| L(s) = 1 | + 2.18·2-s + 2.79·4-s − 0.868·5-s − 7-s + 1.73·8-s − 1.90·10-s − 0.280·11-s + 3.69·13-s − 2.18·14-s − 1.78·16-s + 1.92·17-s + 1.42·19-s − 2.42·20-s − 0.614·22-s + 6.52·23-s − 4.24·25-s + 8.07·26-s − 2.79·28-s + 3.07·29-s − 0.0163·31-s − 7.38·32-s + 4.20·34-s + 0.868·35-s + 0.359·37-s + 3.11·38-s − 1.50·40-s − 8.75·41-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 1.39·4-s − 0.388·5-s − 0.377·7-s + 0.613·8-s − 0.601·10-s − 0.0845·11-s + 1.02·13-s − 0.585·14-s − 0.446·16-s + 0.465·17-s + 0.326·19-s − 0.542·20-s − 0.130·22-s + 1.35·23-s − 0.849·25-s + 1.58·26-s − 0.527·28-s + 0.570·29-s − 0.00294·31-s − 1.30·32-s + 0.721·34-s + 0.146·35-s + 0.0590·37-s + 0.505·38-s − 0.238·40-s − 1.36·41-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
| \(L(1)\) |
\(\approx\) |
\(4.676249789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.676249789\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.18T + 2T^{2} \) |
| 5 | \( 1 + 0.868T + 5T^{2} \) |
| 11 | \( 1 + 0.280T + 11T^{2} \) |
| 13 | \( 1 - 3.69T + 13T^{2} \) |
| 17 | \( 1 - 1.92T + 17T^{2} \) |
| 19 | \( 1 - 1.42T + 19T^{2} \) |
| 23 | \( 1 - 6.52T + 23T^{2} \) |
| 29 | \( 1 - 3.07T + 29T^{2} \) |
| 31 | \( 1 + 0.0163T + 31T^{2} \) |
| 37 | \( 1 - 0.359T + 37T^{2} \) |
| 41 | \( 1 + 8.75T + 41T^{2} \) |
| 43 | \( 1 - 4.09T + 43T^{2} \) |
| 47 | \( 1 + 4.32T + 47T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 0.486T + 61T^{2} \) |
| 67 | \( 1 - 0.701T + 67T^{2} \) |
| 71 | \( 1 - 9.14T + 71T^{2} \) |
| 73 | \( 1 + 0.554T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 7.95T + 97T^{2} \) |
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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
−7.57671245454492399944041092099, −6.87258352124585765875868823600, −6.33925234200748648107880306595, −5.55122147340212791849214690060, −5.09487391567062084398981597859, −4.18886786344295764977043521088, −3.58322687845903641582896733307, −3.09294323992743994666460739947, −2.13035268633726926895106932342, −0.849120197728719626077707297502,
0.849120197728719626077707297502, 2.13035268633726926895106932342, 3.09294323992743994666460739947, 3.58322687845903641582896733307, 4.18886786344295764977043521088, 5.09487391567062084398981597859, 5.55122147340212791849214690060, 6.33925234200748648107880306595, 6.87258352124585765875868823600, 7.57671245454492399944041092099
