Dirichlet series
| L(s) = 1 | − 24·2-s + 44·3-s + 192·4-s + 220·5-s − 1.05e3·6-s + 534·7-s − 512·8-s + 2.37e3·9-s − 5.28e3·10-s − 1.28e4·11-s + 8.44e3·12-s − 1.83e3·13-s − 1.28e4·14-s + 9.68e3·15-s + 4.91e4·17-s − 5.70e4·18-s − 2.44e4·19-s + 4.22e4·20-s + 2.34e4·21-s + 3.09e5·22-s + 1.21e3·23-s − 2.25e4·24-s + 2.58e5·25-s + 4.40e4·26-s + 1.06e5·27-s + 1.02e5·28-s + 4.10e5·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 0.940·3-s + 3/2·4-s + 0.787·5-s − 1.99·6-s + 0.588·7-s − 0.353·8-s + 1.08·9-s − 1.66·10-s − 2.91·11-s + 1.41·12-s − 0.231·13-s − 1.24·14-s + 0.740·15-s + 2.42·17-s − 2.30·18-s − 0.817·19-s + 1.18·20-s + 0.553·21-s + 6.18·22-s + 0.0207·23-s − 0.332·24-s + 3.30·25-s + 0.491·26-s + 1.04·27-s + 0.882·28-s + 3.12·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 11^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.11008\times 10^{10}\) |
| Root analytic conductor: | \(2.62153\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 11^{12} ,\ ( \ : [7/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(0.3757753827\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3757753827\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{3} T + p^{6} T^{2} + p^{9} T^{3} + p^{12} T^{4} )^{3} \) |
| 11 | \( 1 + 1171 p T + 264857 p^{2} T^{2} - 231734363 p^{3} T^{3} - 11166285887 p^{5} T^{4} + 153155351582 p^{7} T^{5} + 1828351063318 p^{10} T^{6} + 153155351582 p^{14} T^{7} - 11166285887 p^{19} T^{8} - 231734363 p^{24} T^{9} + 264857 p^{30} T^{10} + 1171 p^{36} T^{11} + p^{42} T^{12} \) | |
| good | 3 | \( 1 - 44 T - 440 T^{2} + 17146 T^{3} + 3280501 T^{4} - 60741716 p^{2} T^{5} + 670343734 p^{3} T^{6} + 8393116172 p^{3} T^{7} + 208467443150 p^{4} T^{8} - 10135608598828 p^{5} T^{9} + 221063078571290 p^{6} T^{10} - 1638946329945110 p^{7} T^{11} - 12818769127169696 p^{8} T^{12} - 1638946329945110 p^{14} T^{13} + 221063078571290 p^{20} T^{14} - 10135608598828 p^{26} T^{15} + 208467443150 p^{32} T^{16} + 8393116172 p^{38} T^{17} + 670343734 p^{45} T^{18} - 60741716 p^{51} T^{19} + 3280501 p^{56} T^{20} + 17146 p^{63} T^{21} - 440 p^{70} T^{22} - 44 p^{77} T^{23} + p^{84} T^{24} \) |
