Dirichlet series
| L(s) = 1 | + 9·2-s + 264·4-s + 180·5-s − 84·7-s + 927·8-s + 1.62e3·10-s + 8.46e3·11-s − 1.84e3·13-s − 756·14-s + 3.06e4·16-s − 3.05e4·17-s + 2.44e4·19-s + 4.75e4·20-s + 7.61e4·22-s + 5.15e4·23-s + 2.52e5·25-s − 1.66e4·26-s − 2.21e4·28-s + 4.14e5·29-s + 8.19e3·31-s + 1.32e5·32-s − 2.75e5·34-s − 1.51e4·35-s + 1.39e5·37-s + 2.19e5·38-s + 1.66e5·40-s + 1.73e6·41-s + ⋯ |
| L(s) = 1 | + 0.795·2-s + 2.06·4-s + 0.643·5-s − 0.0925·7-s + 0.640·8-s + 0.512·10-s + 1.91·11-s − 0.233·13-s − 0.0736·14-s + 1.86·16-s − 1.50·17-s + 0.817·19-s + 1.32·20-s + 1.52·22-s + 0.884·23-s + 3.23·25-s − 0.185·26-s − 0.190·28-s + 3.15·29-s + 0.0494·31-s + 0.715·32-s − 1.20·34-s − 0.0596·35-s + 0.452·37-s + 0.650·38-s + 0.412·40-s + 3.92·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{36}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.29612\times 10^{11}\) |
| Root analytic conductor: | \(2.90420\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{36} ,\ ( \ : [7/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(157.0679351\) |
| \(L(\frac12)\) | \(\approx\) | \(157.0679351\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 9 T - 183 T^{2} + 387 p^{3} T^{3} - 57 p^{5} T^{4} - 30519 p^{4} T^{5} + 246755 p^{4} T^{6} + 800631 p^{6} T^{7} - 176949 p^{12} T^{8} - 21294711 p^{8} T^{9} + 412724205 p^{8} T^{10} + 294337179 p^{10} T^{11} - 3323542503 p^{12} T^{12} + 294337179 p^{17} T^{13} + 412724205 p^{22} T^{14} - 21294711 p^{29} T^{15} - 176949 p^{40} T^{16} + 800631 p^{41} T^{17} + 246755 p^{46} T^{18} - 30519 p^{53} T^{19} - 57 p^{61} T^{20} + 387 p^{66} T^{21} - 183 p^{70} T^{22} - 9 p^{77} T^{23} + p^{84} T^{24} \) |
| 5 | \( 1 - 36 p T - 220548 T^{2} + 16495344 p T^{3} + 18620801904 T^{4} - 2495133938628 p T^{5} + 15309148946708 p^{2} T^{6} + 7067396112700932 p^{3} T^{7} - 54690247841275656 p^{5} T^{8} - 7151119603879528656 p^{5} T^{9} + \)\(71\!\cdots\!88\)\( p^{6} T^{10} - \)\(34\!\cdots\!88\)\( p^{7} T^{11} - \)\(14\!\cdots\!74\)\( p^{8} T^{12} - \)\(34\!\cdots\!88\)\( p^{14} T^{13} + \)\(71\!\cdots\!88\)\( p^{20} T^{14} - 7151119603879528656 p^{26} T^{15} - 54690247841275656 p^{33} T^{16} + 7067396112700932 p^{38} T^{17} + 15309148946708 p^{44} T^{18} - 2495133938628 p^{50} T^{19} + 18620801904 p^{56} T^{20} + 16495344 p^{64} T^{21} - 220548 p^{70} T^{22} - 36 p^{78} T^{23} + p^{84} T^{24} \) | |
