| L(s) = 1 | − 1.44e6·2-s + 1.16e9·3-s + 1.53e12·4-s + 2.64e13·5-s − 1.67e15·6-s − 2.07e16·7-s − 1.42e18·8-s + 1.35e18·9-s − 3.81e19·10-s + 1.84e20·11-s + 1.78e21·12-s − 9.85e21·13-s + 2.99e22·14-s + 3.07e22·15-s + 1.21e24·16-s + 8.06e23·17-s − 1.95e24·18-s − 2.72e24·19-s + 4.05e25·20-s − 2.41e25·21-s − 2.66e26·22-s − 1.86e26·23-s − 1.65e27·24-s − 1.12e27·25-s + 1.42e28·26-s + 1.57e27·27-s − 3.18e28·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 0.577·3-s + 2.79·4-s + 0.619·5-s − 1.12·6-s − 0.688·7-s − 3.49·8-s + 0.333·9-s − 1.20·10-s + 0.909·11-s + 1.61·12-s − 1.86·13-s + 1.34·14-s + 0.357·15-s + 4.01·16-s + 0.817·17-s − 0.649·18-s − 0.315·19-s + 1.73·20-s − 0.397·21-s − 1.77·22-s − 0.520·23-s − 2.01·24-s − 0.616·25-s + 3.64·26-s + 0.192·27-s − 1.92·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+39/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(20)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{41}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 1.16e9T \) |
| good | 2 | \( 1 + 1.44e6T + 5.49e11T^{2} \) |
| 5 | \( 1 - 2.64e13T + 1.81e27T^{2} \) |
| 7 | \( 1 + 2.07e16T + 9.09e32T^{2} \) |
| 11 | \( 1 - 1.84e20T + 4.11e40T^{2} \) |
| 13 | \( 1 + 9.85e21T + 2.77e43T^{2} \) |
| 17 | \( 1 - 8.06e23T + 9.71e47T^{2} \) |
| 19 | \( 1 + 2.72e24T + 7.43e49T^{2} \) |
| 23 | \( 1 + 1.86e26T + 1.28e53T^{2} \) |
| 29 | \( 1 - 6.77e27T + 1.08e57T^{2} \) |
| 31 | \( 1 - 6.32e28T + 1.45e58T^{2} \) |
| 37 | \( 1 - 2.30e30T + 1.44e61T^{2} \) |
| 41 | \( 1 - 2.74e31T + 7.91e62T^{2} \) |
| 43 | \( 1 + 9.25e31T + 5.07e63T^{2} \) |
| 47 | \( 1 + 4.42e32T + 1.62e65T^{2} \) |
| 53 | \( 1 + 2.11e33T + 1.76e67T^{2} \) |
| 59 | \( 1 + 2.60e34T + 1.15e69T^{2} \) |
| 61 | \( 1 + 7.67e34T + 4.24e69T^{2} \) |
| 67 | \( 1 + 9.27e34T + 1.64e71T^{2} \) |
| 71 | \( 1 + 9.43e35T + 1.58e72T^{2} \) |
| 73 | \( 1 + 2.21e36T + 4.67e72T^{2} \) |
| 79 | \( 1 - 6.79e36T + 1.01e74T^{2} \) |
| 83 | \( 1 - 1.24e37T + 6.98e74T^{2} \) |
| 89 | \( 1 - 2.84e37T + 1.06e76T^{2} \) |
| 97 | \( 1 + 1.24e38T + 3.04e77T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.49875119096686892597405639840, −14.76834909222499174948495041678, −12.09886077454862042265790469879, −9.990547159373372136625982680966, −9.413073973529421304969568179580, −7.76704874659622004663545638978, −6.41647931633546772514096122488, −2.82512907640284595445112548168, −1.60769371311266611042485559591, 0,
1.60769371311266611042485559591, 2.82512907640284595445112548168, 6.41647931633546772514096122488, 7.76704874659622004663545638978, 9.413073973529421304969568179580, 9.990547159373372136625982680966, 12.09886077454862042265790469879, 14.76834909222499174948495041678, 16.49875119096686892597405639840