| 5 | \( 1 - 44 p T - 209934 T^{2} + 13924174 p T^{3} + 16494813421 T^{4} - 504193396212 p^{2} T^{5} + 296405396516856 T^{6} + 285457983905150716 p T^{7} - \)\(25\!\cdots\!64\)\( T^{8} - \)\(19\!\cdots\!96\)\( p T^{9} + \)\(15\!\cdots\!06\)\( p^{2} T^{10} + \)\(24\!\cdots\!94\)\( p^{3} T^{11} - \)\(58\!\cdots\!64\)\( p^{4} T^{12} + \)\(24\!\cdots\!94\)\( p^{10} T^{13} + \)\(15\!\cdots\!06\)\( p^{16} T^{14} - \)\(19\!\cdots\!96\)\( p^{22} T^{15} - \)\(25\!\cdots\!64\)\( p^{28} T^{16} + 285457983905150716 p^{36} T^{17} + 296405396516856 p^{42} T^{18} - 504193396212 p^{51} T^{19} + 16494813421 p^{56} T^{20} + 13924174 p^{64} T^{21} - 209934 p^{70} T^{22} - 44 p^{78} T^{23} + p^{84} T^{24} \) | |
| 7 | \( 1 - 534 T - 3233386 T^{2} + 1748551438 T^{3} + 536028262395 p T^{4} - 1228855607491410 T^{5} - 149873989122984092 p T^{6} - \)\(23\!\cdots\!28\)\( T^{7} - \)\(19\!\cdots\!68\)\( p T^{8} + \)\(47\!\cdots\!10\)\( T^{9} + \)\(10\!\cdots\!50\)\( T^{10} - \)\(22\!\cdots\!84\)\( T^{11} - \)\(31\!\cdots\!72\)\( T^{12} - \)\(22\!\cdots\!84\)\( p^{7} T^{13} + \)\(10\!\cdots\!50\)\( p^{14} T^{14} + \)\(47\!\cdots\!10\)\( p^{21} T^{15} - \)\(19\!\cdots\!68\)\( p^{29} T^{16} - \)\(23\!\cdots\!28\)\( p^{35} T^{17} - 149873989122984092 p^{43} T^{18} - 1228855607491410 p^{49} T^{19} + 536028262395 p^{57} T^{20} + 1748551438 p^{63} T^{21} - 3233386 p^{70} T^{22} - 534 p^{77} T^{23} + p^{84} T^{24} \) | |
| 13 | \( 1 + 1834 T - 38398018 T^{2} + 400006966948 T^{3} + 699874007844425 T^{4} - 29586615816557339730 T^{5} + \)\(30\!\cdots\!48\)\( T^{6} + \)\(14\!\cdots\!48\)\( T^{7} - \)\(23\!\cdots\!88\)\( T^{8} + \)\(94\!\cdots\!30\)\( p T^{9} + \)\(70\!\cdots\!50\)\( T^{10} - \)\(15\!\cdots\!74\)\( T^{11} + \)\(12\!\cdots\!92\)\( T^{12} - \)\(15\!\cdots\!74\)\( p^{7} T^{13} + \)\(70\!\cdots\!50\)\( p^{14} T^{14} + \)\(94\!\cdots\!30\)\( p^{22} T^{15} - \)\(23\!\cdots\!88\)\( p^{28} T^{16} + \)\(14\!\cdots\!48\)\( p^{35} T^{17} + \)\(30\!\cdots\!48\)\( p^{42} T^{18} - 29586615816557339730 p^{49} T^{19} + 699874007844425 p^{56} T^{20} + 400006966948 p^{63} T^{21} - 38398018 p^{70} T^{22} + 1834 p^{77} T^{23} + p^{84} T^{24} \) | |
| 17 | \( 1 - 49128 T + 101676934 p T^{2} - 69959301741424 T^{3} + 2425329709110755825 T^{4} - \)\(69\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!56\)\( p T^{6} - \)\(53\!\cdots\!64\)\( T^{7} + \)\(13\!\cdots\!92\)\( T^{8} - \)\(31\!\cdots\!00\)\( T^{9} + \)\(71\!\cdots\!70\)\( T^{10} - \)\(15\!\cdots\!88\)\( T^{11} + \)\(31\!\cdots\!52\)\( T^{12} - \)\(15\!\cdots\!88\)\( p^{7} T^{13} + \)\(71\!\cdots\!70\)\( p^{14} T^{14} - \)\(31\!\cdots\!00\)\( p^{21} T^{15} + \)\(13\!\cdots\!92\)\( p^{28} T^{16} - \)\(53\!\cdots\!64\)\( p^{35} T^{17} + \)\(11\!\cdots\!56\)\( p^{43} T^{18} - \)\(69\!\cdots\!80\)\( p^{49} T^{19} + 2425329709110755825 p^{56} T^{20} - 69959301741424 p^{63} T^{21} + 101676934 p^{71} T^{22} - 49128 p^{77} T^{23} + p^{84} T^{24} \) | |
