| L(s) = 1 | + 59.1·2-s + 1.55e4·3-s − 1.27e5·4-s + 1.59e6·5-s + 9.18e5·6-s − 2.66e7·7-s − 1.53e7·8-s + 1.11e8·9-s + 9.41e7·10-s − 1.84e7·11-s − 1.98e9·12-s + 2.82e9·13-s − 1.57e9·14-s + 2.46e10·15-s + 1.58e10·16-s − 1.97e10·17-s + 6.61e9·18-s + 1.30e11·19-s − 2.02e11·20-s − 4.13e11·21-s − 1.08e9·22-s + 5.53e11·23-s − 2.37e11·24-s + 1.76e12·25-s + 1.67e11·26-s − 2.68e11·27-s + 3.39e12·28-s + ⋯ |
| L(s) = 1 | + 0.163·2-s + 1.36·3-s − 0.973·4-s + 1.82·5-s + 0.223·6-s − 1.74·7-s − 0.322·8-s + 0.866·9-s + 0.297·10-s − 0.0258·11-s − 1.32·12-s + 0.959·13-s − 0.285·14-s + 2.48·15-s + 0.920·16-s − 0.685·17-s + 0.141·18-s + 1.76·19-s − 1.77·20-s − 2.38·21-s − 0.00423·22-s + 1.47·23-s − 0.440·24-s + 2.31·25-s + 0.156·26-s − 0.182·27-s + 1.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(3.736629042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.736629042\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 5.00e11T \) |
| good | 2 | \( 1 - 59.1T + 1.31e5T^{2} \) |
| 3 | \( 1 - 1.55e4T + 1.29e8T^{2} \) |
| 5 | \( 1 - 1.59e6T + 7.62e11T^{2} \) |
| 7 | \( 1 + 2.66e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 1.84e7T + 5.05e17T^{2} \) |
| 13 | \( 1 - 2.82e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 1.97e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.30e11T + 5.48e21T^{2} \) |
| 23 | \( 1 - 5.53e11T + 1.41e23T^{2} \) |
| 31 | \( 1 - 3.77e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 2.05e12T + 4.56e26T^{2} \) |
| 41 | \( 1 + 8.70e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.89e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 3.80e13T + 2.66e28T^{2} \) |
| 53 | \( 1 - 6.28e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 6.10e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 2.98e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 4.16e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 2.48e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.08e16T + 4.74e31T^{2} \) |
| 79 | \( 1 - 7.02e15T + 1.81e32T^{2} \) |
| 83 | \( 1 + 1.19e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 1.76e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 8.49e16T + 5.95e33T^{2} \) |
| 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
−13.58404424662073576251295286226, −12.90741240339995162523174554970, −10.06216700498585637666328491410, −9.383816137030682557025324681046, −8.770880847183868599616064015388, −6.66172565843920731566042750308, −5.41964586873290993100240888782, −3.46583617277555522565013365429, −2.70392612883029151063544051933, −1.04563228815666757981741481138,
1.04563228815666757981741481138, 2.70392612883029151063544051933, 3.46583617277555522565013365429, 5.41964586873290993100240888782, 6.66172565843920731566042750308, 8.770880847183868599616064015388, 9.383816137030682557025324681046, 10.06216700498585637666328491410, 12.90741240339995162523174554970, 13.58404424662073576251295286226
