\( T^{7} + \)\(10\!\cdots\!26\)\( T^{8} + \)\(45\!\cdots\!96\)\( T^{9} + \)\(20\!\cdots\!83\)\( T^{10} + \)\(79\!\cdots\!68\)\( T^{11} + \)\(30\!\cdots\!53\)\( T^{12} + \)\(10\!\cdots\!84\)\( T^{13} + \)\(38\!\cdots\!32\)\( T^{14} + \)\(10\!\cdots\!84\)\( p^{7} T^{15} + \)\(30\!\cdots\!53\)\( p^{14} T^{16} + \)\(79\!\cdots\!68\)\( p^{21} T^{17} + \)\(20\!\cdots\!83\)\( p^{28} T^{18} + \)\(45\!\cdots\!96\)\( p^{35} T^{19} + \)\(10\!\cdots\!26\)\( p^{42} T^{20} + \)\(20\!\cdots\!94\)\( p^{49} T^{21} + \)\(40\!\cdots\!08\)\( p^{56} T^{22} + \)\(66\!\cdots\!26\)\( p^{63} T^{23} + \)\(10\!\cdots\!44\)\( p^{70} T^{24} + \)\(13\!\cdots\!56\)\( p^{77} T^{25} + 169858754045429 p^{84} T^{26} + 13052276 p^{91} T^{27} + p^{98} T^{28} \) | |
| 79 | \( 1 + 14911090 T + 195967536612692 T^{2} + \)\(17\!\cdots\!48\)\( T^{3} + \)\(14\!\cdots\!99\)\( T^{4} + \)\(93\!\cdots\!00\)\( T^{5} + \)\(57\!\cdots\!49\)\( T^{6} + \)\(30\!\cdots\!84\)\( T^{7} + \)\(15\!\cdots\!94\)\( T^{8} + \)\(75\!\cdots\!30\)\( T^{9} + \)\(36\!\cdots\!58\)\( T^{10} + \)\(16\!\cdots\!30\)\( T^{11} + \)\(78\!\cdots\!28\)\( T^{12} + \)\(34\!\cdots\!04\)\( T^{13} + \)\(15\!\cdots\!03\)\( T^{14} + \)\(34\!\cdots\!04\)\( p^{7} T^{15} + \)\(78\!\cdots\!28\)\( p^{14} T^{16} + \)\(16\!\cdots\!30\)\( p^{21} T^{17} + \)\(36\!\cdots\!58\)\( p^{28} T^{18} + \)\(75\!\cdots\!30\)\( p^{35} T^{19} + \)\(15\!\cdots\!94\)\( p^{42} T^{20} + \)\(30\!\cdots\!84\)\( p^{49} T^{21} + \)\(57\!\cdots\!49\)\( p^{56} T^{22} + \)\(93\!\cdots\!00\)\( p^{63} T^{23} + \)\(14\!\cdots\!99\)\( p^{70} T^{24} + \)\(17\!\cdots\!48\)\( p^{77} T^{25} + 195967536612692 p^{84} T^{26} + 14911090 p^{91} T^{27} + p^{98} T^{28} \) | |
| 83 | \( 1 + 7068900 T + 209634241812814 T^{2} + \)\(11\!\cdots\!42\)\( T^{3} + \)\(20\!\cdots\!53\)\( T^{4} + \)\(83\!\cdots\!18\)\( T^{5} + \)\(12\!\cdots\!73\)\( T^{6} + \)\(38\!\cdots\!10\)\( T^{7} + \)\(57\!\cdots\!64\)\( T^{8} + \)\(12\!\cdots\!14\)\( T^{9} + \)\(21\!\cdots\!78\)\( T^{10} + \)\(35\!\cdots\!26\)\( T^{11} + \)\(70\!\cdots\!94\)\( T^{12} + \)\(95\!\cdots\!30\)\( T^{13} + \)\(20\!\cdots\!47\)\( T^{14} + \)\(95\!\cdots\!30\)\( p^{7} T^{15} + \)\(70\!\cdots\!94\)\( p^{14} T^{16} + \)\(35\!\cdots\!26\)\( p^{21} T^{17} + \)\(21\!\cdots\!78\)\( p^{28} T^{18} + \)\(12\!\cdots\!14\)\( p^{35} T^{19} + \)\(57\!\cdots\!64\)\( p^{42} T^{20} + \)\(38\!\cdots\!10\)\( p^{49} T^{21} + \)\(12\!\cdots\!73\)\( p^{56} T^{22} + \)\(83\!\cdots\!18\)\( p^{63} T^{23} + \)\(20\!\cdots\!53\)\( p^{70} T^{24} + \)\(11\!\cdots\!42\)\( p^{77} T^{25} + 209634241812814 p^{84} T^{26} + 7068900 p^{91} T^{27} + p^{98} T^{28} \) | |
| 89 | \( 1 - 4800832 T + 335155551643121 T^{2} - \)\(16\!\cdots\!48\)\( T^{3} + \)\(55\!\cdots\!01\)\( T^{4} - \)\(28\!\cdots\!16\)\( T^{5} + \)\(62\!\cdots\!24\)\( T^{6} - \)\(32\!\cdots\!96\)\( T^{7} + \)\(52\!\cdots\!52\)\( T^{8} - \)\(28\!\cdots\!80\)\( T^{9} + \)\(35\!\cdots\!92\)\( T^{10} - \)\(18\!\cdots\!36\)\( T^{11} + \)\(20\!\cdots\!46\)\( T^{12} - \)\(10\!\cdots\!92\)\( T^{13} + \)\(97\!\cdots\!46\)\( T^{14} - \)\(10\!\cdots\!92\)\( p^{7} T^{15} + \)\(20\!\cdots\!46\)\( p^{14} T^{16} - \)\(18\!\cdots\!36\)\( p^{21} T^{17} + \)\(35\!\cdots\!92\)\( p^{28} T^{18} - \)\(28\!\cdots\!80\)\( p^{35} T^{19} + \)\(52\!\cdots\!52\)\( p^{42} T^{20} - \)\(32\!\cdots\!96\)\( p^{49} T^{21} + \)\(62\!\cdots\!24\)\( p^{56} T^{22} - \)\(28\!\cdots\!16\)\( p^{63} T^{23} + \)\(55\!\cdots\!01\)\( p^{70} T^{24} - \)\(16\!\cdots\!48\)\( p^{77} T^{25} + 335155551643121 p^{84} T^{26} - 4800832 p^{91} T^{27} + p^{98} T^{28} \) | |
