Dirichlet series
| L(s) = 1 | − 24·2-s − 36·3-s + 312·4-s − 10·5-s + 864·6-s − 25·7-s − 2.91e3·8-s + 702·9-s + 240·10-s + 132·11-s − 1.12e4·12-s + 12·13-s + 600·14-s + 360·15-s + 2.18e4·16-s − 10·17-s − 1.68e4·18-s − 9·19-s − 3.12e3·20-s + 900·21-s − 3.16e3·22-s − 201·23-s + 1.04e5·24-s − 595·25-s − 288·26-s − 9.82e3·27-s − 7.80e3·28-s + ⋯ |
| L(s) = 1 | − 8.48·2-s − 6.92·3-s + 39·4-s − 0.894·5-s + 58.7·6-s − 1.34·7-s − 128.·8-s + 26·9-s + 7.58·10-s + 3.61·11-s − 270.·12-s + 0.256·13-s + 11.4·14-s + 6.19·15-s + 341.·16-s − 0.142·17-s − 220.·18-s − 0.108·19-s − 34.8·20-s + 9.35·21-s − 30.7·22-s − 1.82·23-s + 891.·24-s − 4.75·25-s − 2.17·26-s − 70.0·27-s − 52.6·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 11^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 11^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{12} \cdot 11^{12} \cdot 37^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.00462\times 10^{25}\) |
| Root analytic conductor: | \(12.0034\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{12} \cdot 11^{12} \cdot 37^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
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 | 2 | \( ( 1 + p T )^{12} \) |
| 3 | \( ( 1 + p T )^{12} \) | |
| 11 | \( ( 1 - p T )^{12} \) | |
| 37 | \( ( 1 + p T )^{12} \) | |
| good | 5 | \( 1 + 2 p T + 139 p T^{2} + 263 p^{2} T^{3} + 49233 p T^{4} + 410541 p T^{5} + 58993793 T^{6} + 421262348 T^{7} + 10795298459 T^{8} + 66309190698 T^{9} + 1630853515848 T^{10} + 8956237028044 T^{11} + 215407070259534 T^{12} + 8956237028044 p^{3} T^{13} + 1630853515848 p^{6} T^{14} + 66309190698 p^{9} T^{15} + 10795298459 p^{12} T^{16} + 421262348 p^{15} T^{17} + 58993793 p^{18} T^{18} + 410541 p^{22} T^{19} + 49233 p^{25} T^{20} + 263 p^{29} T^{21} + 139 p^{31} T^{22} + 2 p^{34} T^{23} + p^{36} T^{24} \) |
| 7 | \( 1 + 25 T + 2111 T^{2} + 7246 p T^{3} + 2239476 T^{4} + 47770400 T^{5} + 1532070007 T^{6} + 28444475657 T^{7} + 755283548351 T^{8} + 12431895701560 T^{9} + 297757527302426 T^{10} + 4549435618494526 T^{11} + 104990941599979544 T^{12} + 4549435618494526 p^{3} T^{13} + 297757527302426 p^{6} T^{14} + 12431895701560 p^{9} T^{15} + 755283548351 p^{12} T^{16} + 28444475657 p^{15} T^{17} + 1532070007 p^{18} T^{18} + 47770400 p^{21} T^{19} + 2239476 p^{24} T^{20} + 7246 p^{28} T^{21} + 2111 p^{30} T^{22} + 25 p^{33} T^{23} + p^{36} T^{24} \) | |
| 13 | \( 1 - 12 T + 11289 T^{2} - 5077 p T^{3} + 74263724 T^{4} - 181092173 T^{5} + 343980436621 T^{6} + 208684190400 T^{7} + 1231266130496919 T^{8} + 3001398432155764 T^{9} + 3573826911403604434 T^{10} + 11637266387934339714 T^{11} + \)\(85\!\cdots\!72\)\( T^{12} + 11637266387934339714 p^{3} T^{13} + 3573826911403604434 p^{6} T^{14} + 3001398432155764 p^{9} T^{15} + 1231266130496919 p^{12} T^{16} + 208684190400 p^{15} T^{17} + 343980436621 p^{18} T^{18} - 181092173 p^{21} T^{19} + 74263724 p^{24} T^{20} - 5077 p^{28} T^{21} + 11289 p^{30} T^{22} - 12 p^{33} T^{23} + p^{36} T^{24} \) | |