| 19 | \( 1 + 1287 p T + 768633170 T^{2} - 1523387586842 p T^{3} - 724283788415291568 T^{4} + \)\(14\!\cdots\!35\)\( T^{5} + \)\(16\!\cdots\!35\)\( T^{6} + \)\(23\!\cdots\!96\)\( p T^{7} - \)\(57\!\cdots\!59\)\( T^{8} - \)\(49\!\cdots\!65\)\( T^{9} - \)\(99\!\cdots\!17\)\( T^{10} + \)\(21\!\cdots\!49\)\( T^{11} + \)\(14\!\cdots\!76\)\( T^{12} + \)\(21\!\cdots\!49\)\( p^{7} T^{13} - \)\(99\!\cdots\!17\)\( p^{14} T^{14} - \)\(49\!\cdots\!65\)\( p^{21} T^{15} - \)\(57\!\cdots\!59\)\( p^{28} T^{16} + \)\(23\!\cdots\!96\)\( p^{36} T^{17} + \)\(16\!\cdots\!35\)\( p^{42} T^{18} + \)\(14\!\cdots\!35\)\( p^{49} T^{19} - 724283788415291568 p^{56} T^{20} - 1523387586842 p^{64} T^{21} + 768633170 p^{70} T^{22} + 1287 p^{78} T^{23} + p^{84} T^{24} \) | |
| 23 | \( ( 1 - 606 T + 10434876374 T^{2} + 93655742869382 T^{3} + 65919337346321696879 T^{4} + \)\(52\!\cdots\!76\)\( T^{5} + \)\(27\!\cdots\!24\)\( T^{6} + \)\(52\!\cdots\!76\)\( p^{7} T^{7} + 65919337346321696879 p^{14} T^{8} + 93655742869382 p^{21} T^{9} + 10434876374 p^{28} T^{10} - 606 p^{35} T^{11} + p^{42} T^{12} )^{2} \) | |
| 29 | \( 1 - 410548 T + 28170614262 T^{2} + 9762728056407896 T^{3} - \)\(11\!\cdots\!23\)\( T^{4} - \)\(18\!\cdots\!72\)\( T^{5} + \)\(28\!\cdots\!72\)\( T^{6} + \)\(42\!\cdots\!68\)\( T^{7} - \)\(78\!\cdots\!72\)\( T^{8} - \)\(61\!\cdots\!32\)\( T^{9} + \)\(17\!\cdots\!62\)\( T^{10} + \)\(28\!\cdots\!04\)\( T^{11} - \)\(30\!\cdots\!44\)\( T^{12} + \)\(28\!\cdots\!04\)\( p^{7} T^{13} + \)\(17\!\cdots\!62\)\( p^{14} T^{14} - \)\(61\!\cdots\!32\)\( p^{21} T^{15} - \)\(78\!\cdots\!72\)\( p^{28} T^{16} + \)\(42\!\cdots\!68\)\( p^{35} T^{17} + \)\(28\!\cdots\!72\)\( p^{42} T^{18} - \)\(18\!\cdots\!72\)\( p^{49} T^{19} - \)\(11\!\cdots\!23\)\( p^{56} T^{20} + 9762728056407896 p^{63} T^{21} + 28170614262 p^{70} T^{22} - 410548 p^{77} T^{23} + p^{84} T^{24} \) | |
