Dirichlet series
| L(s) = 1 | − 6·2-s − 42·3-s − 13·4-s − 4·5-s + 252·6-s + 156·8-s + 945·9-s + 24·10-s − 68·11-s + 546·12-s + 182·13-s + 168·15-s − 172·16-s − 5.67e3·18-s + 24·19-s + 52·20-s + 408·22-s − 64·23-s − 6.55e3·24-s − 926·25-s − 1.09e3·26-s − 1.51e4·27-s − 792·29-s − 1.00e3·30-s + 524·31-s − 1.21e3·32-s + 2.85e3·33-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 8.08·3-s − 1.62·4-s − 0.357·5-s + 17.1·6-s + 6.89·8-s + 35·9-s + 0.758·10-s − 1.86·11-s + 13.1·12-s + 3.88·13-s + 2.89·15-s − 2.68·16-s − 74.2·18-s + 0.289·19-s + 0.581·20-s + 3.95·22-s − 0.580·23-s − 55.7·24-s − 7.40·25-s − 8.23·26-s − 107.·27-s − 5.07·29-s − 6.13·30-s + 3.03·31-s − 6.69·32-s + 15.0·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 7^{28} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 7^{28} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(3^{14} \cdot 7^{28} \cdot 13^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.36753\times 10^{28}\) |
| Root analytic conductor: | \(10.6185\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(14\) |
| Selberg data: | \((28,\ 3^{14} \cdot 7^{28} \cdot 13^{14} ,\ ( \ : [3/2]^{14} ),\ 1 )\) |
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 | 3 | \( ( 1 + p T )^{14} \) |
| 7 | \( 1 \) | |
| 13 | \( ( 1 - p T )^{14} \) | |
| good | 2 | \( 1 + 3 p T + 49 T^{2} + 27 p^{3} T^{3} + 1169 T^{4} + 2211 p T^{5} + 19467 T^{6} + 16701 p^{2} T^{7} + 130007 p T^{8} + 104313 p^{3} T^{9} + 737233 p^{2} T^{10} + 277857 p^{5} T^{11} + 452235 p^{6} T^{12} + 19977 p^{12} T^{13} + 1933421 p^{7} T^{14} + 19977 p^{15} T^{15} + 452235 p^{12} T^{16} + 277857 p^{14} T^{17} + 737233 p^{14} T^{18} + 104313 p^{18} T^{19} + 130007 p^{19} T^{20} + 16701 p^{23} T^{21} + 19467 p^{24} T^{22} + 2211 p^{28} T^{23} + 1169 p^{30} T^{24} + 27 p^{36} T^{25} + 49 p^{36} T^{26} + 3 p^{40} T^{27} + p^{42} T^{28} \) |
| 5 | \( 1 + 4 T + 942 T^{2} + 4896 T^{3} + 91679 p T^{4} + 2702928 T^{5} + 151899368 T^{6} + 947332132 T^{7} + 38143388382 T^{8} + 239701217492 T^{9} + 7650349874836 T^{10} + 9336929979736 p T^{11} + 1259157711436016 T^{12} + 7232271788104996 T^{13} + 172295518457805176 T^{14} + 7232271788104996 p^{3} T^{15} + 1259157711436016 p^{6} T^{16} + 9336929979736 p^{10} T^{17} + 7650349874836 p^{12} T^{18} + 239701217492 p^{15} T^{19} + 38143388382 p^{18} T^{20} + 947332132 p^{21} T^{21} + 151899368 p^{24} T^{22} + 2702928 p^{27} T^{23} + 91679 p^{31} T^{24} + 4896 p^{33} T^{25} + 942 p^{36} T^{26} + 4 p^{39} T^{27} + p^{42} T^{28} \) | |
