Dirichlet series
| L(s) = 1 | − 18·3-s − 24·4-s − 30·5-s − 24·7-s − 17·8-s + 54·9-s − 162·11-s + 432·12-s − 72·13-s + 540·15-s + 240·16-s − 66·19-s + 720·20-s + 432·21-s − 282·23-s + 306·24-s − 78·25-s + 770·27-s + 576·28-s − 648·29-s − 504·31-s + 219·32-s + 2.91e3·33-s + 720·35-s − 1.29e3·36-s + 30·37-s + 1.29e3·39-s + ⋯ |
| L(s) = 1 | − 3.46·3-s − 3·4-s − 2.68·5-s − 1.29·7-s − 0.751·8-s + 2·9-s − 4.44·11-s + 10.3·12-s − 1.53·13-s + 9.29·15-s + 15/4·16-s − 0.796·19-s + 8.04·20-s + 4.48·21-s − 2.55·23-s + 2.60·24-s − 0.623·25-s + 5.48·27-s + 3.88·28-s − 4.14·29-s − 2.92·31-s + 1.20·32-s + 15.3·33-s + 3.47·35-s − 6·36-s + 0.133·37-s + 5.32·39-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(17^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.04180\times 10^{14}\) |
| Root analytic conductor: | \(4.12935\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 17^{24} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 3 p^{3} T^{2} + 17 T^{3} + 21 p^{4} T^{4} + 597 T^{5} + 2083 p T^{6} + 9219 T^{7} + 47463 T^{8} + 99575 T^{9} + 237969 p T^{10} + 29085 p^{5} T^{11} + 516393 p^{3} T^{12} + 29085 p^{8} T^{13} + 237969 p^{7} T^{14} + 99575 p^{9} T^{15} + 47463 p^{12} T^{16} + 9219 p^{15} T^{17} + 2083 p^{19} T^{18} + 597 p^{21} T^{19} + 21 p^{28} T^{20} + 17 p^{27} T^{21} + 3 p^{33} T^{22} + p^{36} T^{24} \) |
| 3 | \( 1 + 2 p^{2} T + 10 p^{3} T^{2} + 3118 T^{3} + 3493 p^{2} T^{4} + 93338 p T^{5} + 2283230 T^{6} + 5663386 p T^{7} + 39342911 p T^{8} + 761340866 T^{9} + 1539722494 p T^{10} + 972382312 p^{3} T^{11} + 46929762953 p T^{12} + 972382312 p^{6} T^{13} + 1539722494 p^{7} T^{14} + 761340866 p^{9} T^{15} + 39342911 p^{13} T^{16} + 5663386 p^{16} T^{17} + 2283230 p^{18} T^{18} + 93338 p^{22} T^{19} + 3493 p^{26} T^{20} + 3118 p^{27} T^{21} + 10 p^{33} T^{22} + 2 p^{35} T^{23} + p^{36} T^{24} \) | |
| 5 | \( 1 + 6 p T + 978 T^{2} + 4402 p T^{3} + 98928 p T^{4} + 8963928 T^{5} + 160010892 T^{6} + 2486105382 T^{7} + 37688092122 T^{8} + 511113233924 T^{9} + 6768898744506 T^{10} + 81309529724886 T^{11} + 952406498268277 T^{12} + 81309529724886 p^{3} T^{13} + 6768898744506 p^{6} T^{14} + 511113233924 p^{9} T^{15} + 37688092122 p^{12} T^{16} + 2486105382 p^{15} T^{17} + 160010892 p^{18} T^{18} + 8963928 p^{21} T^{19} + 98928 p^{25} T^{20} + 4402 p^{28} T^{21} + 978 p^{30} T^{22} + 6 p^{34} T^{23} + p^{36} T^{24} \) | |
| 7 | \( 1 + 24 T + 2631 T^{2} + 49932 T^{3} + 3178503 T^{4} + 50026356 T^{5} + 351286134 p T^{6} + 33717298626 T^{7} + 1420494335676 T^{8} + 17645494346682 T^{9} + 658479284458026 T^{10} + 7488753262542012 T^{11} + 249672863392321825 T^{12} + 7488753262542012 p^{3} T^{13} + 658479284458026 p^{6} T^{14} + 17645494346682 p^{9} T^{15} + 1420494335676 p^{12} T^{16} + 33717298626 p^{15} T^{17} + 351286134 p^{19} T^{18} + 50026356 p^{21} T^{19} + 3178503 p^{24} T^{20} + 49932 p^{27} T^{21} + 2631 p^{30} T^{22} + 24 p^{33} T^{23} + p^{36} T^{24} \) | |
