| L(s) = 1 | + 1.32·2-s − 0.232·4-s + 0.891·5-s + 7-s − 2.96·8-s + 1.18·10-s + 1.74·11-s − 2.35·13-s + 1.32·14-s − 3.48·16-s − 1.98·17-s + 6.32·19-s − 0.206·20-s + 2.31·22-s − 4.16·23-s − 4.20·25-s − 3.12·26-s − 0.232·28-s − 5.78·29-s + 2.29·31-s + 1.30·32-s − 2.63·34-s + 0.891·35-s + 6.24·37-s + 8.40·38-s − 2.64·40-s + 8.30·41-s + ⋯ |
| L(s) = 1 | + 0.940·2-s − 0.116·4-s + 0.398·5-s + 0.377·7-s − 1.04·8-s + 0.374·10-s + 0.524·11-s − 0.652·13-s + 0.355·14-s − 0.870·16-s − 0.480·17-s + 1.45·19-s − 0.0462·20-s + 0.493·22-s − 0.868·23-s − 0.841·25-s − 0.613·26-s − 0.0438·28-s − 1.07·29-s + 0.411·31-s + 0.230·32-s − 0.451·34-s + 0.150·35-s + 1.02·37-s + 1.36·38-s − 0.418·40-s + 1.29·41-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
| \(L(1)\) |
\(\approx\) |
\(2.993337174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.993337174\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.32T + 2T^{2} \) |
| 5 | \( 1 - 0.891T + 5T^{2} \) |
| 11 | \( 1 - 1.74T + 11T^{2} \) |
| 13 | \( 1 + 2.35T + 13T^{2} \) |
| 17 | \( 1 + 1.98T + 17T^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 23 | \( 1 + 4.16T + 23T^{2} \) |
| 29 | \( 1 + 5.78T + 29T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 - 6.24T + 37T^{2} \) |
| 41 | \( 1 - 8.30T + 41T^{2} \) |
| 43 | \( 1 + 6.53T + 43T^{2} \) |
| 47 | \( 1 - 9.52T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 9.30T + 59T^{2} \) |
| 61 | \( 1 - 7.96T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 + 2.24T + 73T^{2} \) |
| 79 | \( 1 + 0.349T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 17.9T + 97T^{2} \) |
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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
−7.70181737154718834106853073561, −7.09148973091210688913932756565, −6.15636028488443371296343344283, −5.63090281320830747517023106879, −5.10233508263120356099905049511, −4.18800461442131657637359534899, −3.81698011616018354520986353750, −2.74173868862279828735818618585, −2.03487830201194498807190464442, −0.73786102016329615616651339483,
0.73786102016329615616651339483, 2.03487830201194498807190464442, 2.74173868862279828735818618585, 3.81698011616018354520986353750, 4.18800461442131657637359534899, 5.10233508263120356099905049511, 5.63090281320830747517023106879, 6.15636028488443371296343344283, 7.09148973091210688913932756565, 7.70181737154718834106853073561
