| L(s) = 1 | − 2.61·2-s − 0.901·3-s + 4.85·4-s − 1.31·5-s + 2.36·6-s − 2.24·7-s − 7.48·8-s − 2.18·9-s + 3.43·10-s + 5.25·11-s − 4.37·12-s − 1.46·13-s + 5.87·14-s + 1.18·15-s + 9.88·16-s − 0.0584·17-s + 5.72·18-s + 1.09·19-s − 6.37·20-s + 2.02·21-s − 13.7·22-s − 6.84·23-s + 6.74·24-s − 3.27·25-s + 3.84·26-s + 4.67·27-s − 10.9·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 0.520·3-s + 2.42·4-s − 0.587·5-s + 0.963·6-s − 0.848·7-s − 2.64·8-s − 0.729·9-s + 1.08·10-s + 1.58·11-s − 1.26·12-s − 0.407·13-s + 1.57·14-s + 0.305·15-s + 2.47·16-s − 0.0141·17-s + 1.35·18-s + 0.250·19-s − 1.42·20-s + 0.441·21-s − 2.93·22-s − 1.42·23-s + 1.37·24-s − 0.655·25-s + 0.754·26-s + 0.899·27-s − 2.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1814501937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1814501937\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 8011 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 + 0.901T + 3T^{2} \) |
| 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 - 5.25T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 0.0584T + 17T^{2} \) |
| 19 | \( 1 - 1.09T + 19T^{2} \) |
| 23 | \( 1 + 6.84T + 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 - 7.80T + 31T^{2} \) |
| 37 | \( 1 + 8.81T + 37T^{2} \) |
| 41 | \( 1 + 5.59T + 41T^{2} \) |
| 43 | \( 1 + 3.28T + 43T^{2} \) |
| 47 | \( 1 - 2.16T + 47T^{2} \) |
| 53 | \( 1 - 7.06T + 53T^{2} \) |
| 59 | \( 1 - 2.06T + 59T^{2} \) |
| 61 | \( 1 - 4.33T + 61T^{2} \) |
| 67 | \( 1 + 0.888T + 67T^{2} \) |
| 71 | \( 1 - 2.46T + 71T^{2} \) |
| 73 | \( 1 + 3.08T + 73T^{2} \) |
| 79 | \( 1 - 0.809T + 79T^{2} \) |
| 83 | \( 1 - 0.294T + 83T^{2} \) |
| 89 | \( 1 - 4.84T + 89T^{2} \) |
| 97 | \( 1 + 8.32T + 97T^{2} \) |
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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
−8.016318128111640459097707785722, −7.20840090467268247777935462390, −6.58955086065427921303524788654, −6.22995253287810432691934967002, −5.37233019121282708841286616710, −4.01697652863623652474021830152, −3.35237846220627527752177470488, −2.35400108880601933470999591829, −1.39272026660816124633109031995, −0.29905167387762101577535787042,
0.29905167387762101577535787042, 1.39272026660816124633109031995, 2.35400108880601933470999591829, 3.35237846220627527752177470488, 4.01697652863623652474021830152, 5.37233019121282708841286616710, 6.22995253287810432691934967002, 6.58955086065427921303524788654, 7.20840090467268247777935462390, 8.016318128111640459097707785722
