| L(s) = 1 | − 0.697·2-s − 1.51·4-s + 0.682·5-s + 7-s + 2.45·8-s − 0.476·10-s + 5.51·11-s − 2.82·13-s − 0.697·14-s + 1.31·16-s + 4.54·17-s − 0.369·19-s − 1.03·20-s − 3.84·22-s − 8.06·23-s − 4.53·25-s + 1.96·26-s − 1.51·28-s − 5.46·29-s + 6.64·31-s − 5.82·32-s − 3.17·34-s + 0.682·35-s − 1.54·37-s + 0.258·38-s + 1.67·40-s − 0.605·41-s + ⋯ |
| L(s) = 1 | − 0.493·2-s − 0.756·4-s + 0.305·5-s + 0.377·7-s + 0.866·8-s − 0.150·10-s + 1.66·11-s − 0.782·13-s − 0.186·14-s + 0.328·16-s + 1.10·17-s − 0.0848·19-s − 0.230·20-s − 0.820·22-s − 1.68·23-s − 0.906·25-s + 0.386·26-s − 0.285·28-s − 1.01·29-s + 1.19·31-s − 1.02·32-s − 0.544·34-s + 0.115·35-s − 0.253·37-s + 0.0418·38-s + 0.264·40-s − 0.0945·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 0.697T + 2T^{2} \) |
| 5 | \( 1 - 0.682T + 5T^{2} \) |
| 11 | \( 1 - 5.51T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 4.54T + 17T^{2} \) |
| 19 | \( 1 + 0.369T + 19T^{2} \) |
| 23 | \( 1 + 8.06T + 23T^{2} \) |
| 29 | \( 1 + 5.46T + 29T^{2} \) |
| 31 | \( 1 - 6.64T + 31T^{2} \) |
| 37 | \( 1 + 1.54T + 37T^{2} \) |
| 41 | \( 1 + 0.605T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 + 0.714T + 47T^{2} \) |
| 53 | \( 1 + 8.91T + 53T^{2} \) |
| 59 | \( 1 + 0.745T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 + 0.584T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 8.50T + 73T^{2} \) |
| 79 | \( 1 - 6.70T + 79T^{2} \) |
| 83 | \( 1 + 5.60T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 1.29T + 97T^{2} \) |
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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
−7.81480288254412487496822206447, −6.90566940294521744881656611929, −5.99172945181763337480962887659, −5.53378897725313196604905169882, −4.41142776610128726045674374403, −4.15539740422866271008391168108, −3.18016052978944631524727092798, −1.86111524910402065999084853111, −1.29448644546565863804629226845, 0,
1.29448644546565863804629226845, 1.86111524910402065999084853111, 3.18016052978944631524727092798, 4.15539740422866271008391168108, 4.41142776610128726045674374403, 5.53378897725313196604905169882, 5.99172945181763337480962887659, 6.90566940294521744881656611929, 7.81480288254412487496822206447
