| L(s) = 1 | + 5.13·2-s − 6.39·3-s + 18.4·4-s + 3.19·5-s − 32.8·6-s − 11.5·7-s + 53.5·8-s + 13.9·9-s + 16.4·10-s − 11·11-s − 117.·12-s − 59.2·14-s − 20.4·15-s + 127.·16-s + 51.2·17-s + 71.7·18-s − 18.2·19-s + 58.8·20-s + 73.7·21-s − 56.5·22-s − 3.69·23-s − 342.·24-s − 114.·25-s + 83.4·27-s − 212.·28-s + 144.·29-s − 105.·30-s + ⋯ |
| L(s) = 1 | + 1.81·2-s − 1.23·3-s + 2.30·4-s + 0.285·5-s − 2.23·6-s − 0.621·7-s + 2.36·8-s + 0.516·9-s + 0.519·10-s − 0.301·11-s − 2.83·12-s − 1.13·14-s − 0.352·15-s + 1.99·16-s + 0.731·17-s + 0.939·18-s − 0.220·19-s + 0.658·20-s + 0.766·21-s − 0.547·22-s − 0.0335·23-s − 2.91·24-s − 0.918·25-s + 0.594·27-s − 1.43·28-s + 0.924·29-s − 0.640·30-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 | \( 1 - 5.13T + 8T^{2} \) |
| 3 | \( 1 + 6.39T + 27T^{2} \) |
| 5 | \( 1 - 3.19T + 125T^{2} \) |
| 7 | \( 1 + 11.5T + 343T^{2} \) |
| 17 | \( 1 - 51.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 18.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 3.69T + 1.21e4T^{2} \) |
| 29 | \( 1 - 144.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 65.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 411.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 212.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 430.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 103.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 164.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 661.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 753.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 439.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 52.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 0.827T + 3.89e5T^{2} \) |
| 79 | \( 1 - 465.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 47.9T + 5.71e5T^{2} \) |
| 89 | \( 1 + 781.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 714.T + 9.12e5T^{2} \) |
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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
−8.204551734979121655437170065686, −7.13294016040065072172268928912, −6.41060963073551241451980248250, −5.95943383472185938352364749101, −5.23247138512862878886009573264, −4.64509521648193161768118612918, −3.57153072452360666029907137919, −2.79379580879396164767976379152, −1.54515813317455910215179467511, 0,
1.54515813317455910215179467511, 2.79379580879396164767976379152, 3.57153072452360666029907137919, 4.64509521648193161768118612918, 5.23247138512862878886009573264, 5.95943383472185938352364749101, 6.41060963073551241451980248250, 7.13294016040065072172268928912, 8.204551734979121655437170065686