| 11 | \( 1 + 68 T + 14196 T^{2} + 859356 T^{3} + 98248566 T^{4} + 5302960468 T^{5} + 437396520958 T^{6} + 21110051801676 T^{7} + 1395363657050090 T^{8} + 60381432271654204 T^{9} + 3369162686154551304 T^{10} + \)\(13\!\cdots\!04\)\( T^{11} + \)\(63\!\cdots\!67\)\( T^{12} + \)\(20\!\cdots\!28\)\( p T^{13} + \)\(94\!\cdots\!24\)\( T^{14} + \)\(20\!\cdots\!28\)\( p^{4} T^{15} + \)\(63\!\cdots\!67\)\( p^{6} T^{16} + \)\(13\!\cdots\!04\)\( p^{9} T^{17} + 3369162686154551304 p^{12} T^{18} + 60381432271654204 p^{15} T^{19} + 1395363657050090 p^{18} T^{20} + 21110051801676 p^{21} T^{21} + 437396520958 p^{24} T^{22} + 5302960468 p^{27} T^{23} + 98248566 p^{30} T^{24} + 859356 p^{33} T^{25} + 14196 p^{36} T^{26} + 68 p^{39} T^{27} + p^{42} T^{28} \) | |
| 17 | \( 1 + 52784 T^{2} + 155076 T^{3} + 1318750360 T^{4} + 8006598408 T^{5} + 20796435466538 T^{6} + 189727149164664 T^{7} + 233123130954801270 T^{8} + 2722732637699491668 T^{9} + \)\(19\!\cdots\!56\)\( T^{10} + \)\(15\!\cdots\!04\)\( p T^{11} + \)\(78\!\cdots\!13\)\( p T^{12} + \)\(17\!\cdots\!88\)\( T^{13} + \)\(72\!\cdots\!32\)\( T^{14} + \)\(17\!\cdots\!88\)\( p^{3} T^{15} + \)\(78\!\cdots\!13\)\( p^{7} T^{16} + \)\(15\!\cdots\!04\)\( p^{10} T^{17} + \)\(19\!\cdots\!56\)\( p^{12} T^{18} + 2722732637699491668 p^{15} T^{19} + 233123130954801270 p^{18} T^{20} + 189727149164664 p^{21} T^{21} + 20796435466538 p^{24} T^{22} + 8006598408 p^{27} T^{23} + 1318750360 p^{30} T^{24} + 155076 p^{33} T^{25} + 52784 p^{36} T^{26} + p^{42} T^{28} \) | |
| 19 | \( 1 - 24 T + 70662 T^{2} - 1866436 T^{3} + 2408009601 T^{4} - 69085805960 T^{5} + 52619509739336 T^{6} - 1604137584582044 T^{7} + 826732564934565920 T^{8} - 25975089958165360956 T^{9} + \)\(99\!\cdots\!44\)\( T^{10} - \)\(30\!\cdots\!16\)\( T^{11} + \)\(93\!\cdots\!18\)\( T^{12} - \)\(27\!\cdots\!88\)\( T^{13} + \)\(71\!\cdots\!60\)\( T^{14} - \)\(27\!\cdots\!88\)\( p^{3} T^{15} + \)\(93\!\cdots\!18\)\( p^{6} T^{16} - \)\(30\!\cdots\!16\)\( p^{9} T^{17} + \)\(99\!\cdots\!44\)\( p^{12} T^{18} - 25975089958165360956 p^{15} T^{19} + 826732564934565920 p^{18} T^{20} - 1604137584582044 p^{21} T^{21} + 52619509739336 p^{24} T^{22} - 69085805960 p^{27} T^{23} + 2408009601 p^{30} T^{24} - 1866436 p^{33} T^{25} + 70662 p^{36} T^{26} - 24 p^{39} T^{27} + p^{42} T^{28} \) | |
| 23 | \( 1 + 64 T + 106522 T^{2} + 5282448 T^{3} + 5478782681 T^{4} + 201514751448 T^{5} + 182633027560256 T^{6} + 4692680371439448 T^{7} + 4469503752235819920 T^{8} + 74982993768020302648 T^{9} + \)\(37\!\cdots\!68\)\( p T^{10} + \)\(90\!\cdots\!64\)\( T^{11} + \)\(13\!\cdots\!82\)\( T^{12} + \)\(97\!\cdots\!48\)\( T^{13} + \)\(18\!\cdots\!64\)\( T^{14} + \)\(97\!\cdots\!48\)\( p^{3} T^{15} + \)\(13\!\cdots\!82\)\( p^{6} T^{16} + \)\(90\!\cdots\!64\)\( p^{9} T^{17} + \)\(37\!\cdots\!68\)\( p^{13} T^{18} + 74982993768020302648 p^{15} T^{19} + 4469503752235819920 p^{18} T^{20} + 4692680371439448 p^{21} T^{21} + 182633027560256 p^{24} T^{22} + 201514751448 p^{27} T^{23} + 5478782681 p^{30} T^{24} + 5282448 p^{33} T^{25} + 106522 p^{36} T^{26} + 64 p^{39} T^{27} + p^{42} T^{28} \) | |