| 11 | \( 1 + 162 T + 20376 T^{2} + 1758520 T^{3} + 128665887 T^{4} + 7549392330 T^{5} + 386047940887 T^{6} + 16321486458276 T^{7} + 600098184489765 T^{8} + 17707891221939426 T^{9} + 444740033046531609 T^{10} + 8519881241999961450 T^{11} + \)\(22\!\cdots\!34\)\( T^{12} + 8519881241999961450 p^{3} T^{13} + 444740033046531609 p^{6} T^{14} + 17707891221939426 p^{9} T^{15} + 600098184489765 p^{12} T^{16} + 16321486458276 p^{15} T^{17} + 386047940887 p^{18} T^{18} + 7549392330 p^{21} T^{19} + 128665887 p^{24} T^{20} + 1758520 p^{27} T^{21} + 20376 p^{30} T^{22} + 162 p^{33} T^{23} + p^{36} T^{24} \) | |
| 13 | \( 1 + 72 T + 14790 T^{2} + 952402 T^{3} + 109939374 T^{4} + 6632292606 T^{5} + 549660962912 T^{6} + 31473741682326 T^{7} + 2066167770898980 T^{8} + 111808560575162252 T^{9} + 6158767151543542860 T^{10} + \)\(30\!\cdots\!90\)\( T^{11} + \)\(14\!\cdots\!37\)\( T^{12} + \)\(30\!\cdots\!90\)\( p^{3} T^{13} + 6158767151543542860 p^{6} T^{14} + 111808560575162252 p^{9} T^{15} + 2066167770898980 p^{12} T^{16} + 31473741682326 p^{15} T^{17} + 549660962912 p^{18} T^{18} + 6632292606 p^{21} T^{19} + 109939374 p^{24} T^{20} + 952402 p^{27} T^{21} + 14790 p^{30} T^{22} + 72 p^{33} T^{23} + p^{36} T^{24} \) | |
| 19 | \( 1 + 66 T + 31476 T^{2} + 2467196 T^{3} + 644635554 T^{4} + 47808045678 T^{5} + 9459081995078 T^{6} + 672750203320836 T^{7} + 107985364016568894 T^{8} + 7139368709734578288 T^{9} + \)\(99\!\cdots\!78\)\( T^{10} + \)\(60\!\cdots\!64\)\( T^{11} + \)\(75\!\cdots\!51\)\( T^{12} + \)\(60\!\cdots\!64\)\( p^{3} T^{13} + \)\(99\!\cdots\!78\)\( p^{6} T^{14} + 7139368709734578288 p^{9} T^{15} + 107985364016568894 p^{12} T^{16} + 672750203320836 p^{15} T^{17} + 9459081995078 p^{18} T^{18} + 47808045678 p^{21} T^{19} + 644635554 p^{24} T^{20} + 2467196 p^{27} T^{21} + 31476 p^{30} T^{22} + 66 p^{33} T^{23} + p^{36} T^{24} \) | |
| 23 | \( 1 + 282 T + 129591 T^{2} + 26870378 T^{3} + 7295872581 T^{4} + 1217919573750 T^{5} + 248562967071774 T^{6} + 35006925558952032 T^{7} + 5876611919308179474 T^{8} + \)\(71\!\cdots\!46\)\( T^{9} + \)\(10\!\cdots\!10\)\( T^{10} + \)\(11\!\cdots\!86\)\( T^{11} + \)\(14\!\cdots\!77\)\( T^{12} + \)\(11\!\cdots\!86\)\( p^{3} T^{13} + \)\(10\!\cdots\!10\)\( p^{6} T^{14} + \)\(71\!\cdots\!46\)\( p^{9} T^{15} + 5876611919308179474 p^{12} T^{16} + 35006925558952032 p^{15} T^{17} + 248562967071774 p^{18} T^{18} + 1217919573750 p^{21} T^{19} + 7295872581 p^{24} T^{20} + 26870378 p^{27} T^{21} + 129591 p^{30} T^{22} + 282 p^{33} T^{23} + p^{36} T^{24} \) | |