| 17 | \( 1 + 10 T + 27667 T^{2} + 39711 T^{3} + 349610449 T^{4} - 2518262727 T^{5} + 2712435033049 T^{6} - 43662052512444 T^{7} + 14938301747325511 T^{8} - 378447063015086510 T^{9} + 67660055388089978420 T^{10} - \)\(22\!\cdots\!68\)\( T^{11} + \)\(30\!\cdots\!78\)\( T^{12} - \)\(22\!\cdots\!68\)\( p^{3} T^{13} + 67660055388089978420 p^{6} T^{14} - 378447063015086510 p^{9} T^{15} + 14938301747325511 p^{12} T^{16} - 43662052512444 p^{15} T^{17} + 2712435033049 p^{18} T^{18} - 2518262727 p^{21} T^{19} + 349610449 p^{24} T^{20} + 39711 p^{27} T^{21} + 27667 p^{30} T^{22} + 10 p^{33} T^{23} + p^{36} T^{24} \) | |
| 19 | \( 1 + 9 T + 42603 T^{2} - 775222 T^{3} + 911870904 T^{4} - 35851104756 T^{5} + 13459109884703 T^{6} - 724967643125547 T^{7} + 152415727748358999 T^{8} - 9421629490624775168 T^{9} + 73141956262443143658 p T^{10} - \)\(87\!\cdots\!90\)\( T^{11} + \)\(10\!\cdots\!96\)\( T^{12} - \)\(87\!\cdots\!90\)\( p^{3} T^{13} + 73141956262443143658 p^{7} T^{14} - 9421629490624775168 p^{9} T^{15} + 152415727748358999 p^{12} T^{16} - 724967643125547 p^{15} T^{17} + 13459109884703 p^{18} T^{18} - 35851104756 p^{21} T^{19} + 911870904 p^{24} T^{20} - 775222 p^{27} T^{21} + 42603 p^{30} T^{22} + 9 p^{33} T^{23} + p^{36} T^{24} \) | |
| 23 | \( 1 + 201 T + 96927 T^{2} + 15894854 T^{3} + 4343736840 T^{4} + 625500585264 T^{5} + 125456527807959 T^{6} + 16452611629882825 T^{7} + 2667508768304429039 T^{8} + \)\(32\!\cdots\!88\)\( T^{9} + \)\(44\!\cdots\!74\)\( T^{10} + \)\(49\!\cdots\!42\)\( T^{11} + \)\(59\!\cdots\!60\)\( T^{12} + \)\(49\!\cdots\!42\)\( p^{3} T^{13} + \)\(44\!\cdots\!74\)\( p^{6} T^{14} + \)\(32\!\cdots\!88\)\( p^{9} T^{15} + 2667508768304429039 p^{12} T^{16} + 16452611629882825 p^{15} T^{17} + 125456527807959 p^{18} T^{18} + 625500585264 p^{21} T^{19} + 4343736840 p^{24} T^{20} + 15894854 p^{27} T^{21} + 96927 p^{30} T^{22} + 201 p^{33} T^{23} + p^{36} T^{24} \) | |
| 29 | \( 1 + 84 T + 117183 T^{2} + 5986099 T^{3} + 7629298562 T^{4} + 314084011333 T^{5} + 367733218250339 T^{6} + 13104885542355402 T^{7} + 479509332808096251 p T^{8} + \)\(44\!\cdots\!86\)\( T^{9} + \)\(43\!\cdots\!98\)\( T^{10} + \)\(12\!\cdots\!96\)\( T^{11} + \)\(11\!\cdots\!36\)\( T^{12} + \)\(12\!\cdots\!96\)\( p^{3} T^{13} + \)\(43\!\cdots\!98\)\( p^{6} T^{14} + \)\(44\!\cdots\!86\)\( p^{9} T^{15} + 479509332808096251 p^{13} T^{16} + 13104885542355402 p^{15} T^{17} + 367733218250339 p^{18} T^{18} + 314084011333 p^{21} T^{19} + 7629298562 p^{24} T^{20} + 5986099 p^{27} T^{21} + 117183 p^{30} T^{22} + 84 p^{33} T^{23} + p^{36} T^{24} \) | |
