| L(s) = 1 | + 0.231·2-s + 8.68·3-s − 7.94·4-s − 0.297·5-s + 2.01·6-s − 7.88·7-s − 3.69·8-s + 48.4·9-s − 0.0690·10-s − 11·11-s − 69.0·12-s − 1.82·14-s − 2.58·15-s + 62.7·16-s − 15.3·17-s + 11.2·18-s − 120.·19-s + 2.36·20-s − 68.4·21-s − 2.55·22-s + 156.·23-s − 32.1·24-s − 124.·25-s + 186.·27-s + 62.6·28-s + 217.·29-s − 0.599·30-s + ⋯ |
| L(s) = 1 | + 0.0819·2-s + 1.67·3-s − 0.993·4-s − 0.0266·5-s + 0.137·6-s − 0.425·7-s − 0.163·8-s + 1.79·9-s − 0.00218·10-s − 0.301·11-s − 1.66·12-s − 0.0349·14-s − 0.0445·15-s + 0.979·16-s − 0.219·17-s + 0.147·18-s − 1.45·19-s + 0.0264·20-s − 0.711·21-s − 0.0247·22-s + 1.41·23-s − 0.273·24-s − 0.999·25-s + 1.33·27-s + 0.422·28-s + 1.39·29-s − 0.00364·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.231T + 8T^{2} \) |
| 3 | \( 1 - 8.68T + 27T^{2} \) |
| 5 | \( 1 + 0.297T + 125T^{2} \) |
| 7 | \( 1 + 7.88T + 343T^{2} \) |
| 17 | \( 1 + 15.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 120.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 156.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 217.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 209.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 182.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 286.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 419.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 196.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 175.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 185.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 140.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 484.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 833.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 147.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 737.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 422.T + 9.12e5T^{2} \) |
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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
−8.355751413449869094815796636329, −8.234270799205112007115804222399, −7.05863634901222320135973396015, −6.20980723854287796996810894946, −4.83524715878985797704519536412, −4.27903677446040746004482084535, −3.29016176690569524472338542910, −2.70735796087711096205712051995, −1.46007588523744801658049136796, 0,
1.46007588523744801658049136796, 2.70735796087711096205712051995, 3.29016176690569524472338542910, 4.27903677446040746004482084535, 4.83524715878985797704519536412, 6.20980723854287796996810894946, 7.05863634901222320135973396015, 8.234270799205112007115804222399, 8.355751413449869094815796636329