| 29 | \( 1 + 648 T + 363351 T^{2} + 141812376 T^{3} + 49113689433 T^{4} + 14196586050282 T^{5} + 3739660471763232 T^{6} + 872788818202113846 T^{7} + \)\(18\!\cdots\!86\)\( T^{8} + \)\(37\!\cdots\!68\)\( T^{9} + \)\(69\!\cdots\!26\)\( T^{10} + \)\(11\!\cdots\!52\)\( T^{11} + \)\(19\!\cdots\!41\)\( T^{12} + \)\(11\!\cdots\!52\)\( p^{3} T^{13} + \)\(69\!\cdots\!26\)\( p^{6} T^{14} + \)\(37\!\cdots\!68\)\( p^{9} T^{15} + \)\(18\!\cdots\!86\)\( p^{12} T^{16} + 872788818202113846 p^{15} T^{17} + 3739660471763232 p^{18} T^{18} + 14196586050282 p^{21} T^{19} + 49113689433 p^{24} T^{20} + 141812376 p^{27} T^{21} + 363351 p^{30} T^{22} + 648 p^{33} T^{23} + p^{36} T^{24} \) | |
| 31 | \( 1 + 504 T + 287307 T^{2} + 103850426 T^{3} + 36701706309 T^{4} + 10521372364698 T^{5} + 2904783892693358 T^{6} + 703577060018454174 T^{7} + \)\(16\!\cdots\!48\)\( T^{8} + \)\(34\!\cdots\!54\)\( T^{9} + \)\(70\!\cdots\!84\)\( T^{10} + \)\(13\!\cdots\!86\)\( T^{11} + \)\(23\!\cdots\!89\)\( T^{12} + \)\(13\!\cdots\!86\)\( p^{3} T^{13} + \)\(70\!\cdots\!84\)\( p^{6} T^{14} + \)\(34\!\cdots\!54\)\( p^{9} T^{15} + \)\(16\!\cdots\!48\)\( p^{12} T^{16} + 703577060018454174 p^{15} T^{17} + 2904783892693358 p^{18} T^{18} + 10521372364698 p^{21} T^{19} + 36701706309 p^{24} T^{20} + 103850426 p^{27} T^{21} + 287307 p^{30} T^{22} + 504 p^{33} T^{23} + p^{36} T^{24} \) | |
| 37 | \( 1 - 30 T + 396441 T^{2} + 4303652 T^{3} + 75535258077 T^{4} + 3306733572444 T^{5} + 9386012826459812 T^{6} + 609429333975224280 T^{7} + \)\(86\!\cdots\!90\)\( T^{8} + \)\(63\!\cdots\!52\)\( T^{9} + \)\(61\!\cdots\!80\)\( T^{10} + \)\(43\!\cdots\!68\)\( T^{11} + \)\(35\!\cdots\!25\)\( T^{12} + \)\(43\!\cdots\!68\)\( p^{3} T^{13} + \)\(61\!\cdots\!80\)\( p^{6} T^{14} + \)\(63\!\cdots\!52\)\( p^{9} T^{15} + \)\(86\!\cdots\!90\)\( p^{12} T^{16} + 609429333975224280 p^{15} T^{17} + 9386012826459812 p^{18} T^{18} + 3306733572444 p^{21} T^{19} + 75535258077 p^{24} T^{20} + 4303652 p^{27} T^{21} + 396441 p^{30} T^{22} - 30 p^{33} T^{23} + p^{36} T^{24} \) | |
| 41 | \( 1 + 318 T + 461895 T^{2} + 140587038 T^{3} + 110124298701 T^{4} + 31262577390408 T^{5} + 431137996827126 p T^{6} + 4635187016534166306 T^{7} + \)\(21\!\cdots\!94\)\( T^{8} + \)\(51\!\cdots\!64\)\( T^{9} + \)\(20\!\cdots\!18\)\( T^{10} + \)\(44\!\cdots\!00\)\( T^{11} + \)\(15\!\cdots\!79\)\( T^{12} + \)\(44\!\cdots\!00\)\( p^{3} T^{13} + \)\(20\!\cdots\!18\)\( p^{6} T^{14} + \)\(51\!\cdots\!64\)\( p^{9} T^{15} + \)\(21\!\cdots\!94\)\( p^{12} T^{16} + 4635187016534166306 p^{15} T^{17} + 431137996827126 p^{19} T^{18} + 31262577390408 p^{21} T^{19} + 110124298701 p^{24} T^{20} + 140587038 p^{27} T^{21} + 461895 p^{30} T^{22} + 318 p^{33} T^{23} + p^{36} T^{24} \) | |