| 31 | \( 1 - 10 T + 90073 T^{2} - 10419617 T^{3} + 160199318 p T^{4} - 538483606995 T^{5} + 223678314670449 T^{6} - 15839277608227226 T^{7} + 6051938274070366623 T^{8} - \)\(19\!\cdots\!20\)\( T^{9} + \)\(13\!\cdots\!74\)\( T^{10} + \)\(71\!\cdots\!98\)\( T^{11} + \)\(33\!\cdots\!84\)\( T^{12} + \)\(71\!\cdots\!98\)\( p^{3} T^{13} + \)\(13\!\cdots\!74\)\( p^{6} T^{14} - \)\(19\!\cdots\!20\)\( p^{9} T^{15} + 6051938274070366623 p^{12} T^{16} - 15839277608227226 p^{15} T^{17} + 223678314670449 p^{18} T^{18} - 538483606995 p^{21} T^{19} + 160199318 p^{25} T^{20} - 10419617 p^{27} T^{21} + 90073 p^{30} T^{22} - 10 p^{33} T^{23} + p^{36} T^{24} \) | |
| 41 | \( 1 - 215 T + 421345 T^{2} - 33710442 T^{3} + 80161926850 T^{4} + 1500074879842 T^{5} + 10627931777185841 T^{6} + 927089144057009269 T^{7} + \)\(11\!\cdots\!95\)\( T^{8} + \)\(14\!\cdots\!98\)\( T^{9} + \)\(10\!\cdots\!46\)\( T^{10} + \)\(13\!\cdots\!28\)\( T^{11} + \)\(77\!\cdots\!24\)\( T^{12} + \)\(13\!\cdots\!28\)\( p^{3} T^{13} + \)\(10\!\cdots\!46\)\( p^{6} T^{14} + \)\(14\!\cdots\!98\)\( p^{9} T^{15} + \)\(11\!\cdots\!95\)\( p^{12} T^{16} + 927089144057009269 p^{15} T^{17} + 10627931777185841 p^{18} T^{18} + 1500074879842 p^{21} T^{19} + 80161926850 p^{24} T^{20} - 33710442 p^{27} T^{21} + 421345 p^{30} T^{22} - 215 p^{33} T^{23} + p^{36} T^{24} \) | |
| 43 | \( 1 + 434 T + 577217 T^{2} + 207932441 T^{3} + 154101671582 T^{4} + 1115637311739 p T^{5} + 26323238550530609 T^{6} + 7403713289762247544 T^{7} + \)\(33\!\cdots\!27\)\( T^{8} + \)\(88\!\cdots\!30\)\( T^{9} + \)\(35\!\cdots\!06\)\( T^{10} + \)\(86\!\cdots\!52\)\( T^{11} + \)\(30\!\cdots\!80\)\( T^{12} + \)\(86\!\cdots\!52\)\( p^{3} T^{13} + \)\(35\!\cdots\!06\)\( p^{6} T^{14} + \)\(88\!\cdots\!30\)\( p^{9} T^{15} + \)\(33\!\cdots\!27\)\( p^{12} T^{16} + 7403713289762247544 p^{15} T^{17} + 26323238550530609 p^{18} T^{18} + 1115637311739 p^{22} T^{19} + 154101671582 p^{24} T^{20} + 207932441 p^{27} T^{21} + 577217 p^{30} T^{22} + 434 p^{33} T^{23} + p^{36} T^{24} \) | |
| 47 | \( 1 + 256 T + 702105 T^{2} + 186648095 T^{3} + 246770822847 T^{4} + 68170129884089 T^{5} + 57136534696368343 T^{6} + 16369730634616800028 T^{7} + \)\(98\!\cdots\!39\)\( T^{8} + \)\(28\!\cdots\!80\)\( T^{9} + \)\(13\!\cdots\!28\)\( T^{10} + \)\(38\!\cdots\!42\)\( T^{11} + \)\(15\!\cdots\!26\)\( T^{12} + \)\(38\!\cdots\!42\)\( p^{3} T^{13} + \)\(13\!\cdots\!28\)\( p^{6} T^{14} + \)\(28\!\cdots\!80\)\( p^{9} T^{15} + \)\(98\!\cdots\!39\)\( p^{12} T^{16} + 16369730634616800028 p^{15} T^{17} + 57136534696368343 p^{18} T^{18} + 68170129884089 p^{21} T^{19} + 246770822847 p^{24} T^{20} + 186648095 p^{27} T^{21} + 702105 p^{30} T^{22} + 256 p^{33} T^{23} + p^{36} T^{24} \) | |