| 29 | \( 1 + 792 T + 513748 T^{2} + 231022704 T^{3} + 90693313353 T^{4} + 29500991737848 T^{5} + 8666701668575824 T^{6} + 2240915057621001312 T^{7} + \)\(53\!\cdots\!88\)\( T^{8} + \)\(11\!\cdots\!88\)\( T^{9} + \)\(23\!\cdots\!40\)\( T^{10} + \)\(44\!\cdots\!80\)\( T^{11} + \)\(79\!\cdots\!58\)\( T^{12} + \)\(13\!\cdots\!52\)\( T^{13} + \)\(21\!\cdots\!04\)\( T^{14} + \)\(13\!\cdots\!52\)\( p^{3} T^{15} + \)\(79\!\cdots\!58\)\( p^{6} T^{16} + \)\(44\!\cdots\!80\)\( p^{9} T^{17} + \)\(23\!\cdots\!40\)\( p^{12} T^{18} + \)\(11\!\cdots\!88\)\( p^{15} T^{19} + \)\(53\!\cdots\!88\)\( p^{18} T^{20} + 2240915057621001312 p^{21} T^{21} + 8666701668575824 p^{24} T^{22} + 29500991737848 p^{27} T^{23} + 90693313353 p^{30} T^{24} + 231022704 p^{33} T^{25} + 513748 p^{36} T^{26} + 792 p^{39} T^{27} + p^{42} T^{28} \) | |
| 31 | \( 1 - 524 T + 338288 T^{2} - 114424856 T^{3} + 42089175592 T^{4} - 10512517905180 T^{5} + 2831350122967258 T^{6} - 573989719999727284 T^{7} + \)\(13\!\cdots\!78\)\( T^{8} - \)\(24\!\cdots\!76\)\( T^{9} + \)\(16\!\cdots\!32\)\( p T^{10} - \)\(94\!\cdots\!84\)\( T^{11} + \)\(62\!\cdots\!15\)\( p T^{12} - \)\(34\!\cdots\!16\)\( T^{13} + \)\(63\!\cdots\!52\)\( T^{14} - \)\(34\!\cdots\!16\)\( p^{3} T^{15} + \)\(62\!\cdots\!15\)\( p^{7} T^{16} - \)\(94\!\cdots\!84\)\( p^{9} T^{17} + \)\(16\!\cdots\!32\)\( p^{13} T^{18} - \)\(24\!\cdots\!76\)\( p^{15} T^{19} + \)\(13\!\cdots\!78\)\( p^{18} T^{20} - 573989719999727284 p^{21} T^{21} + 2831350122967258 p^{24} T^{22} - 10512517905180 p^{27} T^{23} + 42089175592 p^{30} T^{24} - 114424856 p^{33} T^{25} + 338288 p^{36} T^{26} - 524 p^{39} T^{27} + p^{42} T^{28} \) | |
| 37 | \( 1 + 344 T + 450070 T^{2} + 159123320 T^{3} + 102013882688 T^{4} + 35133850065112 T^{5} + 15460735682219692 T^{6} + 4967567130072306008 T^{7} + \)\(17\!\cdots\!82\)\( T^{8} + \)\(50\!\cdots\!04\)\( T^{9} + \)\(15\!\cdots\!22\)\( T^{10} + \)\(39\!\cdots\!20\)\( T^{11} + \)\(10\!\cdots\!13\)\( T^{12} + \)\(25\!\cdots\!00\)\( T^{13} + \)\(59\!\cdots\!72\)\( T^{14} + \)\(25\!\cdots\!00\)\( p^{3} T^{15} + \)\(10\!\cdots\!13\)\( p^{6} T^{16} + \)\(39\!\cdots\!20\)\( p^{9} T^{17} + \)\(15\!\cdots\!22\)\( p^{12} T^{18} + \)\(50\!\cdots\!04\)\( p^{15} T^{19} + \)\(17\!\cdots\!82\)\( p^{18} T^{20} + 4967567130072306008 p^{21} T^{21} + 15460735682219692 p^{24} T^{22} + 35133850065112 p^{27} T^{23} + 102013882688 p^{30} T^{24} + 159123320 p^{33} T^{25} + 450070 p^{36} T^{26} + 344 p^{39} T^{27} + p^{42} T^{28} \) | |
