L(s) = 1 | + 2.28·2-s − 0.471·3-s + 3.23·4-s − 5-s − 1.07·6-s + 0.586·7-s + 2.81·8-s − 2.77·9-s − 2.28·10-s + 11-s − 1.52·12-s − 5.54·13-s + 1.34·14-s + 0.471·15-s − 0.0236·16-s + 3.19·17-s − 6.35·18-s + 3.32·19-s − 3.23·20-s − 0.276·21-s + 2.28·22-s − 7.54·23-s − 1.32·24-s + 25-s − 12.6·26-s + 2.72·27-s + 1.89·28-s + ⋯ |
L(s) = 1 | + 1.61·2-s − 0.272·3-s + 1.61·4-s − 0.447·5-s − 0.440·6-s + 0.221·7-s + 0.995·8-s − 0.925·9-s − 0.723·10-s + 0.301·11-s − 0.439·12-s − 1.53·13-s + 0.358·14-s + 0.121·15-s − 0.00590·16-s + 0.773·17-s − 1.49·18-s + 0.761·19-s − 0.722·20-s − 0.0604·21-s + 0.487·22-s − 1.57·23-s − 0.271·24-s + 0.200·25-s − 2.48·26-s + 0.524·27-s + 0.358·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 2 | \( 1 - 2.28T + 2T^{2} \) |
| 3 | \( 1 + 0.471T + 3T^{2} \) |
| 7 | \( 1 - 0.586T + 7T^{2} \) |
| 13 | \( 1 + 5.54T + 13T^{2} \) |
| 17 | \( 1 - 3.19T + 17T^{2} \) |
| 19 | \( 1 - 3.32T + 19T^{2} \) |
| 23 | \( 1 + 7.54T + 23T^{2} \) |
| 29 | \( 1 - 10.7T + 29T^{2} \) |
| 31 | \( 1 - 0.332T + 31T^{2} \) |
| 37 | \( 1 + 4.54T + 37T^{2} \) |
| 41 | \( 1 + 8.04T + 41T^{2} \) |
| 43 | \( 1 + 8.68T + 43T^{2} \) |
| 47 | \( 1 + 4.96T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 - 9.53T + 59T^{2} \) |
| 61 | \( 1 + 6.99T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 - 8.63T + 71T^{2} \) |
| 79 | \( 1 + 9.24T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 7.44T + 89T^{2} \) |
| 97 | \( 1 - 6.57T + 97T^{2} \) |
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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
−7.945266219638310123292314732263, −7.06071229296444259353594255830, −6.39990207521645461637767124975, −5.65306462951231642849198069246, −4.90774608088815663798220938899, −4.56464569176210303329178990813, −3.33055134608603305103194242798, −2.98643882140629983289184706897, −1.81012246758245517864048738028, 0,
1.81012246758245517864048738028, 2.98643882140629983289184706897, 3.33055134608603305103194242798, 4.56464569176210303329178990813, 4.90774608088815663798220938899, 5.65306462951231642849198069246, 6.39990207521645461637767124975, 7.06071229296444259353594255830, 7.945266219638310123292314732263