| 53 | \( 1 + 448 T + 852779 T^{2} + 304397209 T^{3} + 329197450580 T^{4} + 108267718553923 T^{5} + 79820025726846231 T^{6} + 27356934140522765726 T^{7} + \)\(13\!\cdots\!03\)\( T^{8} + \)\(54\!\cdots\!62\)\( T^{9} + \)\(19\!\cdots\!02\)\( T^{10} + \)\(92\!\cdots\!44\)\( T^{11} + \)\(26\!\cdots\!08\)\( T^{12} + \)\(92\!\cdots\!44\)\( p^{3} T^{13} + \)\(19\!\cdots\!02\)\( p^{6} T^{14} + \)\(54\!\cdots\!62\)\( p^{9} T^{15} + \)\(13\!\cdots\!03\)\( p^{12} T^{16} + 27356934140522765726 p^{15} T^{17} + 79820025726846231 p^{18} T^{18} + 108267718553923 p^{21} T^{19} + 329197450580 p^{24} T^{20} + 304397209 p^{27} T^{21} + 852779 p^{30} T^{22} + 448 p^{33} T^{23} + p^{36} T^{24} \) | |
| 59 | \( 1 + 588 T + 1484280 T^{2} + 822766096 T^{3} + 1140262384462 T^{4} + 586280887529800 T^{5} + 585574876911601496 T^{6} + \)\(27\!\cdots\!12\)\( T^{7} + \)\(22\!\cdots\!51\)\( T^{8} + \)\(95\!\cdots\!36\)\( T^{9} + \)\(64\!\cdots\!00\)\( T^{10} + \)\(25\!\cdots\!00\)\( T^{11} + \)\(14\!\cdots\!60\)\( T^{12} + \)\(25\!\cdots\!00\)\( p^{3} T^{13} + \)\(64\!\cdots\!00\)\( p^{6} T^{14} + \)\(95\!\cdots\!36\)\( p^{9} T^{15} + \)\(22\!\cdots\!51\)\( p^{12} T^{16} + \)\(27\!\cdots\!12\)\( p^{15} T^{17} + 585574876911601496 p^{18} T^{18} + 586280887529800 p^{21} T^{19} + 1140262384462 p^{24} T^{20} + 822766096 p^{27} T^{21} + 1484280 p^{30} T^{22} + 588 p^{33} T^{23} + p^{36} T^{24} \) | |
| 61 | \( 1 - 735 T + 1428722 T^{2} - 1062190012 T^{3} + 1113644224992 T^{4} - 778555443895792 T^{5} + 605287141557600778 T^{6} - \)\(38\!\cdots\!03\)\( T^{7} + \)\(24\!\cdots\!19\)\( T^{8} - \)\(13\!\cdots\!86\)\( T^{9} + \)\(79\!\cdots\!56\)\( T^{10} - \)\(39\!\cdots\!24\)\( T^{11} + \)\(20\!\cdots\!68\)\( T^{12} - \)\(39\!\cdots\!24\)\( p^{3} T^{13} + \)\(79\!\cdots\!56\)\( p^{6} T^{14} - \)\(13\!\cdots\!86\)\( p^{9} T^{15} + \)\(24\!\cdots\!19\)\( p^{12} T^{16} - \)\(38\!\cdots\!03\)\( p^{15} T^{17} + 605287141557600778 p^{18} T^{18} - 778555443895792 p^{21} T^{19} + 1113644224992 p^{24} T^{20} - 1062190012 p^{27} T^{21} + 1428722 p^{30} T^{22} - 735 p^{33} T^{23} + p^{36} T^{24} \) | |
| 67 | \( 1 + 288 T + 1955739 T^{2} + 693951755 T^{3} + 1961942869220 T^{4} + 756030313980721 T^{5} + 1309499293586745175 T^{6} + \)\(52\!\cdots\!72\)\( T^{7} + \)\(64\!\cdots\!43\)\( T^{8} + \)\(26\!\cdots\!44\)\( T^{9} + \)\(25\!\cdots\!14\)\( T^{10} + \)\(10\!\cdots\!22\)\( T^{11} + \)\(83\!\cdots\!52\)\( T^{12} + \)\(10\!\cdots\!22\)\( p^{3} T^{13} + \)\(25\!\cdots\!14\)\( p^{6} T^{14} + \)\(26\!\cdots\!44\)\( p^{9} T^{15} + \)\(64\!\cdots\!43\)\( p^{12} T^{16} + \)\(52\!\cdots\!72\)\( p^{15} T^{17} + 1309499293586745175 p^{18} T^{18} + 756030313980721 p^{21} T^{19} + 1961942869220 p^{24} T^{20} + 693951755 p^{27} T^{21} + 1955739 p^{30} T^{22} + 288 p^{33} T^{23} + p^{36} T^{24} \) | |
| 71 | \( 1 - 1019 T + 2675593 T^{2} - 2526211999 T^{3} + 3618973713293 T^{4} - 3048622473089575 T^{5} + 3229096994785801263 T^{6} - \)\(24\!\cdots\!03\)\( T^{7} + \)\(21\!\cdots\!55\)\( T^{8} - \)\(14\!\cdots\!24\)\( T^{9} + \)\(10\!\cdots\!24\)\( T^{10} - \)\(63\!\cdots\!76\)\( T^{11} + \)\(42\!\cdots\!94\)\( T^{12} - \)\(63\!\cdots\!76\)\( p^{3} T^{13} + \)\(10\!\cdots\!24\)\( p^{6} T^{14} - \)\(14\!\cdots\!24\)\( p^{9} T^{15} + \)\(21\!\cdots\!55\)\( p^{12} T^{16} - \)\(24\!\cdots\!03\)\( p^{15} T^{17} + 3229096994785801263 p^{18} T^{18} - 3048622473089575 p^{21} T^{19} + 3618973713293 p^{24} T^{20} - 2526211999 p^{27} T^{21} + 2675593 p^{30} T^{22} - 1019 p^{33} T^{23} + p^{36} T^{24} \) | |