| 41 | \( 1 - 44 T + 544292 T^{2} + 45871588 T^{3} + 3299709981 p T^{4} + 29296103335800 T^{5} + 22419653990126056 T^{6} + 7231760732266499832 T^{7} + \)\(30\!\cdots\!11\)\( T^{8} + \)\(10\!\cdots\!00\)\( T^{9} + \)\(35\!\cdots\!08\)\( T^{10} + \)\(11\!\cdots\!68\)\( T^{11} + \)\(35\!\cdots\!99\)\( T^{12} + \)\(95\!\cdots\!80\)\( T^{13} + \)\(66\!\cdots\!08\)\( p T^{14} + \)\(95\!\cdots\!80\)\( p^{3} T^{15} + \)\(35\!\cdots\!99\)\( p^{6} T^{16} + \)\(11\!\cdots\!68\)\( p^{9} T^{17} + \)\(35\!\cdots\!08\)\( p^{12} T^{18} + \)\(10\!\cdots\!00\)\( p^{15} T^{19} + \)\(30\!\cdots\!11\)\( p^{18} T^{20} + 7231760732266499832 p^{21} T^{21} + 22419653990126056 p^{24} T^{22} + 29296103335800 p^{27} T^{23} + 3299709981 p^{31} T^{24} + 45871588 p^{33} T^{25} + 544292 p^{36} T^{26} - 44 p^{39} T^{27} + p^{42} T^{28} \) | |
| 43 | \( 1 - 144 T + 717078 T^{2} - 111934792 T^{3} + 257993225097 T^{4} - 41169308846624 T^{5} + 61443914016378620 T^{6} - 9665258844454173128 T^{7} + \)\(10\!\cdots\!16\)\( T^{8} - \)\(16\!\cdots\!60\)\( T^{9} + \)\(14\!\cdots\!28\)\( T^{10} - \)\(20\!\cdots\!20\)\( T^{11} + \)\(15\!\cdots\!90\)\( T^{12} - \)\(20\!\cdots\!88\)\( T^{13} + \)\(14\!\cdots\!52\)\( T^{14} - \)\(20\!\cdots\!88\)\( p^{3} T^{15} + \)\(15\!\cdots\!90\)\( p^{6} T^{16} - \)\(20\!\cdots\!20\)\( p^{9} T^{17} + \)\(14\!\cdots\!28\)\( p^{12} T^{18} - \)\(16\!\cdots\!60\)\( p^{15} T^{19} + \)\(10\!\cdots\!16\)\( p^{18} T^{20} - 9665258844454173128 p^{21} T^{21} + 61443914016378620 p^{24} T^{22} - 41169308846624 p^{27} T^{23} + 257993225097 p^{30} T^{24} - 111934792 p^{33} T^{25} + 717078 p^{36} T^{26} - 144 p^{39} T^{27} + p^{42} T^{28} \) | |
| 47 | \( 1 + 236 T + 686722 T^{2} + 198812064 T^{3} + 252053547554 T^{4} + 82991698637028 T^{5} + 64998448973641424 T^{6} + 22700570028276474708 T^{7} + \)\(13\!\cdots\!06\)\( T^{8} + \)\(45\!\cdots\!64\)\( T^{9} + \)\(21\!\cdots\!42\)\( T^{10} + \)\(71\!\cdots\!80\)\( T^{11} + \)\(29\!\cdots\!87\)\( T^{12} + \)\(90\!\cdots\!36\)\( T^{13} + \)\(33\!\cdots\!20\)\( T^{14} + \)\(90\!\cdots\!36\)\( p^{3} T^{15} + \)\(29\!\cdots\!87\)\( p^{6} T^{16} + \)\(71\!\cdots\!80\)\( p^{9} T^{17} + \)\(21\!\cdots\!42\)\( p^{12} T^{18} + \)\(45\!\cdots\!64\)\( p^{15} T^{19} + \)\(13\!\cdots\!06\)\( p^{18} T^{20} + 22700570028276474708 p^{21} T^{21} + 64998448973641424 p^{24} T^{22} + 82991698637028 p^{27} T^{23} + 252053547554 p^{30} T^{24} + 198812064 p^{33} T^{25} + 686722 p^{36} T^{26} + 236 p^{39} T^{27} + p^{42} T^{28} \) | |