| 31 | \( 1 + 285894 T - 77692428146 T^{2} - 32767885675187454 T^{3} + \)\(20\!\cdots\!21\)\( T^{4} + \)\(20\!\cdots\!46\)\( T^{5} + \)\(92\!\cdots\!72\)\( T^{6} - \)\(26\!\cdots\!16\)\( p T^{7} - \)\(10\!\cdots\!32\)\( T^{8} + \)\(21\!\cdots\!66\)\( T^{9} + \)\(55\!\cdots\!06\)\( T^{10} - \)\(24\!\cdots\!40\)\( T^{11} - \)\(18\!\cdots\!16\)\( T^{12} - \)\(24\!\cdots\!40\)\( p^{7} T^{13} + \)\(55\!\cdots\!06\)\( p^{14} T^{14} + \)\(21\!\cdots\!66\)\( p^{21} T^{15} - \)\(10\!\cdots\!32\)\( p^{28} T^{16} - \)\(26\!\cdots\!16\)\( p^{36} T^{17} + \)\(92\!\cdots\!72\)\( p^{42} T^{18} + \)\(20\!\cdots\!46\)\( p^{49} T^{19} + \)\(20\!\cdots\!21\)\( p^{56} T^{20} - 32767885675187454 p^{63} T^{21} - 77692428146 p^{70} T^{22} + 285894 p^{77} T^{23} + p^{84} T^{24} \) | |
| 37 | \( 1 + 229440 T - 109122948886 T^{2} - 41677645837231338 T^{3} - \)\(21\!\cdots\!51\)\( p T^{4} - \)\(77\!\cdots\!24\)\( T^{5} + \)\(14\!\cdots\!12\)\( T^{6} + \)\(13\!\cdots\!84\)\( T^{7} + \)\(30\!\cdots\!28\)\( T^{8} + \)\(21\!\cdots\!28\)\( T^{9} + \)\(13\!\cdots\!34\)\( T^{10} - \)\(75\!\cdots\!46\)\( T^{11} - \)\(53\!\cdots\!16\)\( T^{12} - \)\(75\!\cdots\!46\)\( p^{7} T^{13} + \)\(13\!\cdots\!34\)\( p^{14} T^{14} + \)\(21\!\cdots\!28\)\( p^{21} T^{15} + \)\(30\!\cdots\!28\)\( p^{28} T^{16} + \)\(13\!\cdots\!84\)\( p^{35} T^{17} + \)\(14\!\cdots\!12\)\( p^{42} T^{18} - \)\(77\!\cdots\!24\)\( p^{49} T^{19} - \)\(21\!\cdots\!51\)\( p^{57} T^{20} - 41677645837231338 p^{63} T^{21} - 109122948886 p^{70} T^{22} + 229440 p^{77} T^{23} + p^{84} T^{24} \) | |
| 41 | \( 1 + 838292 T - 159497637510 T^{2} - 132666414987652932 T^{3} + \)\(12\!\cdots\!29\)\( T^{4} + \)\(96\!\cdots\!88\)\( T^{5} + \)\(28\!\cdots\!36\)\( T^{6} - \)\(82\!\cdots\!00\)\( T^{7} - \)\(40\!\cdots\!16\)\( T^{8} + \)\(65\!\cdots\!32\)\( T^{9} + \)\(32\!\cdots\!10\)\( T^{10} + \)\(77\!\cdots\!80\)\( T^{11} - \)\(18\!\cdots\!80\)\( T^{12} + \)\(77\!\cdots\!80\)\( p^{7} T^{13} + \)\(32\!\cdots\!10\)\( p^{14} T^{14} + \)\(65\!\cdots\!32\)\( p^{21} T^{15} - \)\(40\!\cdots\!16\)\( p^{28} T^{16} - \)\(82\!\cdots\!00\)\( p^{35} T^{17} + \)\(28\!\cdots\!36\)\( p^{42} T^{18} + \)\(96\!\cdots\!88\)\( p^{49} T^{19} + \)\(12\!\cdots\!29\)\( p^{56} T^{20} - 132666414987652932 p^{63} T^{21} - 159497637510 p^{70} T^{22} + 838292 p^{77} T^{23} + p^{84} T^{24} \) | |
| 43 | \( ( 1 + 387349 T + 852313111631 T^{2} + 277386530147406595 T^{3} + \)\(34\!\cdots\!87\)\( T^{4} + \)\(10\!\cdots\!90\)\( T^{5} + \)\(10\!\cdots\!66\)\( T^{6} + \)\(10\!\cdots\!90\)\( p^{7} T^{7} + \)\(34\!\cdots\!87\)\( p^{14} T^{8} + 277386530147406595 p^{21} T^{9} + 852313111631 p^{28} T^{10} + 387349 p^{35} T^{11} + p^{42} T^{12} )^{2} \) | |