| 7 | \( 1 + 12 p T - 2377596 T^{2} + 213453616 p T^{3} + 2757308182656 T^{4} - 454435277572572 p T^{5} - 114699785351547668 p T^{6} + \)\(40\!\cdots\!84\)\( p T^{7} - \)\(92\!\cdots\!04\)\( T^{8} - \)\(14\!\cdots\!16\)\( p T^{9} + \)\(62\!\cdots\!40\)\( T^{10} - \)\(45\!\cdots\!76\)\( p T^{11} - \)\(22\!\cdots\!70\)\( T^{12} - \)\(45\!\cdots\!76\)\( p^{8} T^{13} + \)\(62\!\cdots\!40\)\( p^{14} T^{14} - \)\(14\!\cdots\!16\)\( p^{22} T^{15} - \)\(92\!\cdots\!04\)\( p^{28} T^{16} + \)\(40\!\cdots\!84\)\( p^{36} T^{17} - 114699785351547668 p^{43} T^{18} - 454435277572572 p^{50} T^{19} + 2757308182656 p^{56} T^{20} + 213453616 p^{64} T^{21} - 2377596 p^{70} T^{22} + 12 p^{78} T^{23} + p^{84} T^{24} \) | |
| 11 | \( 1 - 8460 T - 25736151 T^{2} + 55078409556 T^{3} + 2824084004151891 T^{4} - 1614161825654271912 T^{5} - \)\(53\!\cdots\!24\)\( T^{6} - \)\(27\!\cdots\!76\)\( T^{7} + \)\(11\!\cdots\!77\)\( T^{8} + \)\(47\!\cdots\!28\)\( T^{9} + \)\(12\!\cdots\!63\)\( T^{10} - \)\(92\!\cdots\!92\)\( T^{11} - \)\(20\!\cdots\!86\)\( T^{12} - \)\(92\!\cdots\!92\)\( p^{7} T^{13} + \)\(12\!\cdots\!63\)\( p^{14} T^{14} + \)\(47\!\cdots\!28\)\( p^{21} T^{15} + \)\(11\!\cdots\!77\)\( p^{28} T^{16} - \)\(27\!\cdots\!76\)\( p^{35} T^{17} - \)\(53\!\cdots\!24\)\( p^{42} T^{18} - 1614161825654271912 p^{49} T^{19} + 2824084004151891 p^{56} T^{20} + 55078409556 p^{63} T^{21} - 25736151 p^{70} T^{22} - 8460 p^{77} T^{23} + p^{84} T^{24} \) | |
| 13 | \( 1 + 1848 T - 185188728 T^{2} - 432313748216 T^{3} + 15543298148736264 T^{4} + 54936881684405686344 T^{5} - \)\(72\!\cdots\!88\)\( T^{6} - \)\(63\!\cdots\!88\)\( T^{7} + \)\(18\!\cdots\!88\)\( T^{8} + \)\(36\!\cdots\!60\)\( p T^{9} - \)\(54\!\cdots\!88\)\( T^{10} - \)\(13\!\cdots\!60\)\( T^{11} + \)\(44\!\cdots\!50\)\( T^{12} - \)\(13\!\cdots\!60\)\( p^{7} T^{13} - \)\(54\!\cdots\!88\)\( p^{14} T^{14} + \)\(36\!\cdots\!60\)\( p^{22} T^{15} + \)\(18\!\cdots\!88\)\( p^{28} T^{16} - \)\(63\!\cdots\!88\)\( p^{35} T^{17} - \)\(72\!\cdots\!88\)\( p^{42} T^{18} + 54936881684405686344 p^{49} T^{19} + 15543298148736264 p^{56} T^{20} - 432313748216 p^{63} T^{21} - 185188728 p^{70} T^{22} + 1848 p^{77} T^{23} + p^{84} T^{24} \) | |
| 17 | \( ( 1 + 15282 T + 1977260727 T^{2} + 24425219624706 T^{3} + 1753804802788120635 T^{4} + \)\(17\!\cdots\!92\)\( T^{5} + \)\(91\!\cdots\!02\)\( T^{6} + \)\(17\!\cdots\!92\)\( p^{7} T^{7} + 1753804802788120635 p^{14} T^{8} + 24425219624706 p^{21} T^{9} + 1977260727 p^{28} T^{10} + 15282 p^{35} T^{11} + p^{42} T^{12} )^{2} \) | |
| 19 | \( ( 1 - 12216 T + 3184630329 T^{2} - 6663360860248 T^{3} + 4189366163547965259 T^{4} + \)\(26\!\cdots\!24\)\( T^{5} + \)\(38\!\cdots\!22\)\( T^{6} + \)\(26\!\cdots\!24\)\( p^{7} T^{7} + 4189366163547965259 p^{14} T^{8} - 6663360860248 p^{21} T^{9} + 3184630329 p^{28} T^{10} - 12216 p^{35} T^{11} + p^{42} T^{12} )^{2} \) | |