| 53 | \( 1 + 1556 T + 2468862 T^{2} + 2500806496 T^{3} + 2417330354432 T^{4} + 1874214261595388 T^{5} + 1378134097010393356 T^{6} + \)\(87\!\cdots\!96\)\( T^{7} + \)\(53\!\cdots\!46\)\( T^{8} + \)\(28\!\cdots\!44\)\( T^{9} + \)\(15\!\cdots\!26\)\( T^{10} + \)\(71\!\cdots\!08\)\( T^{11} + \)\(32\!\cdots\!69\)\( T^{12} + \)\(13\!\cdots\!52\)\( T^{13} + \)\(54\!\cdots\!84\)\( T^{14} + \)\(13\!\cdots\!52\)\( p^{3} T^{15} + \)\(32\!\cdots\!69\)\( p^{6} T^{16} + \)\(71\!\cdots\!08\)\( p^{9} T^{17} + \)\(15\!\cdots\!26\)\( p^{12} T^{18} + \)\(28\!\cdots\!44\)\( p^{15} T^{19} + \)\(53\!\cdots\!46\)\( p^{18} T^{20} + \)\(87\!\cdots\!96\)\( p^{21} T^{21} + 1378134097010393356 p^{24} T^{22} + 1874214261595388 p^{27} T^{23} + 2417330354432 p^{30} T^{24} + 2500806496 p^{33} T^{25} + 2468862 p^{36} T^{26} + 1556 p^{39} T^{27} + p^{42} T^{28} \) | |
| 59 | \( 1 - 1244 T + 2391910 T^{2} - 2202443172 T^{3} + 2479564713289 T^{4} - 1840973765972008 T^{5} + 1559025126038602120 T^{6} - \)\(98\!\cdots\!88\)\( T^{7} + \)\(68\!\cdots\!39\)\( T^{8} - \)\(38\!\cdots\!96\)\( T^{9} + \)\(23\!\cdots\!38\)\( T^{10} - \)\(11\!\cdots\!32\)\( T^{11} + \)\(63\!\cdots\!35\)\( T^{12} - \)\(28\!\cdots\!92\)\( T^{13} + \)\(14\!\cdots\!64\)\( T^{14} - \)\(28\!\cdots\!92\)\( p^{3} T^{15} + \)\(63\!\cdots\!35\)\( p^{6} T^{16} - \)\(11\!\cdots\!32\)\( p^{9} T^{17} + \)\(23\!\cdots\!38\)\( p^{12} T^{18} - \)\(38\!\cdots\!96\)\( p^{15} T^{19} + \)\(68\!\cdots\!39\)\( p^{18} T^{20} - \)\(98\!\cdots\!88\)\( p^{21} T^{21} + 1559025126038602120 p^{24} T^{22} - 1840973765972008 p^{27} T^{23} + 2479564713289 p^{30} T^{24} - 2202443172 p^{33} T^{25} + 2391910 p^{36} T^{26} - 1244 p^{39} T^{27} + p^{42} T^{28} \) | |
| 61 | \( 1 - 984 T + 2448012 T^{2} - 1984332868 T^{3} + 2783932032756 T^{4} - 1931917648550032 T^{5} + 1987583372515078046 T^{6} - \)\(12\!\cdots\!20\)\( T^{7} + \)\(10\!\cdots\!18\)\( T^{8} - \)\(54\!\cdots\!32\)\( T^{9} + \)\(39\!\cdots\!84\)\( T^{10} - \)\(19\!\cdots\!12\)\( T^{11} + \)\(12\!\cdots\!65\)\( T^{12} - \)\(53\!\cdots\!52\)\( T^{13} + \)\(30\!\cdots\!72\)\( T^{14} - \)\(53\!\cdots\!52\)\( p^{3} T^{15} + \)\(12\!\cdots\!65\)\( p^{6} T^{16} - \)\(19\!\cdots\!12\)\( p^{9} T^{17} + \)\(39\!\cdots\!84\)\( p^{12} T^{18} - \)\(54\!\cdots\!32\)\( p^{15} T^{19} + \)\(10\!\cdots\!18\)\( p^{18} T^{20} - \)\(12\!\cdots\!20\)\( p^{21} T^{21} + 1987583372515078046 p^{24} T^{22} - 1931917648550032 p^{27} T^{23} + 2783932032756 p^{30} T^{24} - 1984332868 p^{33} T^{25} + 2448012 p^{36} T^{26} - 984 p^{39} T^{27} + p^{42} T^{28} \) | |
