L(s) = 1 | − 2-s − 2.04·3-s + 4-s − 4.09·5-s + 2.04·6-s + 1.49·7-s − 8-s + 1.18·9-s + 4.09·10-s + 4.36·11-s − 2.04·12-s − 2.07·13-s − 1.49·14-s + 8.37·15-s + 16-s + 2.71·17-s − 1.18·18-s − 19-s − 4.09·20-s − 3.06·21-s − 4.36·22-s − 8.72·23-s + 2.04·24-s + 11.7·25-s + 2.07·26-s + 3.71·27-s + 1.49·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.18·3-s + 0.5·4-s − 1.83·5-s + 0.835·6-s + 0.566·7-s − 0.353·8-s + 0.395·9-s + 1.29·10-s + 1.31·11-s − 0.590·12-s − 0.576·13-s − 0.400·14-s + 2.16·15-s + 0.250·16-s + 0.658·17-s − 0.279·18-s − 0.229·19-s − 0.915·20-s − 0.668·21-s − 0.930·22-s − 1.81·23-s + 0.417·24-s + 2.35·25-s + 0.407·26-s + 0.714·27-s + 0.283·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 + 2.04T + 3T^{2} \) |
| 5 | \( 1 + 4.09T + 5T^{2} \) |
| 7 | \( 1 - 1.49T + 7T^{2} \) |
| 11 | \( 1 - 4.36T + 11T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 17 | \( 1 - 2.71T + 17T^{2} \) |
| 23 | \( 1 + 8.72T + 23T^{2} \) |
| 29 | \( 1 - 1.76T + 29T^{2} \) |
| 31 | \( 1 + 3.93T + 31T^{2} \) |
| 37 | \( 1 + 9.86T + 37T^{2} \) |
| 41 | \( 1 - 5.20T + 41T^{2} \) |
| 43 | \( 1 - 4.35T + 43T^{2} \) |
| 47 | \( 1 + 5.27T + 47T^{2} \) |
| 53 | \( 1 + 2.90T + 53T^{2} \) |
| 59 | \( 1 - 7.50T + 59T^{2} \) |
| 61 | \( 1 + 4.68T + 61T^{2} \) |
| 67 | \( 1 + 8.57T + 67T^{2} \) |
| 71 | \( 1 - 1.09T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 7.46T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 7.24T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−7.63067013637624980659618813646, −6.83035511134184003671617893336, −6.32781066246640520788511568340, −5.41127269962562439250556124571, −4.64467724394778478550893568773, −3.96728104202722070213390645597, −3.29598020292521397541170974860, −1.87963281992997566057202134114, −0.828515127013870510752212403165, 0,
0.828515127013870510752212403165, 1.87963281992997566057202134114, 3.29598020292521397541170974860, 3.96728104202722070213390645597, 4.64467724394778478550893568773, 5.41127269962562439250556124571, 6.32781066246640520788511568340, 6.83035511134184003671617893336, 7.63067013637624980659618813646