| 47 | \( 1 - 1107540 T - 553339421838 T^{2} + 1374119702008679280 T^{3} - \)\(67\!\cdots\!95\)\( T^{4} - \)\(30\!\cdots\!80\)\( T^{5} + \)\(61\!\cdots\!80\)\( T^{6} - \)\(34\!\cdots\!00\)\( T^{7} - \)\(28\!\cdots\!04\)\( T^{8} + \)\(22\!\cdots\!40\)\( T^{9} - \)\(13\!\cdots\!10\)\( T^{10} - \)\(42\!\cdots\!20\)\( T^{11} + \)\(88\!\cdots\!36\)\( T^{12} - \)\(42\!\cdots\!20\)\( p^{7} T^{13} - \)\(13\!\cdots\!10\)\( p^{14} T^{14} + \)\(22\!\cdots\!40\)\( p^{21} T^{15} - \)\(28\!\cdots\!04\)\( p^{28} T^{16} - \)\(34\!\cdots\!00\)\( p^{35} T^{17} + \)\(61\!\cdots\!80\)\( p^{42} T^{18} - \)\(30\!\cdots\!80\)\( p^{49} T^{19} - \)\(67\!\cdots\!95\)\( p^{56} T^{20} + 1374119702008679280 p^{63} T^{21} - 553339421838 p^{70} T^{22} - 1107540 p^{77} T^{23} + p^{84} T^{24} \) | |
| 53 | \( 1 + 3013626 T + 1871482610366 T^{2} - 2402602653160407410 T^{3} - \)\(42\!\cdots\!31\)\( T^{4} - \)\(48\!\cdots\!94\)\( T^{5} - \)\(28\!\cdots\!28\)\( T^{6} + \)\(55\!\cdots\!64\)\( T^{7} + \)\(79\!\cdots\!20\)\( T^{8} + \)\(17\!\cdots\!54\)\( T^{9} + \)\(49\!\cdots\!62\)\( T^{10} - \)\(70\!\cdots\!56\)\( p T^{11} - \)\(10\!\cdots\!32\)\( T^{12} - \)\(70\!\cdots\!56\)\( p^{8} T^{13} + \)\(49\!\cdots\!62\)\( p^{14} T^{14} + \)\(17\!\cdots\!54\)\( p^{21} T^{15} + \)\(79\!\cdots\!20\)\( p^{28} T^{16} + \)\(55\!\cdots\!64\)\( p^{35} T^{17} - \)\(28\!\cdots\!28\)\( p^{42} T^{18} - \)\(48\!\cdots\!94\)\( p^{49} T^{19} - \)\(42\!\cdots\!31\)\( p^{56} T^{20} - 2402602653160407410 p^{63} T^{21} + 1871482610366 p^{70} T^{22} + 3013626 p^{77} T^{23} + p^{84} T^{24} \) | |
| 59 | \( 1 - 2583885 T - 1458993155458 T^{2} + 4162884874868362150 T^{3} + \)\(10\!\cdots\!12\)\( T^{4} - \)\(15\!\cdots\!35\)\( T^{5} - \)\(70\!\cdots\!49\)\( T^{6} + \)\(28\!\cdots\!40\)\( T^{7} - \)\(58\!\cdots\!75\)\( T^{8} - \)\(24\!\cdots\!95\)\( T^{9} + \)\(21\!\cdots\!83\)\( T^{10} + \)\(12\!\cdots\!55\)\( T^{11} - \)\(89\!\cdots\!48\)\( T^{12} + \)\(12\!\cdots\!55\)\( p^{7} T^{13} + \)\(21\!\cdots\!83\)\( p^{14} T^{14} - \)\(24\!\cdots\!95\)\( p^{21} T^{15} - \)\(58\!\cdots\!75\)\( p^{28} T^{16} + \)\(28\!\cdots\!40\)\( p^{35} T^{17} - \)\(70\!\cdots\!49\)\( p^{42} T^{18} - \)\(15\!\cdots\!35\)\( p^{49} T^{19} + \)\(10\!\cdots\!12\)\( p^{56} T^{20} + 4162884874868362150 p^{63} T^{21} - 1458993155458 p^{70} T^{22} - 2583885 p^{77} T^{23} + p^{84} T^{24} \) | |