| 67 | \( 1 + 1396 T + 3402988 T^{2} + 3888035772 T^{3} + 5484174477923 T^{4} + 5282839515523304 T^{5} + 5572013970556145416 T^{6} + \)\(46\!\cdots\!76\)\( T^{7} + \)\(40\!\cdots\!65\)\( T^{8} + \)\(29\!\cdots\!52\)\( T^{9} + \)\(21\!\cdots\!20\)\( T^{10} + \)\(14\!\cdots\!92\)\( T^{11} + \)\(91\!\cdots\!67\)\( T^{12} + \)\(53\!\cdots\!44\)\( T^{13} + \)\(30\!\cdots\!04\)\( T^{14} + \)\(53\!\cdots\!44\)\( p^{3} T^{15} + \)\(91\!\cdots\!67\)\( p^{6} T^{16} + \)\(14\!\cdots\!92\)\( p^{9} T^{17} + \)\(21\!\cdots\!20\)\( p^{12} T^{18} + \)\(29\!\cdots\!52\)\( p^{15} T^{19} + \)\(40\!\cdots\!65\)\( p^{18} T^{20} + \)\(46\!\cdots\!76\)\( p^{21} T^{21} + 5572013970556145416 p^{24} T^{22} + 5282839515523304 p^{27} T^{23} + 5484174477923 p^{30} T^{24} + 3888035772 p^{33} T^{25} + 3402988 p^{36} T^{26} + 1396 p^{39} T^{27} + p^{42} T^{28} \) | |
| 71 | \( 1 + 1216 T + 3409482 T^{2} + 3605266904 T^{3} + 5528118823041 T^{4} + 5091091002378368 T^{5} + 5648874463608575128 T^{6} + \)\(45\!\cdots\!52\)\( T^{7} + \)\(40\!\cdots\!47\)\( T^{8} + \)\(41\!\cdots\!00\)\( p T^{9} + \)\(22\!\cdots\!38\)\( T^{10} + \)\(14\!\cdots\!56\)\( T^{11} + \)\(10\!\cdots\!59\)\( T^{12} + \)\(60\!\cdots\!40\)\( T^{13} + \)\(39\!\cdots\!36\)\( T^{14} + \)\(60\!\cdots\!40\)\( p^{3} T^{15} + \)\(10\!\cdots\!59\)\( p^{6} T^{16} + \)\(14\!\cdots\!56\)\( p^{9} T^{17} + \)\(22\!\cdots\!38\)\( p^{12} T^{18} + \)\(41\!\cdots\!00\)\( p^{16} T^{19} + \)\(40\!\cdots\!47\)\( p^{18} T^{20} + \)\(45\!\cdots\!52\)\( p^{21} T^{21} + 5648874463608575128 p^{24} T^{22} + 5091091002378368 p^{27} T^{23} + 5528118823041 p^{30} T^{24} + 3605266904 p^{33} T^{25} + 3409482 p^{36} T^{26} + 1216 p^{39} T^{27} + p^{42} T^{28} \) | |
| 73 | \( 1 - 1768 T + 4077314 T^{2} - 5320540348 T^{3} + 7541003493673 T^{4} - 8140550594031056 T^{5} + 8947020469187567864 T^{6} - \)\(83\!\cdots\!56\)\( T^{7} + \)\(77\!\cdots\!84\)\( T^{8} - \)\(63\!\cdots\!92\)\( T^{9} + \)\(51\!\cdots\!28\)\( T^{10} - \)\(38\!\cdots\!32\)\( T^{11} + \)\(27\!\cdots\!86\)\( T^{12} - \)\(18\!\cdots\!36\)\( T^{13} + \)\(11\!\cdots\!56\)\( T^{14} - \)\(18\!\cdots\!36\)\( p^{3} T^{15} + \)\(27\!\cdots\!86\)\( p^{6} T^{16} - \)\(38\!\cdots\!32\)\( p^{9} T^{17} + \)\(51\!\cdots\!28\)\( p^{12} T^{18} - \)\(63\!\cdots\!92\)\( p^{15} T^{19} + \)\(77\!\cdots\!84\)\( p^{18} T^{20} - \)\(83\!\cdots\!56\)\( p^{21} T^{21} + 8947020469187567864 p^{24} T^{22} - 8140550594031056 p^{27} T^{23} + 7541003493673 p^{30} T^{24} - 5320540348 p^{33} T^{25} + 4077314 p^{36} T^{26} - 1768 p^{39} T^{27} + p^{42} T^{28} \) | |