| 43 | \( 1 - 486 T + 537816 T^{2} - 234810144 T^{3} + 149332540095 T^{4} - 58372416122220 T^{5} + 28190059917868464 T^{6} - 9841117987706330736 T^{7} + \)\(39\!\cdots\!47\)\( T^{8} - \)\(12\!\cdots\!82\)\( T^{9} + \)\(43\!\cdots\!10\)\( T^{10} - \)\(28\!\cdots\!64\)\( p T^{11} + \)\(38\!\cdots\!75\)\( T^{12} - \)\(28\!\cdots\!64\)\( p^{4} T^{13} + \)\(43\!\cdots\!10\)\( p^{6} T^{14} - \)\(12\!\cdots\!82\)\( p^{9} T^{15} + \)\(39\!\cdots\!47\)\( p^{12} T^{16} - 9841117987706330736 p^{15} T^{17} + 28190059917868464 p^{18} T^{18} - 58372416122220 p^{21} T^{19} + 149332540095 p^{24} T^{20} - 234810144 p^{27} T^{21} + 537816 p^{30} T^{22} - 486 p^{33} T^{23} + p^{36} T^{24} \) | |
| 47 | \( 1 + 888 T + 926349 T^{2} + 650503016 T^{3} + 421950918861 T^{4} + 238133559715818 T^{5} + 122068250658909346 T^{6} + 57411272664754757682 T^{7} + \)\(24\!\cdots\!54\)\( T^{8} + \)\(10\!\cdots\!46\)\( T^{9} + \)\(38\!\cdots\!66\)\( T^{10} + \)\(13\!\cdots\!86\)\( T^{11} + \)\(44\!\cdots\!93\)\( T^{12} + \)\(13\!\cdots\!86\)\( p^{3} T^{13} + \)\(38\!\cdots\!66\)\( p^{6} T^{14} + \)\(10\!\cdots\!46\)\( p^{9} T^{15} + \)\(24\!\cdots\!54\)\( p^{12} T^{16} + 57411272664754757682 p^{15} T^{17} + 122068250658909346 p^{18} T^{18} + 238133559715818 p^{21} T^{19} + 421950918861 p^{24} T^{20} + 650503016 p^{27} T^{21} + 926349 p^{30} T^{22} + 888 p^{33} T^{23} + p^{36} T^{24} \) | |
| 53 | \( 1 - 1026 T + 1814271 T^{2} - 26381752 p T^{3} + 1427421860244 T^{4} - 890940789162870 T^{5} + 671398014573953230 T^{6} - \)\(35\!\cdots\!16\)\( T^{7} + \)\(21\!\cdots\!45\)\( T^{8} - \)\(96\!\cdots\!94\)\( T^{9} + \)\(49\!\cdots\!87\)\( T^{10} - \)\(19\!\cdots\!78\)\( T^{11} + \)\(85\!\cdots\!88\)\( T^{12} - \)\(19\!\cdots\!78\)\( p^{3} T^{13} + \)\(49\!\cdots\!87\)\( p^{6} T^{14} - \)\(96\!\cdots\!94\)\( p^{9} T^{15} + \)\(21\!\cdots\!45\)\( p^{12} T^{16} - \)\(35\!\cdots\!16\)\( p^{15} T^{17} + 671398014573953230 p^{18} T^{18} - 890940789162870 p^{21} T^{19} + 1427421860244 p^{24} T^{20} - 26381752 p^{28} T^{21} + 1814271 p^{30} T^{22} - 1026 p^{33} T^{23} + p^{36} T^{24} \) | |
