| L(s) = 1 | + 1.05e10·2-s − 1.85e15·3-s + 7.41e19·4-s − 4.74e22·5-s − 1.95e25·6-s + 5.36e26·7-s + 3.92e29·8-s + 3.43e30·9-s − 5.00e32·10-s − 5.04e33·11-s − 1.37e35·12-s + 3.00e36·13-s + 5.65e36·14-s + 8.80e37·15-s + 1.39e39·16-s − 5.13e39·17-s + 3.61e40·18-s − 6.06e40·19-s − 3.52e42·20-s − 9.94e41·21-s − 5.31e43·22-s − 3.21e44·23-s − 7.27e44·24-s − 4.54e44·25-s + 3.17e46·26-s − 6.36e45·27-s + 3.97e46·28-s + ⋯ |
| L(s) = 1 | + 1.73·2-s − 0.577·3-s + 2.00·4-s − 0.912·5-s − 1.00·6-s + 0.183·7-s + 1.75·8-s + 0.333·9-s − 1.58·10-s − 0.720·11-s − 1.16·12-s + 1.88·13-s + 0.318·14-s + 0.526·15-s + 1.02·16-s − 0.525·17-s + 0.578·18-s − 0.167·19-s − 1.83·20-s − 0.106·21-s − 1.24·22-s − 1.78·23-s − 1.01·24-s − 0.167·25-s + 3.26·26-s − 0.192·27-s + 0.368·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(66-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+65/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(33)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{67}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.85e15T \) |
| good | 2 | \( 1 - 1.05e10T + 3.68e19T^{2} \) |
| 5 | \( 1 + 4.74e22T + 2.71e45T^{2} \) |
| 7 | \( 1 - 5.36e26T + 8.53e54T^{2} \) |
| 11 | \( 1 + 5.04e33T + 4.90e67T^{2} \) |
| 13 | \( 1 - 3.00e36T + 2.54e72T^{2} \) |
| 17 | \( 1 + 5.13e39T + 9.53e79T^{2} \) |
| 19 | \( 1 + 6.06e40T + 1.31e83T^{2} \) |
| 23 | \( 1 + 3.21e44T + 3.25e88T^{2} \) |
| 29 | \( 1 - 1.91e47T + 1.13e95T^{2} \) |
| 31 | \( 1 + 3.95e48T + 8.67e96T^{2} \) |
| 37 | \( 1 + 3.75e50T + 8.57e101T^{2} \) |
| 41 | \( 1 + 5.03e52T + 6.77e104T^{2} \) |
| 43 | \( 1 - 2.04e53T + 1.49e106T^{2} \) |
| 47 | \( 1 + 1.77e54T + 4.85e108T^{2} \) |
| 53 | \( 1 + 4.07e55T + 1.19e112T^{2} \) |
| 59 | \( 1 + 1.40e57T + 1.27e115T^{2} \) |
| 61 | \( 1 + 2.35e57T + 1.11e116T^{2} \) |
| 67 | \( 1 - 3.07e59T + 4.95e118T^{2} \) |
| 71 | \( 1 + 2.32e59T + 2.14e120T^{2} \) |
| 73 | \( 1 - 5.07e60T + 1.30e121T^{2} \) |
| 79 | \( 1 + 2.67e61T + 2.21e123T^{2} \) |
| 83 | \( 1 + 3.16e61T + 5.49e124T^{2} \) |
| 89 | \( 1 + 3.47e63T + 5.13e126T^{2} \) |
| 97 | \( 1 - 9.55e63T + 1.38e129T^{2} \) |
| 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
−12.80063207968675753237278409612, −11.63464354614113082942996696384, −10.80925823166306139888813235390, −8.084383175908926741312620176157, −6.55396388932621647879139635009, −5.52533743881252544852735623552, −4.24778429472003558914529476965, −3.48159308895923988403890001035, −1.81319116593979706001177169350, 0,
1.81319116593979706001177169350, 3.48159308895923988403890001035, 4.24778429472003558914529476965, 5.52533743881252544852735623552, 6.55396388932621647879139635009, 8.084383175908926741312620176157, 10.80925823166306139888813235390, 11.63464354614113082942996696384, 12.80063207968675753237278409612