| 23 | \( 1 - 51588 T - 7731166620 T^{2} - 13934509937568 T^{3} + 37240413272987354256 T^{4} + \)\(17\!\cdots\!76\)\( T^{5} - \)\(93\!\cdots\!56\)\( T^{6} - \)\(21\!\cdots\!32\)\( T^{7} + \)\(10\!\cdots\!68\)\( T^{8} - \)\(18\!\cdots\!44\)\( T^{9} - \)\(18\!\cdots\!76\)\( T^{10} + \)\(41\!\cdots\!52\)\( T^{11} + \)\(12\!\cdots\!78\)\( T^{12} + \)\(41\!\cdots\!52\)\( p^{7} T^{13} - \)\(18\!\cdots\!76\)\( p^{14} T^{14} - \)\(18\!\cdots\!44\)\( p^{21} T^{15} + \)\(10\!\cdots\!68\)\( p^{28} T^{16} - \)\(21\!\cdots\!32\)\( p^{35} T^{17} - \)\(93\!\cdots\!56\)\( p^{42} T^{18} + \)\(17\!\cdots\!76\)\( p^{49} T^{19} + 37240413272987354256 p^{56} T^{20} - 13934509937568 p^{63} T^{21} - 7731166620 p^{70} T^{22} - 51588 p^{77} T^{23} + p^{84} T^{24} \) | |
| 29 | \( 1 - 414648 T + 31769341512 T^{2} + 6581786545827864 T^{3} - \)\(27\!\cdots\!48\)\( T^{4} - \)\(19\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!16\)\( T^{6} + \)\(37\!\cdots\!64\)\( T^{7} + \)\(80\!\cdots\!20\)\( T^{8} - \)\(41\!\cdots\!76\)\( T^{9} - \)\(50\!\cdots\!04\)\( T^{10} + \)\(10\!\cdots\!12\)\( T^{11} + \)\(15\!\cdots\!66\)\( T^{12} + \)\(10\!\cdots\!12\)\( p^{7} T^{13} - \)\(50\!\cdots\!04\)\( p^{14} T^{14} - \)\(41\!\cdots\!76\)\( p^{21} T^{15} + \)\(80\!\cdots\!20\)\( p^{28} T^{16} + \)\(37\!\cdots\!64\)\( p^{35} T^{17} + \)\(10\!\cdots\!16\)\( p^{42} T^{18} - \)\(19\!\cdots\!16\)\( p^{49} T^{19} - \)\(27\!\cdots\!48\)\( p^{56} T^{20} + 6581786545827864 p^{63} T^{21} + 31769341512 p^{70} T^{22} - 414648 p^{77} T^{23} + p^{84} T^{24} \) | |
| 31 | \( 1 - 8196 T - 95383966224 T^{2} + 7353543400934752 T^{3} + \)\(49\!\cdots\!00\)\( T^{4} - \)\(62\!\cdots\!60\)\( T^{5} - \)\(13\!\cdots\!24\)\( T^{6} + \)\(31\!\cdots\!04\)\( T^{7} + \)\(92\!\cdots\!00\)\( T^{8} - \)\(89\!\cdots\!56\)\( T^{9} + \)\(90\!\cdots\!24\)\( T^{10} + \)\(11\!\cdots\!32\)\( T^{11} - \)\(38\!\cdots\!26\)\( T^{12} + \)\(11\!\cdots\!32\)\( p^{7} T^{13} + \)\(90\!\cdots\!24\)\( p^{14} T^{14} - \)\(89\!\cdots\!56\)\( p^{21} T^{15} + \)\(92\!\cdots\!00\)\( p^{28} T^{16} + \)\(31\!\cdots\!04\)\( p^{35} T^{17} - \)\(13\!\cdots\!24\)\( p^{42} T^{18} - \)\(62\!\cdots\!60\)\( p^{49} T^{19} + \)\(49\!\cdots\!00\)\( p^{56} T^{20} + 7353543400934752 p^{63} T^{21} - 95383966224 p^{70} T^{22} - 8196 p^{77} T^{23} + p^{84} T^{24} \) | |
| 37 | \( ( 1 - 69672 T + 428372792310 T^{2} - 21254579002480072 T^{3} + \)\(84\!\cdots\!59\)\( T^{4} - \)\(34\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!52\)\( T^{6} - \)\(34\!\cdots\!16\)\( p^{7} T^{7} + \)\(84\!\cdots\!59\)\( p^{14} T^{8} - 21254579002480072 p^{21} T^{9} + 428372792310 p^{28} T^{10} - 69672 p^{35} T^{11} + p^{42} T^{12} )^{2} \) | |