| 61 | \( 1 - 418618 T - 3028105566874 T^{2} + 12054304841373643570 T^{3} + \)\(96\!\cdots\!77\)\( T^{4} - \)\(15\!\cdots\!58\)\( T^{5} + \)\(30\!\cdots\!08\)\( T^{6} + \)\(61\!\cdots\!08\)\( T^{7} + \)\(17\!\cdots\!48\)\( T^{8} + \)\(23\!\cdots\!22\)\( T^{9} - \)\(17\!\cdots\!82\)\( T^{10} + \)\(60\!\cdots\!52\)\( T^{11} + \)\(31\!\cdots\!32\)\( T^{12} + \)\(60\!\cdots\!52\)\( p^{7} T^{13} - \)\(17\!\cdots\!82\)\( p^{14} T^{14} + \)\(23\!\cdots\!22\)\( p^{21} T^{15} + \)\(17\!\cdots\!48\)\( p^{28} T^{16} + \)\(61\!\cdots\!08\)\( p^{35} T^{17} + \)\(30\!\cdots\!08\)\( p^{42} T^{18} - \)\(15\!\cdots\!58\)\( p^{49} T^{19} + \)\(96\!\cdots\!77\)\( p^{56} T^{20} + 12054304841373643570 p^{63} T^{21} - 3028105566874 p^{70} T^{22} - 418618 p^{77} T^{23} + p^{84} T^{24} \) | |
| 67 | \( ( 1 + 9177277 T + 58026212499961 T^{2} + \)\(25\!\cdots\!79\)\( T^{3} + \)\(93\!\cdots\!39\)\( T^{4} + \)\(28\!\cdots\!18\)\( T^{5} + \)\(75\!\cdots\!06\)\( T^{6} + \)\(28\!\cdots\!18\)\( p^{7} T^{7} + \)\(93\!\cdots\!39\)\( p^{14} T^{8} + \)\(25\!\cdots\!79\)\( p^{21} T^{9} + 58026212499961 p^{28} T^{10} + 9177277 p^{35} T^{11} + p^{42} T^{12} )^{2} \) | |
| 71 | \( 1 - 22870340 T + 249133435922786 T^{2} - \)\(17\!\cdots\!50\)\( T^{3} + \)\(90\!\cdots\!57\)\( T^{4} - \)\(38\!\cdots\!00\)\( T^{5} + \)\(19\!\cdots\!52\)\( p T^{6} - \)\(44\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!08\)\( T^{8} - \)\(27\!\cdots\!00\)\( T^{9} + \)\(46\!\cdots\!82\)\( T^{10} - \)\(58\!\cdots\!10\)\( T^{11} + \)\(90\!\cdots\!08\)\( T^{12} - \)\(58\!\cdots\!10\)\( p^{7} T^{13} + \)\(46\!\cdots\!82\)\( p^{14} T^{14} - \)\(27\!\cdots\!00\)\( p^{21} T^{15} + \)\(12\!\cdots\!08\)\( p^{28} T^{16} - \)\(44\!\cdots\!00\)\( p^{35} T^{17} + \)\(19\!\cdots\!52\)\( p^{43} T^{18} - \)\(38\!\cdots\!00\)\( p^{49} T^{19} + \)\(90\!\cdots\!57\)\( p^{56} T^{20} - \)\(17\!\cdots\!50\)\( p^{63} T^{21} + 249133435922786 p^{70} T^{22} - 22870340 p^{77} T^{23} + p^{84} T^{24} \) | |
| 73 | \( 1 + 7374584 T + 56366040461474 T^{2} + \)\(23\!\cdots\!08\)\( T^{3} + \)\(12\!\cdots\!65\)\( T^{4} + \)\(51\!\cdots\!00\)\( T^{5} + \)\(24\!\cdots\!40\)\( T^{6} + \)\(96\!\cdots\!40\)\( T^{7} + \)\(39\!\cdots\!56\)\( T^{8} + \)\(14\!\cdots\!00\)\( T^{9} + \)\(53\!\cdots\!50\)\( T^{10} + \)\(18\!\cdots\!00\)\( T^{11} + \)\(63\!\cdots\!16\)\( T^{12} + \)\(18\!\cdots\!00\)\( p^{7} T^{13} + \)\(53\!\cdots\!50\)\( p^{14} T^{14} + \)\(14\!\cdots\!00\)\( p^{21} T^{15} + \)\(39\!\cdots\!56\)\( p^{28} T^{16} + \)\(96\!\cdots\!40\)\( p^{35} T^{17} + \)\(24\!\cdots\!40\)\( p^{42} T^{18} + \)\(51\!\cdots\!00\)\( p^{49} T^{19} + \)\(12\!\cdots\!65\)\( p^{56} T^{20} + \)\(23\!\cdots\!08\)\( p^{63} T^{21} + 56366040461474 p^{70} T^{22} + 7374584 p^{77} T^{23} + p^{84} T^{24} \) | |
