| L(s) = 1 | + 2.97·3-s − 4.20·5-s + 0.274·7-s + 5.83·9-s − 2.12·11-s − 6.94·13-s − 12.4·15-s − 17-s + 3.01·19-s + 0.816·21-s + 6.46·23-s + 12.6·25-s + 8.41·27-s + 3.91·29-s + 4.63·31-s − 6.32·33-s − 1.15·35-s + 1.57·37-s − 20.6·39-s + 10.7·41-s + 5.33·43-s − 24.4·45-s + 5.38·47-s − 6.92·49-s − 2.97·51-s − 6.66·53-s + 8.94·55-s + ⋯ |
| L(s) = 1 | + 1.71·3-s − 1.87·5-s + 0.103·7-s + 1.94·9-s − 0.642·11-s − 1.92·13-s − 3.22·15-s − 0.242·17-s + 0.690·19-s + 0.178·21-s + 1.34·23-s + 2.52·25-s + 1.61·27-s + 0.726·29-s + 0.831·31-s − 1.10·33-s − 0.195·35-s + 0.259·37-s − 3.30·39-s + 1.68·41-s + 0.813·43-s − 3.65·45-s + 0.785·47-s − 0.989·49-s − 0.416·51-s − 0.915·53-s + 1.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.245213846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.245213846\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 2.97T + 3T^{2} \) |
| 5 | \( 1 + 4.20T + 5T^{2} \) |
| 7 | \( 1 - 0.274T + 7T^{2} \) |
| 11 | \( 1 + 2.12T + 11T^{2} \) |
| 13 | \( 1 + 6.94T + 13T^{2} \) |
| 19 | \( 1 - 3.01T + 19T^{2} \) |
| 23 | \( 1 - 6.46T + 23T^{2} \) |
| 29 | \( 1 - 3.91T + 29T^{2} \) |
| 31 | \( 1 - 4.63T + 31T^{2} \) |
| 37 | \( 1 - 1.57T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 5.33T + 43T^{2} \) |
| 47 | \( 1 - 5.38T + 47T^{2} \) |
| 53 | \( 1 + 6.66T + 53T^{2} \) |
| 61 | \( 1 + 6.84T + 61T^{2} \) |
| 67 | \( 1 + 4.54T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 3.83T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 - 3.68T + 97T^{2} \) |
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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
−8.193215516345731574336480572029, −7.71668670141739370076120082101, −7.49151407084107083143359799079, −6.71248583911623576773442993684, −4.90452793874504439414035513810, −4.62125184690246832069736049671, −3.67202747932276992855860936292, −2.89497831980569127018135648222, −2.47959099265333261063427425571, −0.77402734504339462006245372678,
0.77402734504339462006245372678, 2.47959099265333261063427425571, 2.89497831980569127018135648222, 3.67202747932276992855860936292, 4.62125184690246832069736049671, 4.90452793874504439414035513810, 6.71248583911623576773442993684, 7.49151407084107083143359799079, 7.71668670141739370076120082101, 8.193215516345731574336480572029