| 73 | \( 1 + 309 T + 1990140 T^{2} + 112698726 T^{3} + 2035236130234 T^{4} - 67383058270746 T^{5} + 1612851640509005404 T^{6} - 77341960290299734783 T^{7} + \)\(98\!\cdots\!39\)\( T^{8} - \)\(65\!\cdots\!54\)\( T^{9} + \)\(49\!\cdots\!48\)\( T^{10} - \)\(30\!\cdots\!04\)\( T^{11} + \)\(21\!\cdots\!44\)\( T^{12} - \)\(30\!\cdots\!04\)\( p^{3} T^{13} + \)\(49\!\cdots\!48\)\( p^{6} T^{14} - \)\(65\!\cdots\!54\)\( p^{9} T^{15} + \)\(98\!\cdots\!39\)\( p^{12} T^{16} - 77341960290299734783 p^{15} T^{17} + 1612851640509005404 p^{18} T^{18} - 67383058270746 p^{21} T^{19} + 2035236130234 p^{24} T^{20} + 112698726 p^{27} T^{21} + 1990140 p^{30} T^{22} + 309 p^{33} T^{23} + p^{36} T^{24} \) | |
| 79 | \( 1 - 1116 T + 3674841 T^{2} - 3277627887 T^{3} + 6376580835445 T^{4} - 4824091130060677 T^{5} + 7204518282412451013 T^{6} - \)\(47\!\cdots\!56\)\( T^{7} + \)\(60\!\cdots\!27\)\( T^{8} - \)\(35\!\cdots\!56\)\( T^{9} + \)\(39\!\cdots\!94\)\( T^{10} - \)\(21\!\cdots\!34\)\( T^{11} + \)\(21\!\cdots\!98\)\( T^{12} - \)\(21\!\cdots\!34\)\( p^{3} T^{13} + \)\(39\!\cdots\!94\)\( p^{6} T^{14} - \)\(35\!\cdots\!56\)\( p^{9} T^{15} + \)\(60\!\cdots\!27\)\( p^{12} T^{16} - \)\(47\!\cdots\!56\)\( p^{15} T^{17} + 7204518282412451013 p^{18} T^{18} - 4824091130060677 p^{21} T^{19} + 6376580835445 p^{24} T^{20} - 3277627887 p^{27} T^{21} + 3674841 p^{30} T^{22} - 1116 p^{33} T^{23} + p^{36} T^{24} \) | |
| 83 | \( 1 + 662 T + 4710321 T^{2} + 2895282419 T^{3} + 10409888186807 T^{4} + 6088502477211081 T^{5} + 14582244119054253881 T^{6} + \)\(82\!\cdots\!82\)\( T^{7} + \)\(14\!\cdots\!43\)\( T^{8} + \)\(81\!\cdots\!72\)\( T^{9} + \)\(11\!\cdots\!06\)\( T^{10} + \)\(60\!\cdots\!50\)\( T^{11} + \)\(73\!\cdots\!70\)\( T^{12} + \)\(60\!\cdots\!50\)\( p^{3} T^{13} + \)\(11\!\cdots\!06\)\( p^{6} T^{14} + \)\(81\!\cdots\!72\)\( p^{9} T^{15} + \)\(14\!\cdots\!43\)\( p^{12} T^{16} + \)\(82\!\cdots\!82\)\( p^{15} T^{17} + 14582244119054253881 p^{18} T^{18} + 6088502477211081 p^{21} T^{19} + 10409888186807 p^{24} T^{20} + 2895282419 p^{27} T^{21} + 4710321 p^{30} T^{22} + 662 p^{33} T^{23} + p^{36} T^{24} \) | |
| 89 | \( 1 - 1603 T + 6068922 T^{2} - 7442346332 T^{3} + 16374941922044 T^{4} - 16226006304027372 T^{5} + 27030431798943843958 T^{6} - \)\(22\!\cdots\!31\)\( T^{7} + \)\(31\!\cdots\!51\)\( T^{8} - \)\(22\!\cdots\!38\)\( T^{9} + \)\(28\!\cdots\!84\)\( T^{10} - \)\(18\!\cdots\!16\)\( T^{11} + \)\(21\!\cdots\!60\)\( T^{12} - \)\(18\!\cdots\!16\)\( p^{3} T^{13} + \)\(28\!\cdots\!84\)\( p^{6} T^{14} - \)\(22\!\cdots\!38\)\( p^{9} T^{15} + \)\(31\!\cdots\!51\)\( p^{12} T^{16} - \)\(22\!\cdots\!31\)\( p^{15} T^{17} + 27030431798943843958 p^{18} T^{18} - 16226006304027372 p^{21} T^{19} + 16374941922044 p^{24} T^{20} - 7442346332 p^{27} T^{21} + 6068922 p^{30} T^{22} - 1603 p^{33} T^{23} + p^{36} T^{24} \) | |