| 79 | \( 1 - 4715278 T - 17785975482010 T^{2} + \)\(17\!\cdots\!18\)\( T^{3} - \)\(37\!\cdots\!63\)\( T^{4} - \)\(45\!\cdots\!50\)\( T^{5} + \)\(90\!\cdots\!40\)\( T^{6} + \)\(99\!\cdots\!76\)\( T^{7} - \)\(36\!\cdots\!84\)\( T^{8} - \)\(16\!\cdots\!90\)\( T^{9} + \)\(12\!\cdots\!18\)\( T^{10} + \)\(90\!\cdots\!96\)\( T^{11} - \)\(26\!\cdots\!44\)\( T^{12} + \)\(90\!\cdots\!96\)\( p^{7} T^{13} + \)\(12\!\cdots\!18\)\( p^{14} T^{14} - \)\(16\!\cdots\!90\)\( p^{21} T^{15} - \)\(36\!\cdots\!84\)\( p^{28} T^{16} + \)\(99\!\cdots\!76\)\( p^{35} T^{17} + \)\(90\!\cdots\!40\)\( p^{42} T^{18} - \)\(45\!\cdots\!50\)\( p^{49} T^{19} - \)\(37\!\cdots\!63\)\( p^{56} T^{20} + \)\(17\!\cdots\!18\)\( p^{63} T^{21} - 17785975482010 p^{70} T^{22} - 4715278 p^{77} T^{23} + p^{84} T^{24} \) | |
| 83 | \( 1 + 13257515 T + 90327411213148 T^{2} + \)\(54\!\cdots\!10\)\( T^{3} + \)\(31\!\cdots\!10\)\( T^{4} + \)\(20\!\cdots\!25\)\( T^{5} + \)\(14\!\cdots\!15\)\( T^{6} + \)\(93\!\cdots\!60\)\( T^{7} + \)\(57\!\cdots\!01\)\( T^{8} + \)\(35\!\cdots\!65\)\( T^{9} + \)\(18\!\cdots\!05\)\( T^{10} + \)\(89\!\cdots\!15\)\( T^{11} + \)\(45\!\cdots\!96\)\( T^{12} + \)\(89\!\cdots\!15\)\( p^{7} T^{13} + \)\(18\!\cdots\!05\)\( p^{14} T^{14} + \)\(35\!\cdots\!65\)\( p^{21} T^{15} + \)\(57\!\cdots\!01\)\( p^{28} T^{16} + \)\(93\!\cdots\!60\)\( p^{35} T^{17} + \)\(14\!\cdots\!15\)\( p^{42} T^{18} + \)\(20\!\cdots\!25\)\( p^{49} T^{19} + \)\(31\!\cdots\!10\)\( p^{56} T^{20} + \)\(54\!\cdots\!10\)\( p^{63} T^{21} + 90327411213148 p^{70} T^{22} + 13257515 p^{77} T^{23} + p^{84} T^{24} \) | |
| 89 | \( ( 1 + 17432397 T + 285777780450333 T^{2} + \)\(32\!\cdots\!93\)\( T^{3} + \)\(32\!\cdots\!75\)\( T^{4} + \)\(26\!\cdots\!66\)\( T^{5} + \)\(19\!\cdots\!02\)\( T^{6} + \)\(26\!\cdots\!66\)\( p^{7} T^{7} + \)\(32\!\cdots\!75\)\( p^{14} T^{8} + \)\(32\!\cdots\!93\)\( p^{21} T^{9} + 285777780450333 p^{28} T^{10} + 17432397 p^{35} T^{11} + p^{42} T^{12} )^{2} \) | |