| 41 | \( 1 - 1731582 T + 663188299215 T^{2} + 113380453022686974 T^{3} + \)\(30\!\cdots\!47\)\( T^{4} - \)\(38\!\cdots\!68\)\( T^{5} + \)\(43\!\cdots\!08\)\( T^{6} + \)\(14\!\cdots\!48\)\( T^{7} + \)\(54\!\cdots\!45\)\( T^{8} - \)\(25\!\cdots\!50\)\( T^{9} - \)\(26\!\cdots\!07\)\( T^{10} - \)\(20\!\cdots\!78\)\( T^{11} + \)\(34\!\cdots\!10\)\( T^{12} - \)\(20\!\cdots\!78\)\( p^{7} T^{13} - \)\(26\!\cdots\!07\)\( p^{14} T^{14} - \)\(25\!\cdots\!50\)\( p^{21} T^{15} + \)\(54\!\cdots\!45\)\( p^{28} T^{16} + \)\(14\!\cdots\!48\)\( p^{35} T^{17} + \)\(43\!\cdots\!08\)\( p^{42} T^{18} - \)\(38\!\cdots\!68\)\( p^{49} T^{19} + \)\(30\!\cdots\!47\)\( p^{56} T^{20} + 113380453022686974 p^{63} T^{21} + 663188299215 p^{70} T^{22} - 1731582 p^{77} T^{23} + p^{84} T^{24} \) | |
| 43 | \( 1 - 408372 T - 752243449935 T^{2} + 2875479618164836 p T^{3} + \)\(37\!\cdots\!11\)\( T^{4} + \)\(20\!\cdots\!64\)\( T^{5} - \)\(88\!\cdots\!76\)\( T^{6} - \)\(53\!\cdots\!08\)\( T^{7} + \)\(10\!\cdots\!13\)\( T^{8} + \)\(19\!\cdots\!04\)\( T^{9} + \)\(35\!\cdots\!59\)\( T^{10} - \)\(29\!\cdots\!92\)\( T^{11} - \)\(13\!\cdots\!42\)\( T^{12} - \)\(29\!\cdots\!92\)\( p^{7} T^{13} + \)\(35\!\cdots\!59\)\( p^{14} T^{14} + \)\(19\!\cdots\!04\)\( p^{21} T^{15} + \)\(10\!\cdots\!13\)\( p^{28} T^{16} - \)\(53\!\cdots\!08\)\( p^{35} T^{17} - \)\(88\!\cdots\!76\)\( p^{42} T^{18} + \)\(20\!\cdots\!64\)\( p^{49} T^{19} + \)\(37\!\cdots\!11\)\( p^{56} T^{20} + 2875479618164836 p^{64} T^{21} - 752243449935 p^{70} T^{22} - 408372 p^{77} T^{23} + p^{84} T^{24} \) | |
| 47 | \( 1 - 1631484 T + 160944084708 T^{2} + 713488342682427552 T^{3} + \)\(26\!\cdots\!36\)\( T^{4} - \)\(76\!\cdots\!60\)\( T^{5} + \)\(22\!\cdots\!84\)\( T^{6} + \)\(20\!\cdots\!60\)\( T^{7} - \)\(13\!\cdots\!56\)\( T^{8} - \)\(14\!\cdots\!16\)\( T^{9} + \)\(22\!\cdots\!48\)\( T^{10} - \)\(20\!\cdots\!32\)\( T^{11} - \)\(83\!\cdots\!02\)\( T^{12} - \)\(20\!\cdots\!32\)\( p^{7} T^{13} + \)\(22\!\cdots\!48\)\( p^{14} T^{14} - \)\(14\!\cdots\!16\)\( p^{21} T^{15} - \)\(13\!\cdots\!56\)\( p^{28} T^{16} + \)\(20\!\cdots\!60\)\( p^{35} T^{17} + \)\(22\!\cdots\!84\)\( p^{42} T^{18} - \)\(76\!\cdots\!60\)\( p^{49} T^{19} + \)\(26\!\cdots\!36\)\( p^{56} T^{20} + 713488342682427552 p^{63} T^{21} + 160944084708 p^{70} T^{22} - 1631484 p^{77} T^{23} + p^{84} T^{24} \) | |