| 59 | \( 1 + 792 T + 32886 p T^{2} + 1404683820 T^{3} + 1829013156759 T^{4} + 1178531621654760 T^{5} + 1092756627068295240 T^{6} + \)\(62\!\cdots\!28\)\( T^{7} + \)\(45\!\cdots\!01\)\( T^{8} + \)\(23\!\cdots\!76\)\( T^{9} + \)\(14\!\cdots\!54\)\( T^{10} + \)\(62\!\cdots\!40\)\( T^{11} + \)\(33\!\cdots\!35\)\( T^{12} + \)\(62\!\cdots\!40\)\( p^{3} T^{13} + \)\(14\!\cdots\!54\)\( p^{6} T^{14} + \)\(23\!\cdots\!76\)\( p^{9} T^{15} + \)\(45\!\cdots\!01\)\( p^{12} T^{16} + \)\(62\!\cdots\!28\)\( p^{15} T^{17} + 1092756627068295240 p^{18} T^{18} + 1178531621654760 p^{21} T^{19} + 1829013156759 p^{24} T^{20} + 1404683820 p^{27} T^{21} + 32886 p^{31} T^{22} + 792 p^{33} T^{23} + p^{36} T^{24} \) | |
| 61 | \( 1 + 1212 T + 2239122 T^{2} + 1950405742 T^{3} + 2103563986245 T^{4} + 1463801267573628 T^{5} + 1192783852845337544 T^{6} + \)\(70\!\cdots\!50\)\( T^{7} + \)\(48\!\cdots\!71\)\( T^{8} + \)\(25\!\cdots\!88\)\( T^{9} + \)\(15\!\cdots\!48\)\( T^{10} + \)\(71\!\cdots\!12\)\( T^{11} + \)\(38\!\cdots\!85\)\( T^{12} + \)\(71\!\cdots\!12\)\( p^{3} T^{13} + \)\(15\!\cdots\!48\)\( p^{6} T^{14} + \)\(25\!\cdots\!88\)\( p^{9} T^{15} + \)\(48\!\cdots\!71\)\( p^{12} T^{16} + \)\(70\!\cdots\!50\)\( p^{15} T^{17} + 1192783852845337544 p^{18} T^{18} + 1463801267573628 p^{21} T^{19} + 2103563986245 p^{24} T^{20} + 1950405742 p^{27} T^{21} + 2239122 p^{30} T^{22} + 1212 p^{33} T^{23} + p^{36} T^{24} \) | |
| 67 | \( 1 - 624 T + 2227497 T^{2} - 1020875822 T^{3} + 2240105622237 T^{4} - 734700728026800 T^{5} + 1396377708613323076 T^{6} - \)\(30\!\cdots\!92\)\( T^{7} + \)\(62\!\cdots\!56\)\( T^{8} - \)\(81\!\cdots\!54\)\( T^{9} + \)\(22\!\cdots\!32\)\( T^{10} - \)\(17\!\cdots\!30\)\( T^{11} + \)\(72\!\cdots\!51\)\( T^{12} - \)\(17\!\cdots\!30\)\( p^{3} T^{13} + \)\(22\!\cdots\!32\)\( p^{6} T^{14} - \)\(81\!\cdots\!54\)\( p^{9} T^{15} + \)\(62\!\cdots\!56\)\( p^{12} T^{16} - \)\(30\!\cdots\!92\)\( p^{15} T^{17} + 1396377708613323076 p^{18} T^{18} - 734700728026800 p^{21} T^{19} + 2240105622237 p^{24} T^{20} - 1020875822 p^{27} T^{21} + 2227497 p^{30} T^{22} - 624 p^{33} T^{23} + p^{36} T^{24} \) | |
| 71 | \( 1 + 2802 T + 5337834 T^{2} + 7404701484 T^{3} + 8651498332659 T^{4} + 8740808299670118 T^{5} + 7989057437172554298 T^{6} + \)\(66\!\cdots\!52\)\( T^{7} + \)\(51\!\cdots\!71\)\( T^{8} + \)\(37\!\cdots\!02\)\( T^{9} + \)\(25\!\cdots\!20\)\( T^{10} + \)\(16\!\cdots\!84\)\( T^{11} + \)\(10\!\cdots\!89\)\( T^{12} + \)\(16\!\cdots\!84\)\( p^{3} T^{13} + \)\(25\!\cdots\!20\)\( p^{6} T^{14} + \)\(37\!\cdots\!02\)\( p^{9} T^{15} + \)\(51\!\cdots\!71\)\( p^{12} T^{16} + \)\(66\!\cdots\!52\)\( p^{15} T^{17} + 7989057437172554298 p^{18} T^{18} + 8740808299670118 p^{21} T^{19} + 8651498332659 p^{24} T^{20} + 7404701484 p^{27} T^{21} + 5337834 p^{30} T^{22} + 2802 p^{33} T^{23} + p^{36} T^{24} \) | |