| 97 | \( 1 - 349 T + 8005194 T^{2} - 3567007288 T^{3} + 30664296810796 T^{4} - 16071993333155708 T^{5} + 74819693905074286482 T^{6} - \)\(43\!\cdots\!05\)\( T^{7} + \)\(13\!\cdots\!67\)\( p T^{8} - \)\(77\!\cdots\!30\)\( T^{9} + \)\(17\!\cdots\!96\)\( T^{10} - \)\(97\!\cdots\!04\)\( T^{11} + \)\(17\!\cdots\!32\)\( T^{12} - \)\(97\!\cdots\!04\)\( p^{3} T^{13} + \)\(17\!\cdots\!96\)\( p^{6} T^{14} - \)\(77\!\cdots\!30\)\( p^{9} T^{15} + \)\(13\!\cdots\!67\)\( p^{13} T^{16} - \)\(43\!\cdots\!05\)\( p^{15} T^{17} + 74819693905074286482 p^{18} T^{18} - 16071993333155708 p^{21} T^{19} + 30664296810796 p^{24} T^{20} - 3567007288 p^{27} T^{21} + 8005194 p^{30} T^{22} - 349 p^{33} T^{23} + p^{36} T^{24} \) | |
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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
−2.78761667237605464113412025254, −2.26794401234863648196936193394, −2.21090910867065753125175844382, −2.18542602712602282985103740681, −2.13679691529029389142475790400, −2.08702424886437363564333508742, −2.04425385286698435407081551166, −2.03659520034807278732420985878, −2.02070598518135154301909979286, −1.91283461461433911563003411066, −1.89834181309450420956397491676, −1.83801186556053101835896509774, −1.82394940212659550957932355322, −1.27348436654210685541479764159, −1.25743265106341161299017477627, −1.20992554882746181973408088983, −1.19412315990276152957177029079, −1.13858179193870004903260774192, −1.10765722069248408030823971402, −1.10019112572532313938189542341, −1.02578839012028026852437279800, −0.887469980194265983265687872784, −0.862863324267224004364871055960, −0.73662306483684975610875694500, −0.71103670600311487968008117695, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.71103670600311487968008117695, 0.73662306483684975610875694500, 0.862863324267224004364871055960, 0.887469980194265983265687872784, 1.02578839012028026852437279800, 1.10019112572532313938189542341, 1.10765722069248408030823971402, 1.13858179193870004903260774192, 1.19412315990276152957177029079, 1.20992554882746181973408088983, 1.25743265106341161299017477627, 1.27348436654210685541479764159, 1.82394940212659550957932355322, 1.83801186556053101835896509774, 1.89834181309450420956397491676, 1.91283461461433911563003411066, 2.02070598518135154301909979286, 2.03659520034807278732420985878, 2.04425385286698435407081551166, 2.08702424886437363564333508742, 2.13679691529029389142475790400, 2.18542602712602282985103740681, 2.21090910867065753125175844382, 2.26794401234863648196936193394, 2.78761667237605464113412025254
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.