| 53 | \( ( 1 + 1417824 T + 5329846489782 T^{2} + 6873564019198682304 T^{3} + \)\(13\!\cdots\!35\)\( T^{4} + \)\(14\!\cdots\!48\)\( T^{5} + \)\(19\!\cdots\!24\)\( T^{6} + \)\(14\!\cdots\!48\)\( p^{7} T^{7} + \)\(13\!\cdots\!35\)\( p^{14} T^{8} + 6873564019198682304 p^{21} T^{9} + 5329846489782 p^{28} T^{10} + 1417824 p^{35} T^{11} + p^{42} T^{12} )^{2} \) | |
| 59 | \( 1 - 2055636 T - 5809797682743 T^{2} + 2746105158756095820 T^{3} + \)\(67\!\cdots\!53\)\( p T^{4} + \)\(87\!\cdots\!60\)\( T^{5} - \)\(10\!\cdots\!72\)\( T^{6} - \)\(17\!\cdots\!84\)\( T^{7} + \)\(17\!\cdots\!17\)\( T^{8} + \)\(47\!\cdots\!32\)\( T^{9} + \)\(40\!\cdots\!39\)\( T^{10} - \)\(78\!\cdots\!40\)\( T^{11} - \)\(13\!\cdots\!98\)\( T^{12} - \)\(78\!\cdots\!40\)\( p^{7} T^{13} + \)\(40\!\cdots\!39\)\( p^{14} T^{14} + \)\(47\!\cdots\!32\)\( p^{21} T^{15} + \)\(17\!\cdots\!17\)\( p^{28} T^{16} - \)\(17\!\cdots\!84\)\( p^{35} T^{17} - \)\(10\!\cdots\!72\)\( p^{42} T^{18} + \)\(87\!\cdots\!60\)\( p^{49} T^{19} + \)\(67\!\cdots\!53\)\( p^{57} T^{20} + 2746105158756095820 p^{63} T^{21} - 5809797682743 p^{70} T^{22} - 2055636 p^{77} T^{23} + p^{84} T^{24} \) | |
| 61 | \( 1 + 2723196 T - 6806305437204 T^{2} - 17706905500131240704 T^{3} + \)\(38\!\cdots\!36\)\( T^{4} + \)\(63\!\cdots\!52\)\( T^{5} - \)\(15\!\cdots\!88\)\( T^{6} - \)\(10\!\cdots\!48\)\( T^{7} + \)\(54\!\cdots\!32\)\( T^{8} + \)\(54\!\cdots\!52\)\( T^{9} - \)\(13\!\cdots\!52\)\( T^{10} + \)\(93\!\cdots\!88\)\( T^{11} + \)\(36\!\cdots\!98\)\( T^{12} + \)\(93\!\cdots\!88\)\( p^{7} T^{13} - \)\(13\!\cdots\!52\)\( p^{14} T^{14} + \)\(54\!\cdots\!52\)\( p^{21} T^{15} + \)\(54\!\cdots\!32\)\( p^{28} T^{16} - \)\(10\!\cdots\!48\)\( p^{35} T^{17} - \)\(15\!\cdots\!88\)\( p^{42} T^{18} + \)\(63\!\cdots\!52\)\( p^{49} T^{19} + \)\(38\!\cdots\!36\)\( p^{56} T^{20} - 17706905500131240704 p^{63} T^{21} - 6806305437204 p^{70} T^{22} + 2723196 p^{77} T^{23} + p^{84} T^{24} \) | |
| 67 | \( 1 - 3806556 T - 9134617912023 T^{2} + 66340343180276661124 T^{3} - \)\(22\!\cdots\!29\)\( T^{4} - \)\(40\!\cdots\!96\)\( T^{5} + \)\(94\!\cdots\!00\)\( T^{6} - \)\(62\!\cdots\!64\)\( T^{7} - \)\(37\!\cdots\!79\)\( T^{8} + \)\(22\!\cdots\!16\)\( T^{9} - \)\(35\!\cdots\!45\)\( T^{10} - \)\(80\!\cdots\!36\)\( T^{11} + \)\(43\!\cdots\!42\)\( T^{12} - \)\(80\!\cdots\!36\)\( p^{7} T^{13} - \)\(35\!\cdots\!45\)\( p^{14} T^{14} + \)\(22\!\cdots\!16\)\( p^{21} T^{15} - \)\(37\!\cdots\!79\)\( p^{28} T^{16} - \)\(62\!\cdots\!64\)\( p^{35} T^{17} + \)\(94\!\cdots\!00\)\( p^{42} T^{18} - \)\(40\!\cdots\!96\)\( p^{49} T^{19} - \)\(22\!\cdots\!29\)\( p^{56} T^{20} + 66340343180276661124 p^{63} T^{21} - 9134617912023 p^{70} T^{22} - 3806556 p^{77} T^{23} + p^{84} T^{24} \) | |