| 97 | \( 1 + 11578773 T + 500689587811442 T^{2} + \)\(31\!\cdots\!04\)\( T^{3} + \)\(92\!\cdots\!32\)\( T^{4} + \)\(59\!\cdots\!57\)\( T^{5} + \)\(81\!\cdots\!63\)\( T^{6} - \)\(78\!\cdots\!04\)\( T^{7} + \)\(45\!\cdots\!55\)\( T^{8} - \)\(12\!\cdots\!87\)\( T^{9} + \)\(66\!\cdots\!13\)\( T^{10} - \)\(97\!\cdots\!71\)\( T^{11} + \)\(12\!\cdots\!28\)\( T^{12} - \)\(44\!\cdots\!51\)\( T^{13} + \)\(13\!\cdots\!49\)\( T^{14} - \)\(44\!\cdots\!51\)\( p^{7} T^{15} + \)\(12\!\cdots\!28\)\( p^{14} T^{16} - \)\(97\!\cdots\!71\)\( p^{21} T^{17} + \)\(66\!\cdots\!13\)\( p^{28} T^{18} - \)\(12\!\cdots\!87\)\( p^{35} T^{19} + \)\(45\!\cdots\!55\)\( p^{42} T^{20} - \)\(78\!\cdots\!04\)\( p^{49} T^{21} + \)\(81\!\cdots\!63\)\( p^{56} T^{22} + \)\(59\!\cdots\!57\)\( p^{63} T^{23} + \)\(92\!\cdots\!32\)\( p^{70} T^{24} + \)\(31\!\cdots\!04\)\( p^{77} T^{25} + 500689587811442 p^{84} T^{26} + 11578773 p^{91} T^{27} + p^{98} T^{28} \) | |
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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
−3.00767048291531021894842373411, −2.76280944015801313301949505333, −2.73354421418563840616816095944, −2.63334034507469629195322186047, −2.41211449561878759395688354223, −2.38764085781232366381635547565, −2.35593811402278173279038224403, −2.27620205318970154766087910770, −2.25023820043809453282514338294, −2.15032955943229640675669997924, −2.02926686267831389540881537539, −2.02338232771774846655521662089, −2.01301876338942582184365519724, −1.94862856077960390623117415873, −1.61244219001054657713877706033, −1.49766742522953151798129901661, −1.47149781430235305181988039793, −1.42720416792317801592150424259, −1.41279766236296321323253915426, −1.21136786800230336423124736983, −1.21009666982804796945949999569, −1.15636083935061643718101165588, −1.11811457856444674968113690443, −0.906040062722935065392603651007, −0.821336464083524446196762826862, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.821336464083524446196762826862, 0.906040062722935065392603651007, 1.11811457856444674968113690443, 1.15636083935061643718101165588, 1.21009666982804796945949999569, 1.21136786800230336423124736983, 1.41279766236296321323253915426, 1.42720416792317801592150424259, 1.47149781430235305181988039793, 1.49766742522953151798129901661, 1.61244219001054657713877706033, 1.94862856077960390623117415873, 2.01301876338942582184365519724, 2.02338232771774846655521662089, 2.02926686267831389540881537539, 2.15032955943229640675669997924, 2.25023820043809453282514338294, 2.27620205318970154766087910770, 2.35593811402278173279038224403, 2.38764085781232366381635547565, 2.41211449561878759395688354223, 2.63334034507469629195322186047, 2.73354421418563840616816095944, 2.76280944015801313301949505333, 3.00767048291531021894842373411
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.