| L(s) = 1 | + 3.34·2-s − 6.73·3-s + 3.19·4-s + 0.513·5-s − 22.5·6-s − 16.0·8-s + 18.4·9-s + 1.71·10-s − 26.8·11-s − 21.5·12-s − 25.6·13-s − 3.46·15-s − 79.3·16-s + 14.6·17-s + 61.6·18-s − 47.0·19-s + 1.64·20-s − 89.9·22-s − 165.·23-s + 108.·24-s − 124.·25-s − 85.9·26-s + 57.7·27-s + 265.·29-s − 11.5·30-s − 297.·31-s − 136.·32-s + ⋯ |
| L(s) = 1 | + 1.18·2-s − 1.29·3-s + 0.399·4-s + 0.0459·5-s − 1.53·6-s − 0.710·8-s + 0.682·9-s + 0.0543·10-s − 0.736·11-s − 0.517·12-s − 0.548·13-s − 0.0596·15-s − 1.23·16-s + 0.209·17-s + 0.807·18-s − 0.568·19-s + 0.0183·20-s − 0.871·22-s − 1.49·23-s + 0.921·24-s − 0.997·25-s − 0.648·26-s + 0.411·27-s + 1.70·29-s − 0.0705·30-s − 1.72·31-s − 0.755·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6010187420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6010187420\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 - 3.34T + 8T^{2} \) |
| 3 | \( 1 + 6.73T + 27T^{2} \) |
| 5 | \( 1 - 0.513T + 125T^{2} \) |
| 11 | \( 1 + 26.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 25.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 14.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 47.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 165.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 265.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 297.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 305.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 233.T + 7.95e4T^{2} \) |
| 53 | \( 1 + 628.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 271.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 321.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 242.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 536.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 804.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.21e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.42e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 872.T + 9.12e5T^{2} \) |
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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
−8.570227697657623953070986052996, −7.68144638084617513959485886429, −6.67780036179520252774222305157, −5.99713095139015069144088605914, −5.49599564487896498684819971931, −4.78201661049192531245868415486, −4.11567065813787987291108230408, −3.04320382908771806164317250630, −1.96593388201686416398544690605, −0.29177529121723774560445340787,
0.29177529121723774560445340787, 1.96593388201686416398544690605, 3.04320382908771806164317250630, 4.11567065813787987291108230408, 4.78201661049192531245868415486, 5.49599564487896498684819971931, 5.99713095139015069144088605914, 6.67780036179520252774222305157, 7.68144638084617513959485886429, 8.570227697657623953070986052996
