| L(s) = 1 | + 2.14·2-s − 0.619·3-s + 2.60·4-s + 2.51·5-s − 1.33·6-s − 4.78·7-s + 1.29·8-s − 2.61·9-s + 5.39·10-s + 2.09·11-s − 1.61·12-s + 3.38·13-s − 10.2·14-s − 1.55·15-s − 2.42·16-s + 0.479·17-s − 5.61·18-s + 7.16·19-s + 6.55·20-s + 2.96·21-s + 4.49·22-s + 7.76·23-s − 0.805·24-s + 1.32·25-s + 7.26·26-s + 3.48·27-s − 12.4·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s − 0.357·3-s + 1.30·4-s + 1.12·5-s − 0.543·6-s − 1.80·7-s + 0.459·8-s − 0.871·9-s + 1.70·10-s + 0.632·11-s − 0.466·12-s + 0.939·13-s − 2.74·14-s − 0.402·15-s − 0.605·16-s + 0.116·17-s − 1.32·18-s + 1.64·19-s + 1.46·20-s + 0.647·21-s + 0.959·22-s + 1.61·23-s − 0.164·24-s + 0.264·25-s + 1.42·26-s + 0.670·27-s − 2.35·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.108328412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.108328412\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4001 | \( 1+O(T) \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 3 | \( 1 + 0.619T + 3T^{2} \) |
| 5 | \( 1 - 2.51T + 5T^{2} \) |
| 7 | \( 1 + 4.78T + 7T^{2} \) |
| 11 | \( 1 - 2.09T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 - 0.479T + 17T^{2} \) |
| 19 | \( 1 - 7.16T + 19T^{2} \) |
| 23 | \( 1 - 7.76T + 23T^{2} \) |
| 29 | \( 1 - 5.00T + 29T^{2} \) |
| 31 | \( 1 - 0.822T + 31T^{2} \) |
| 37 | \( 1 + 2.72T + 37T^{2} \) |
| 41 | \( 1 + 1.05T + 41T^{2} \) |
| 43 | \( 1 + 5.14T + 43T^{2} \) |
| 47 | \( 1 - 1.83T + 47T^{2} \) |
| 53 | \( 1 - 9.33T + 53T^{2} \) |
| 59 | \( 1 + 4.44T + 59T^{2} \) |
| 61 | \( 1 - 4.37T + 61T^{2} \) |
| 67 | \( 1 + 2.89T + 67T^{2} \) |
| 71 | \( 1 - 5.72T + 71T^{2} \) |
| 73 | \( 1 + 1.46T + 73T^{2} \) |
| 79 | \( 1 - 9.67T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 5.31T + 89T^{2} \) |
| 97 | \( 1 - 6.21T + 97T^{2} \) |
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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
−8.713766316116241859457247051956, −7.13668881518928320562030597864, −6.52355716913778429979835621600, −6.11631741490990152646406344529, −5.53006771941824696594887902001, −4.93622990869904441680384546285, −3.57982437195430743737072187702, −3.25378932497765497548871308760, −2.48919324605588768410135445003, −0.953677812210124417163641517817,
0.953677812210124417163641517817, 2.48919324605588768410135445003, 3.25378932497765497548871308760, 3.57982437195430743737072187702, 4.93622990869904441680384546285, 5.53006771941824696594887902001, 6.11631741490990152646406344529, 6.52355716913778429979835621600, 7.13668881518928320562030597864, 8.713766316116241859457247051956
