| L(s) = 1 | + 1.83·2-s + 1.36·4-s − 0.363·5-s − 2.69·7-s − 1.16·8-s − 0.666·10-s + 1.67·11-s + 1.75·13-s − 4.94·14-s − 4.86·16-s − 2.30·17-s + 1.71·19-s − 0.495·20-s + 3.06·22-s − 6.12·23-s − 4.86·25-s + 3.22·26-s − 3.67·28-s − 7.88·29-s + 5.12·31-s − 6.59·32-s − 4.22·34-s + 0.980·35-s − 2.89·37-s + 3.14·38-s + 0.424·40-s − 1.16·41-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.681·4-s − 0.162·5-s − 1.01·7-s − 0.412·8-s − 0.210·10-s + 0.503·11-s + 0.487·13-s − 1.32·14-s − 1.21·16-s − 0.558·17-s + 0.394·19-s − 0.110·20-s + 0.653·22-s − 1.27·23-s − 0.973·25-s + 0.631·26-s − 0.695·28-s − 1.46·29-s + 0.919·31-s − 1.16·32-s − 0.724·34-s + 0.165·35-s − 0.476·37-s + 0.510·38-s + 0.0670·40-s − 0.181·41-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 | \( 1 - 1.83T + 2T^{2} \) |
| 5 | \( 1 + 0.363T + 5T^{2} \) |
| 7 | \( 1 + 2.69T + 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 - 1.75T + 13T^{2} \) |
| 17 | \( 1 + 2.30T + 17T^{2} \) |
| 19 | \( 1 - 1.71T + 19T^{2} \) |
| 23 | \( 1 + 6.12T + 23T^{2} \) |
| 29 | \( 1 + 7.88T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 + 2.89T + 37T^{2} \) |
| 41 | \( 1 + 1.16T + 41T^{2} \) |
| 43 | \( 1 + 2.33T + 43T^{2} \) |
| 47 | \( 1 + 6.40T + 47T^{2} \) |
| 53 | \( 1 - 5.06T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 1.91T + 61T^{2} \) |
| 67 | \( 1 - 8.88T + 67T^{2} \) |
| 71 | \( 1 + 6.34T + 71T^{2} \) |
| 73 | \( 1 + 1.95T + 73T^{2} \) |
| 79 | \( 1 - 3.03T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 2.92T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−8.771016611667020460578998031512, −7.77523668965717435460846856515, −6.74075904110718769371899212498, −6.19009591926792557766391231560, −5.56148182126247324974508954858, −4.47423309764397449586305248118, −3.76492330178559977145251397365, −3.18619948965999239473806934290, −1.96488847615216388004574996815, 0,
1.96488847615216388004574996815, 3.18619948965999239473806934290, 3.76492330178559977145251397365, 4.47423309764397449586305248118, 5.56148182126247324974508954858, 6.19009591926792557766391231560, 6.74075904110718769371899212498, 7.77523668965717435460846856515, 8.771016611667020460578998031512