| 79 | \( 1 + 1888 T + 5582500 T^{2} + 7622480076 T^{3} + 13227924814316 T^{4} + 14283656562480600 T^{5} + 18553799098249813426 T^{6} + \)\(16\!\cdots\!36\)\( T^{7} + \)\(17\!\cdots\!38\)\( T^{8} + \)\(13\!\cdots\!28\)\( T^{9} + \)\(12\!\cdots\!92\)\( T^{10} + \)\(84\!\cdots\!08\)\( T^{11} + \)\(73\!\cdots\!53\)\( T^{12} + \)\(45\!\cdots\!68\)\( T^{13} + \)\(37\!\cdots\!92\)\( T^{14} + \)\(45\!\cdots\!68\)\( p^{3} T^{15} + \)\(73\!\cdots\!53\)\( p^{6} T^{16} + \)\(84\!\cdots\!08\)\( p^{9} T^{17} + \)\(12\!\cdots\!92\)\( p^{12} T^{18} + \)\(13\!\cdots\!28\)\( p^{15} T^{19} + \)\(17\!\cdots\!38\)\( p^{18} T^{20} + \)\(16\!\cdots\!36\)\( p^{21} T^{21} + 18553799098249813426 p^{24} T^{22} + 14283656562480600 p^{27} T^{23} + 13227924814316 p^{30} T^{24} + 7622480076 p^{33} T^{25} + 5582500 p^{36} T^{26} + 1888 p^{39} T^{27} + p^{42} T^{28} \) | |
| 83 | \( 1 + 1008 T + 6056992 T^{2} + 4982292420 T^{3} + 17037449747270 T^{4} + 140733019132608 p T^{5} + 30104979489705076366 T^{6} + \)\(17\!\cdots\!80\)\( T^{7} + \)\(37\!\cdots\!22\)\( T^{8} + \)\(19\!\cdots\!44\)\( T^{9} + \)\(36\!\cdots\!88\)\( T^{10} + \)\(16\!\cdots\!44\)\( T^{11} + \)\(28\!\cdots\!47\)\( T^{12} + \)\(11\!\cdots\!56\)\( T^{13} + \)\(17\!\cdots\!16\)\( T^{14} + \)\(11\!\cdots\!56\)\( p^{3} T^{15} + \)\(28\!\cdots\!47\)\( p^{6} T^{16} + \)\(16\!\cdots\!44\)\( p^{9} T^{17} + \)\(36\!\cdots\!88\)\( p^{12} T^{18} + \)\(19\!\cdots\!44\)\( p^{15} T^{19} + \)\(37\!\cdots\!22\)\( p^{18} T^{20} + \)\(17\!\cdots\!80\)\( p^{21} T^{21} + 30104979489705076366 p^{24} T^{22} + 140733019132608 p^{28} T^{23} + 17037449747270 p^{30} T^{24} + 4982292420 p^{33} T^{25} + 6056992 p^{36} T^{26} + 1008 p^{39} T^{27} + p^{42} T^{28} \) | |
| 89 | \( 1 + 864 T + 6798614 T^{2} + 4294796676 T^{3} + 21508606198342 T^{4} + 9870470113033632 T^{5} + 486515007414972700 p T^{6} + \)\(14\!\cdots\!88\)\( T^{7} + \)\(63\!\cdots\!98\)\( T^{8} + \)\(14\!\cdots\!40\)\( T^{9} + \)\(72\!\cdots\!78\)\( T^{10} + \)\(11\!\cdots\!64\)\( T^{11} + \)\(67\!\cdots\!23\)\( T^{12} + \)\(81\!\cdots\!28\)\( T^{13} + \)\(51\!\cdots\!92\)\( T^{14} + \)\(81\!\cdots\!28\)\( p^{3} T^{15} + \)\(67\!\cdots\!23\)\( p^{6} T^{16} + \)\(11\!\cdots\!64\)\( p^{9} T^{17} + \)\(72\!\cdots\!78\)\( p^{12} T^{18} + \)\(14\!\cdots\!40\)\( p^{15} T^{19} + \)\(63\!\cdots\!98\)\( p^{18} T^{20} + \)\(14\!\cdots\!88\)\( p^{21} T^{21} + 486515007414972700 p^{25} T^{22} + 9870470113033632 p^{27} T^{23} + 21508606198342 p^{30} T^{24} + 4294796676 p^{33} T^{25} + 6798614 p^{36} T^{26} + 864 p^{39} T^{27} + p^{42} T^{28} \) | |