| 97 | \( 1 + 28563397 T + 498735971273852 T^{2} + \)\(74\!\cdots\!56\)\( T^{3} + \)\(92\!\cdots\!54\)\( T^{4} + \)\(89\!\cdots\!37\)\( T^{5} + \)\(77\!\cdots\!99\)\( T^{6} + \)\(65\!\cdots\!64\)\( T^{7} + \)\(50\!\cdots\!57\)\( T^{8} + \)\(43\!\cdots\!63\)\( T^{9} + \)\(48\!\cdots\!17\)\( T^{10} + \)\(49\!\cdots\!39\)\( T^{11} + \)\(44\!\cdots\!84\)\( T^{12} + \)\(49\!\cdots\!39\)\( p^{7} T^{13} + \)\(48\!\cdots\!17\)\( p^{14} T^{14} + \)\(43\!\cdots\!63\)\( p^{21} T^{15} + \)\(50\!\cdots\!57\)\( p^{28} T^{16} + \)\(65\!\cdots\!64\)\( p^{35} T^{17} + \)\(77\!\cdots\!99\)\( p^{42} T^{18} + \)\(89\!\cdots\!37\)\( p^{49} T^{19} + \)\(92\!\cdots\!54\)\( p^{56} T^{20} + \)\(74\!\cdots\!56\)\( p^{63} T^{21} + 498735971273852 p^{70} T^{22} + 28563397 p^{77} T^{23} + p^{84} T^{24} \) | |
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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
−5.19584481744430826391260331614, −5.12208523039564588040553118141, −5.07126636284053143075259577926, −4.85510645277357988437619596582, −4.62520557582643795807429928089, −4.40258321921233845535116843596, −4.36985314830412692191155787267, −4.01564080783179611658209502699, −3.98558585414131858563147245595, −3.33844115720863634386339187972, −3.21253078113605496904419547270, −3.07213581759994691324461804866, −2.95728940321106980491257364237, −2.94978829573732824680901051041, −2.62502930937829908675955356765, −2.35930406983726569920719650478, −1.99855380872789294968006667796, −1.92782028084337411148158524874, −1.69594881240732968381740448838, −1.25321258207684987313657706521, −1.05876089230965476071749674699, −0.897400232771789526194452154272, −0.854667717212688796529406444565, −0.30710754866221368796161984675, −0.10289510317743249888455890365, 0.10289510317743249888455890365, 0.30710754866221368796161984675, 0.854667717212688796529406444565, 0.897400232771789526194452154272, 1.05876089230965476071749674699, 1.25321258207684987313657706521, 1.69594881240732968381740448838, 1.92782028084337411148158524874, 1.99855380872789294968006667796, 2.35930406983726569920719650478, 2.62502930937829908675955356765, 2.94978829573732824680901051041, 2.95728940321106980491257364237, 3.07213581759994691324461804866, 3.21253078113605496904419547270, 3.33844115720863634386339187972, 3.98558585414131858563147245595, 4.01564080783179611658209502699, 4.36985314830412692191155787267, 4.40258321921233845535116843596, 4.62520557582643795807429928089, 4.85510645277357988437619596582, 5.07126636284053143075259577926, 5.12208523039564588040553118141, 5.19584481744430826391260331614
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.