| L(s) = 1 | − 48·2-s − 1.95e5·3-s − 3.35e7·4-s − 7.41e8·5-s + 9.39e6·6-s + 3.90e10·7-s + 3.22e9·8-s − 8.08e11·9-s + 3.56e10·10-s + 8.41e12·11-s + 6.56e12·12-s − 8.16e13·13-s − 1.87e12·14-s + 1.45e14·15-s + 1.12e15·16-s − 2.51e15·17-s + 3.88e13·18-s − 6.08e15·19-s + 2.48e16·20-s − 7.65e15·21-s − 4.04e14·22-s − 9.49e16·23-s − 6.30e14·24-s + 2.52e17·25-s + 3.91e15·26-s + 3.24e17·27-s − 1.31e18·28-s + ⋯ |
| L(s) = 1 | − 0.00828·2-s − 0.212·3-s − 0.999·4-s − 1.35·5-s + 0.00176·6-s + 1.06·7-s + 0.0165·8-s − 0.954·9-s + 0.0112·10-s + 0.808·11-s + 0.212·12-s − 0.972·13-s − 0.00884·14-s + 0.289·15-s + 0.999·16-s − 1.04·17-s + 0.00791·18-s − 0.630·19-s + 1.35·20-s − 0.227·21-s − 0.00670·22-s − 0.903·23-s − 0.00352·24-s + 0.847·25-s + 0.00805·26-s + 0.415·27-s − 1.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(26-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+25/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(13)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 3 p^{4} T + p^{25} T^{2} \) |
| 3 | \( 1 + 7252 p^{3} T + p^{25} T^{2} \) |
| 5 | \( 1 + 29679594 p^{2} T + p^{25} T^{2} \) |
| 7 | \( 1 - 797563208 p^{2} T + p^{25} T^{2} \) |
| 11 | \( 1 - 765410481732 p T + p^{25} T^{2} \) |
| 13 | \( 1 + 6280849641178 p T + p^{25} T^{2} \) |
| 17 | \( 1 + 148229413467534 p T + p^{25} T^{2} \) |
| 19 | \( 1 + 320108230016260 p T + p^{25} T^{2} \) |
| 23 | \( 1 + 4130229578100888 p T + p^{25} T^{2} \) |
| 29 | \( 1 + 271246959476737410 T + p^{25} T^{2} \) |
| 31 | \( 1 - 4291666067521509152 T + p^{25} T^{2} \) |
| 37 | \( 1 - 20301484446109126982 T + p^{25} T^{2} \) |
| 41 | \( 1 + \)\(18\!\cdots\!98\)\( T + p^{25} T^{2} \) |
| 43 | \( 1 - \)\(30\!\cdots\!56\)\( T + p^{25} T^{2} \) |
| 47 | \( 1 + \)\(92\!\cdots\!88\)\( T + p^{25} T^{2} \) |
| 53 | \( 1 + \)\(99\!\cdots\!54\)\( T + p^{25} T^{2} \) |
| 59 | \( 1 - \)\(13\!\cdots\!80\)\( T + p^{25} T^{2} \) |
| 61 | \( 1 - \)\(90\!\cdots\!02\)\( T + p^{25} T^{2} \) |
| 67 | \( 1 + \)\(26\!\cdots\!28\)\( T + p^{25} T^{2} \) |
| 71 | \( 1 + \)\(19\!\cdots\!48\)\( T + p^{25} T^{2} \) |
| 73 | \( 1 - \)\(42\!\cdots\!26\)\( T + p^{25} T^{2} \) |
| 79 | \( 1 + \)\(27\!\cdots\!60\)\( T + p^{25} T^{2} \) |
| 83 | \( 1 + \)\(93\!\cdots\!84\)\( T + p^{25} T^{2} \) |
| 89 | \( 1 + \)\(17\!\cdots\!30\)\( T + p^{25} T^{2} \) |
| 97 | \( 1 - \)\(28\!\cdots\!62\)\( T + p^{25} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−26.88537206297050000566551198613, −23.93312251573463892822942793783, −22.42268203005764508318554782850, −19.62814349222831481270313941415, −17.43082052812684484241584138314, −14.61160856189401032583901840016, −11.70089941202410050445650230556, −8.332583170673278606893499008029, −4.43532131875266390947498555989, 0,
4.43532131875266390947498555989, 8.332583170673278606893499008029, 11.70089941202410050445650230556, 14.61160856189401032583901840016, 17.43082052812684484241584138314, 19.62814349222831481270313941415, 22.42268203005764508318554782850, 23.93312251573463892822942793783, 26.88537206297050000566551198613