| 73 | \( 1 + 726 T + 2680023 T^{2} + 1620396282 T^{3} + 3655194247101 T^{4} + 1949122934768184 T^{5} + 3380858402717245766 T^{6} + \)\(16\!\cdots\!78\)\( T^{7} + \)\(23\!\cdots\!90\)\( T^{8} + \)\(10\!\cdots\!28\)\( T^{9} + \)\(12\!\cdots\!42\)\( T^{10} + \)\(49\!\cdots\!64\)\( T^{11} + \)\(75\!\cdots\!51\)\( p T^{12} + \)\(49\!\cdots\!64\)\( p^{3} T^{13} + \)\(12\!\cdots\!42\)\( p^{6} T^{14} + \)\(10\!\cdots\!28\)\( p^{9} T^{15} + \)\(23\!\cdots\!90\)\( p^{12} T^{16} + \)\(16\!\cdots\!78\)\( p^{15} T^{17} + 3380858402717245766 p^{18} T^{18} + 1949122934768184 p^{21} T^{19} + 3655194247101 p^{24} T^{20} + 1620396282 p^{27} T^{21} + 2680023 p^{30} T^{22} + 726 p^{33} T^{23} + p^{36} T^{24} \) | |
| 79 | \( 1 - 444 T + 5072793 T^{2} - 1964463704 T^{3} + 12036963512007 T^{4} - 4076410208752380 T^{5} + 17743948207785774412 T^{6} - \)\(52\!\cdots\!64\)\( T^{7} + \)\(18\!\cdots\!41\)\( T^{8} - \)\(47\!\cdots\!40\)\( T^{9} + \)\(13\!\cdots\!71\)\( T^{10} - \)\(31\!\cdots\!72\)\( T^{11} + \)\(77\!\cdots\!46\)\( T^{12} - \)\(31\!\cdots\!72\)\( p^{3} T^{13} + \)\(13\!\cdots\!71\)\( p^{6} T^{14} - \)\(47\!\cdots\!40\)\( p^{9} T^{15} + \)\(18\!\cdots\!41\)\( p^{12} T^{16} - \)\(52\!\cdots\!64\)\( p^{15} T^{17} + 17743948207785774412 p^{18} T^{18} - 4076410208752380 p^{21} T^{19} + 12036963512007 p^{24} T^{20} - 1964463704 p^{27} T^{21} + 5072793 p^{30} T^{22} - 444 p^{33} T^{23} + p^{36} T^{24} \) | |
| 83 | \( 1 - 672 T + 4281114 T^{2} - 2904905872 T^{3} + 9248883258969 T^{4} - 6065735288241552 T^{5} + 13237198106428122734 T^{6} - \)\(81\!\cdots\!04\)\( T^{7} + \)\(13\!\cdots\!82\)\( T^{8} - \)\(79\!\cdots\!68\)\( T^{9} + \)\(11\!\cdots\!30\)\( T^{10} - \)\(58\!\cdots\!12\)\( T^{11} + \)\(72\!\cdots\!57\)\( T^{12} - \)\(58\!\cdots\!12\)\( p^{3} T^{13} + \)\(11\!\cdots\!30\)\( p^{6} T^{14} - \)\(79\!\cdots\!68\)\( p^{9} T^{15} + \)\(13\!\cdots\!82\)\( p^{12} T^{16} - \)\(81\!\cdots\!04\)\( p^{15} T^{17} + 13237198106428122734 p^{18} T^{18} - 6065735288241552 p^{21} T^{19} + 9248883258969 p^{24} T^{20} - 2904905872 p^{27} T^{21} + 4281114 p^{30} T^{22} - 672 p^{33} T^{23} + p^{36} T^{24} \) | |
| 89 | \( 1 + 906 T + 3296823 T^{2} + 1683077954 T^{3} + 4256546856249 T^{4} + 876481176210408 T^{5} + 3114301150610841606 T^{6} - \)\(24\!\cdots\!86\)\( T^{7} + \)\(15\!\cdots\!58\)\( T^{8} - \)\(59\!\cdots\!64\)\( T^{9} + \)\(69\!\cdots\!66\)\( T^{10} - \)\(47\!\cdots\!40\)\( T^{11} + \)\(37\!\cdots\!23\)\( T^{12} - \)\(47\!\cdots\!40\)\( p^{3} T^{13} + \)\(69\!\cdots\!66\)\( p^{6} T^{14} - \)\(59\!\cdots\!64\)\( p^{9} T^{15} + \)\(15\!\cdots\!58\)\( p^{12} T^{16} - \)\(24\!\cdots\!86\)\( p^{15} T^{17} + 3114301150610841606 p^{18} T^{18} + 876481176210408 p^{21} T^{19} + 4256546856249 p^{24} T^{20} + 1683077954 p^{27} T^{21} + 3296823 p^{30} T^{22} + 906 p^{33} T^{23} + p^{36} T^{24} \) | |
