| L(s) = 1 | + 2-s + 0.836·3-s + 4-s − 3.34·5-s + 0.836·6-s − 0.767·7-s + 8-s − 2.30·9-s − 3.34·10-s − 6.48·11-s + 0.836·12-s + 1.07·13-s − 0.767·14-s − 2.79·15-s + 16-s − 0.0698·17-s − 2.30·18-s − 19-s − 3.34·20-s − 0.641·21-s − 6.48·22-s + 6.21·23-s + 0.836·24-s + 6.18·25-s + 1.07·26-s − 4.43·27-s − 0.767·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.482·3-s + 0.5·4-s − 1.49·5-s + 0.341·6-s − 0.290·7-s + 0.353·8-s − 0.766·9-s − 1.05·10-s − 1.95·11-s + 0.241·12-s + 0.297·13-s − 0.205·14-s − 0.722·15-s + 0.250·16-s − 0.0169·17-s − 0.542·18-s − 0.229·19-s − 0.747·20-s − 0.140·21-s − 1.38·22-s + 1.29·23-s + 0.170·24-s + 1.23·25-s + 0.210·26-s − 0.853·27-s − 0.145·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.409429775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.409429775\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 0.836T + 3T^{2} \) |
| 5 | \( 1 + 3.34T + 5T^{2} \) |
| 7 | \( 1 + 0.767T + 7T^{2} \) |
| 11 | \( 1 + 6.48T + 11T^{2} \) |
| 13 | \( 1 - 1.07T + 13T^{2} \) |
| 17 | \( 1 + 0.0698T + 17T^{2} \) |
| 23 | \( 1 - 6.21T + 23T^{2} \) |
| 29 | \( 1 + 5.02T + 29T^{2} \) |
| 31 | \( 1 + 4.55T + 31T^{2} \) |
| 37 | \( 1 - 9.11T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 9.28T + 43T^{2} \) |
| 47 | \( 1 + 3.90T + 47T^{2} \) |
| 53 | \( 1 - 1.97T + 53T^{2} \) |
| 59 | \( 1 + 5.39T + 59T^{2} \) |
| 61 | \( 1 - 9.57T + 61T^{2} \) |
| 67 | \( 1 - 8.23T + 67T^{2} \) |
| 71 | \( 1 - 6.56T + 71T^{2} \) |
| 73 | \( 1 - 1.31T + 73T^{2} \) |
| 79 | \( 1 - 9.27T + 79T^{2} \) |
| 83 | \( 1 + 7.18T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 9.54T + 97T^{2} \) |
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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
−7.85359794844351966057500934445, −7.30586656402759400543166659184, −6.48988863142033206237082292930, −5.47884043340788939555738728949, −5.08514686803978713835771333083, −4.16303089354402680246124436196, −3.40910588745114651051002197033, −2.97170953974756755634805812397, −2.18029572193767702689849035549, −0.47855091485073462279581707985,
0.47855091485073462279581707985, 2.18029572193767702689849035549, 2.97170953974756755634805812397, 3.40910588745114651051002197033, 4.16303089354402680246124436196, 5.08514686803978713835771333083, 5.47884043340788939555738728949, 6.48988863142033206237082292930, 7.30586656402759400543166659184, 7.85359794844351966057500934445