| 97 | \( 1 - 1368 T + 6805636 T^{2} - 7623857676 T^{3} + 21659216185900 T^{4} - 19978094594608624 T^{5} + 43707663529813303126 T^{6} - \)\(33\!\cdots\!08\)\( T^{7} + \)\(65\!\cdots\!62\)\( T^{8} - \)\(43\!\cdots\!20\)\( T^{9} + \)\(81\!\cdots\!84\)\( T^{10} - \)\(48\!\cdots\!56\)\( T^{11} + \)\(88\!\cdots\!69\)\( T^{12} - \)\(49\!\cdots\!64\)\( T^{13} + \)\(86\!\cdots\!08\)\( T^{14} - \)\(49\!\cdots\!64\)\( p^{3} T^{15} + \)\(88\!\cdots\!69\)\( p^{6} T^{16} - \)\(48\!\cdots\!56\)\( p^{9} T^{17} + \)\(81\!\cdots\!84\)\( p^{12} T^{18} - \)\(43\!\cdots\!20\)\( p^{15} T^{19} + \)\(65\!\cdots\!62\)\( p^{18} T^{20} - \)\(33\!\cdots\!08\)\( p^{21} T^{21} + 43707663529813303126 p^{24} T^{22} - 19978094594608624 p^{27} T^{23} + 21659216185900 p^{30} T^{24} - 7623857676 p^{33} T^{25} + 6805636 p^{36} T^{26} - 1368 p^{39} T^{27} + p^{42} T^{28} \) | |
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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
−2.57975394183153751720603640637, −2.51096223854669310137336112550, −2.41305314797380592370268819232, −2.21777411997027165831406531039, −2.13322057846563626260936707730, −2.03406403276042144629797895360, −2.01982395544726912580253611653, −1.97755601761455855304196848335, −1.91600800776634928997402357273, −1.82916750602556523318737098452, −1.80389499370133213003704329004, −1.70442548670268643104901088616, −1.67348620332356708744111514847, −1.32291309999629811128355803616, −1.25268563657859176778331990688, −1.22583477028210366413334370214, −1.19975748175706052301167904602, −1.12676541497763642336004733723, −1.10168558761786491765647908098, −1.01133478639080669019514682793, −0.961619689769373570645263239562, −0.954725813012965154610656187785, −0.887856806229641752943057361777, −0.77945684081857270507593957510, −0.69484736958534199942046293653, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.69484736958534199942046293653, 0.77945684081857270507593957510, 0.887856806229641752943057361777, 0.954725813012965154610656187785, 0.961619689769373570645263239562, 1.01133478639080669019514682793, 1.10168558761786491765647908098, 1.12676541497763642336004733723, 1.19975748175706052301167904602, 1.22583477028210366413334370214, 1.25268563657859176778331990688, 1.32291309999629811128355803616, 1.67348620332356708744111514847, 1.70442548670268643104901088616, 1.80389499370133213003704329004, 1.82916750602556523318737098452, 1.91600800776634928997402357273, 1.97755601761455855304196848335, 2.01982395544726912580253611653, 2.03406403276042144629797895360, 2.13322057846563626260936707730, 2.21777411997027165831406531039, 2.41305314797380592370268819232, 2.51096223854669310137336112550, 2.57975394183153751720603640637
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.