| 97 | \( 1 - 3246 T + 11906295 T^{2} - 26951325746 T^{3} + 59212806180915 T^{4} - 104655424107839712 T^{5} + \)\(17\!\cdots\!52\)\( T^{6} - \)\(25\!\cdots\!04\)\( T^{7} + \)\(34\!\cdots\!41\)\( T^{8} - \)\(42\!\cdots\!90\)\( T^{9} + \)\(48\!\cdots\!29\)\( T^{10} - \)\(51\!\cdots\!22\)\( T^{11} + \)\(51\!\cdots\!30\)\( T^{12} - \)\(51\!\cdots\!22\)\( p^{3} T^{13} + \)\(48\!\cdots\!29\)\( p^{6} T^{14} - \)\(42\!\cdots\!90\)\( p^{9} T^{15} + \)\(34\!\cdots\!41\)\( p^{12} T^{16} - \)\(25\!\cdots\!04\)\( p^{15} T^{17} + \)\(17\!\cdots\!52\)\( p^{18} T^{18} - 104655424107839712 p^{21} T^{19} + 59212806180915 p^{24} T^{20} - 26951325746 p^{27} T^{21} + 11906295 p^{30} T^{22} - 3246 p^{33} T^{23} + p^{36} T^{24} \) | |
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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
−4.08995932751701900968821899048, −3.94585058445318839524627756981, −3.90610803385001792551608934247, −3.86104316188538243498343269302, −3.68998379763064704941300761933, −3.65879524289308241778556071881, −3.63426010047432223921482666656, −3.58511338042728606061281075872, −3.37916915246236202328194421329, −3.37531954117928163134716772537, −3.11927256052460660064871427172, −2.82323790110148593813724749730, −2.79779114379239568143387187704, −2.66941105442484189568232135199, −2.64652265527151294360049913959, −2.60116633966459032028945946288, −2.30686478130574781048998629109, −2.16872836955996155159496255373, −1.87972748143915294414097184787, −1.87940627769928505063215704472, −1.79076213772651482353341510891, −1.57884593107119191998622730356, −1.42765235796776634997395105303, −0.977322334493618334195441898523, −0.915905877343383039859197628306, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.915905877343383039859197628306, 0.977322334493618334195441898523, 1.42765235796776634997395105303, 1.57884593107119191998622730356, 1.79076213772651482353341510891, 1.87940627769928505063215704472, 1.87972748143915294414097184787, 2.16872836955996155159496255373, 2.30686478130574781048998629109, 2.60116633966459032028945946288, 2.64652265527151294360049913959, 2.66941105442484189568232135199, 2.79779114379239568143387187704, 2.82323790110148593813724749730, 3.11927256052460660064871427172, 3.37531954117928163134716772537, 3.37916915246236202328194421329, 3.58511338042728606061281075872, 3.63426010047432223921482666656, 3.65879524289308241778556071881, 3.68998379763064704941300761933, 3.86104316188538243498343269302, 3.90610803385001792551608934247, 3.94585058445318839524627756981, 4.08995932751701900968821899048
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.