| L(s) = 1 | − 1.24·2-s − 0.445·4-s + 1.44·5-s − 7-s + 3.04·8-s − 1.80·10-s + 0.445·11-s − 2.44·13-s + 1.24·14-s − 2.91·16-s + 0.554·17-s + 1.29·19-s − 0.643·20-s − 0.554·22-s − 2.69·23-s − 2.91·25-s + 3.04·26-s + 0.445·28-s − 0.423·29-s − 5.60·31-s − 2.46·32-s − 0.692·34-s − 1.44·35-s + 3.74·37-s − 1.61·38-s + 4.40·40-s + 6.52·41-s + ⋯ |
| L(s) = 1 | − 0.881·2-s − 0.222·4-s + 0.646·5-s − 0.377·7-s + 1.07·8-s − 0.569·10-s + 0.134·11-s − 0.678·13-s + 0.333·14-s − 0.727·16-s + 0.134·17-s + 0.297·19-s − 0.143·20-s − 0.118·22-s − 0.561·23-s − 0.582·25-s + 0.597·26-s + 0.0841·28-s − 0.0785·29-s − 1.00·31-s − 0.436·32-s − 0.118·34-s − 0.244·35-s + 0.615·37-s − 0.262·38-s + 0.696·40-s + 1.01·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - T \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 5 | \( 1 - 1.44T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 0.445T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 - 0.554T + 17T^{2} \) |
| 19 | \( 1 - 1.29T + 19T^{2} \) |
| 23 | \( 1 + 2.69T + 23T^{2} \) |
| 29 | \( 1 + 0.423T + 29T^{2} \) |
| 31 | \( 1 + 5.60T + 31T^{2} \) |
| 37 | \( 1 - 3.74T + 37T^{2} \) |
| 41 | \( 1 - 6.52T + 41T^{2} \) |
| 43 | \( 1 - 3.24T + 43T^{2} \) |
| 47 | \( 1 + 1.57T + 47T^{2} \) |
| 53 | \( 1 + 0.149T + 53T^{2} \) |
| 59 | \( 1 + 1.08T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 0.527T + 67T^{2} \) |
| 71 | \( 1 - 5.76T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 1.82T + 79T^{2} \) |
| 83 | \( 1 - 8.81T + 83T^{2} \) |
| 89 | \( 1 - 0.408T + 89T^{2} \) |
| 97 | \( 1 + 2.12T + 97T^{2} \) |
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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
−8.958011006068867419311668435779, −7.87465045184272656608890260796, −7.45824890743226174420393549297, −6.39018760722924869712735064219, −5.59291399767965885186434123465, −4.68458618865351635189401075630, −3.74163540201153102960216447744, −2.45815976340180140474737061980, −1.42344710524852237044104350320, 0,
1.42344710524852237044104350320, 2.45815976340180140474737061980, 3.74163540201153102960216447744, 4.68458618865351635189401075630, 5.59291399767965885186434123465, 6.39018760722924869712735064219, 7.45824890743226174420393549297, 7.87465045184272656608890260796, 8.958011006068867419311668435779