| 71 | \( ( 1 + 1204200 T + 40680637951386 T^{2} + 13974404037087114936 T^{3} + \)\(70\!\cdots\!75\)\( T^{4} - \)\(11\!\cdots\!52\)\( T^{5} + \)\(75\!\cdots\!84\)\( T^{6} - \)\(11\!\cdots\!52\)\( p^{7} T^{7} + \)\(70\!\cdots\!75\)\( p^{14} T^{8} + 13974404037087114936 p^{21} T^{9} + 40680637951386 p^{28} T^{10} + 1204200 p^{35} T^{11} + p^{42} T^{12} )^{2} \) | |
| 73 | \( ( 1 + 5335026 T + 43738542053175 T^{2} + \)\(16\!\cdots\!14\)\( T^{3} + \)\(85\!\cdots\!59\)\( T^{4} + \)\(26\!\cdots\!08\)\( T^{5} + \)\(11\!\cdots\!58\)\( T^{6} + \)\(26\!\cdots\!08\)\( p^{7} T^{7} + \)\(85\!\cdots\!59\)\( p^{14} T^{8} + \)\(16\!\cdots\!14\)\( p^{21} T^{9} + 43738542053175 p^{28} T^{10} + 5335026 p^{35} T^{11} + p^{42} T^{12} )^{2} \) | |
| 79 | \( 1 - 6020916 T - 70122599265648 T^{2} + \)\(42\!\cdots\!20\)\( T^{3} + \)\(32\!\cdots\!32\)\( T^{4} - \)\(17\!\cdots\!00\)\( T^{5} - \)\(10\!\cdots\!12\)\( T^{6} + \)\(47\!\cdots\!56\)\( T^{7} + \)\(28\!\cdots\!52\)\( T^{8} - \)\(83\!\cdots\!08\)\( T^{9} - \)\(66\!\cdots\!16\)\( T^{10} + \)\(63\!\cdots\!60\)\( T^{11} + \)\(13\!\cdots\!02\)\( T^{12} + \)\(63\!\cdots\!60\)\( p^{7} T^{13} - \)\(66\!\cdots\!16\)\( p^{14} T^{14} - \)\(83\!\cdots\!08\)\( p^{21} T^{15} + \)\(28\!\cdots\!52\)\( p^{28} T^{16} + \)\(47\!\cdots\!56\)\( p^{35} T^{17} - \)\(10\!\cdots\!12\)\( p^{42} T^{18} - \)\(17\!\cdots\!00\)\( p^{49} T^{19} + \)\(32\!\cdots\!32\)\( p^{56} T^{20} + \)\(42\!\cdots\!20\)\( p^{63} T^{21} - 70122599265648 p^{70} T^{22} - 6020916 p^{77} T^{23} + p^{84} T^{24} \) | |
| 83 | \( 1 + 9605052 T - 97427446819608 T^{2} - \)\(78\!\cdots\!44\)\( T^{3} + \)\(95\!\cdots\!96\)\( T^{4} + \)\(51\!\cdots\!40\)\( T^{5} - \)\(58\!\cdots\!44\)\( T^{6} - \)\(18\!\cdots\!60\)\( T^{7} + \)\(29\!\cdots\!44\)\( T^{8} + \)\(52\!\cdots\!28\)\( T^{9} - \)\(11\!\cdots\!48\)\( T^{10} - \)\(49\!\cdots\!36\)\( T^{11} + \)\(34\!\cdots\!58\)\( T^{12} - \)\(49\!\cdots\!36\)\( p^{7} T^{13} - \)\(11\!\cdots\!48\)\( p^{14} T^{14} + \)\(52\!\cdots\!28\)\( p^{21} T^{15} + \)\(29\!\cdots\!44\)\( p^{28} T^{16} - \)\(18\!\cdots\!60\)\( p^{35} T^{17} - \)\(58\!\cdots\!44\)\( p^{42} T^{18} + \)\(51\!\cdots\!40\)\( p^{49} T^{19} + \)\(95\!\cdots\!96\)\( p^{56} T^{20} - \)\(78\!\cdots\!44\)\( p^{63} T^{21} - 97427446819608 p^{70} T^{22} + 9605052 p^{77} T^{23} + p^{84} T^{24} \) | |
| 89 | \( ( 1 - 12315132 T + 262975895847282 T^{2} - \)\(24\!\cdots\!52\)\( T^{3} + \)\(29\!\cdots\!99\)\( T^{4} - \)\(20\!\cdots\!44\)\( T^{5} + \)\(17\!\cdots\!96\)\( T^{6} - \)\(20\!\cdots\!44\)\( p^{7} T^{7} + \)\(29\!\cdots\!99\)\( p^{14} T^{8} - \)\(24\!\cdots\!52\)\( p^{21} T^{9} + 262975895847282 p^{28} T^{10} - 12315132 p^{35} T^{11} + p^{42} T^{12} )^{2} \) | |
| 97 | \( 1 - 102858 p T - 243565972578345 T^{2} + \)\(21\!\cdots\!66\)\( T^{3} + \)\(32\!\cdots\!31\)\( T^{4} - \)\(19\!\cdots\!68\)\( T^{5} - \)\(41\!\cdots\!24\)\( T^{6} + \)\(11\!\cdots\!64\)\( T^{7} + \)\(49\!\cdots\!13\)\( T^{8} - \)\(89\!\cdots\!18\)\( T^{9} - \)\(43\!\cdots\!19\)\( T^{10} + \)\(38\!\cdots\!66\)\( T^{11} + \)\(33\!\cdots\!98\)\( T^{12} + \)\(38\!\cdots\!66\)\( p^{7} T^{13} - \)\(43\!\cdots\!19\)\( p^{14} T^{14} - \)\(89\!\cdots\!18\)\( p^{21} T^{15} + \)\(49\!\cdots\!13\)\( p^{28} T^{16} + \)\(11\!\cdots\!64\)\( p^{35} T^{17} - \)\(41\!\cdots\!24\)\( p^{42} T^{18} - \)\(19\!\cdots\!68\)\( p^{49} T^{19} + \)\(32\!\cdots\!31\)\( p^{56} T^{20} + \)\(21\!\cdots\!66\)\( p^{63} T^{21} - 243565972578345 p^{70} T^{22} - 102858 p^{78} T^{23} + p^{84} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.89262494199627318691227345441, −4.62304789085843018865209206816, −4.54094893402121111555745794736, −4.37437244935278899092905566766, −4.31869183302133458731619646778, −4.30586777205236860918703343570, −4.26348792682242192329613420885, −3.67937152837211937726356755605, −3.41820769535435579000164424120, −3.26646087005394778022548003631, −3.09877728602284930825096950188, −2.99990404956195678371369026175, −2.83552683462618708942594630337, −2.79621864186572152702597260937, −2.33761097148625353062640224967, −2.26434947898053426175958650163, −2.17344646477674092709092965650, −1.99809531518282661039187242748, −1.52780775808144603467658367125, −1.27930418734725997511411275023, −0.971517904559833672087795532531, −0.874364088891315031493422015662, −0.805578594831086751356085132335, −0.69782075374072603430680690711, −0.43415453425917737568884053680, 0.43415453425917737568884053680, 0.69782075374072603430680690711, 0.805578594831086751356085132335, 0.874364088891315031493422015662, 0.971517904559833672087795532531, 1.27930418734725997511411275023, 1.52780775808144603467658367125, 1.99809531518282661039187242748, 2.17344646477674092709092965650, 2.26434947898053426175958650163, 2.33761097148625353062640224967, 2.79621864186572152702597260937, 2.83552683462618708942594630337, 2.99990404956195678371369026175, 3.09877728602284930825096950188, 3.26646087005394778022548003631, 3.41820769535435579000164424120, 3.67937152837211937726356755605, 4.26348792682242192329613420885, 4.30586777205236860918703343570, 4.31869183302133458731619646778, 4.37437244935278899092905566766, 4.54094893402121111555745794736, 4.62304789085843018865209206816, 4.89262494199627318691227345441
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.