| L(s) = 1 | − 0.395·2-s − 1.84·4-s − 1.95·5-s + 7-s + 1.52·8-s + 0.772·10-s + 5.34·11-s + 6.49·13-s − 0.395·14-s + 3.08·16-s − 7.92·17-s − 5.50·19-s + 3.59·20-s − 2.11·22-s + 3.09·23-s − 1.18·25-s − 2.56·26-s − 1.84·28-s + 3.39·29-s + 10.2·31-s − 4.26·32-s + 3.13·34-s − 1.95·35-s + 7.25·37-s + 2.17·38-s − 2.96·40-s + 10.1·41-s + ⋯ |
| L(s) = 1 | − 0.279·2-s − 0.921·4-s − 0.873·5-s + 0.377·7-s + 0.537·8-s + 0.244·10-s + 1.61·11-s + 1.80·13-s − 0.105·14-s + 0.771·16-s − 1.92·17-s − 1.26·19-s + 0.804·20-s − 0.450·22-s + 0.646·23-s − 0.237·25-s − 0.503·26-s − 0.348·28-s + 0.630·29-s + 1.83·31-s − 0.753·32-s + 0.537·34-s − 0.329·35-s + 1.19·37-s + 0.353·38-s − 0.469·40-s + 1.59·41-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
| \(L(1)\) |
\(\approx\) |
\(1.415749998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.415749998\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.395T + 2T^{2} \) |
| 5 | \( 1 + 1.95T + 5T^{2} \) |
| 11 | \( 1 - 5.34T + 11T^{2} \) |
| 13 | \( 1 - 6.49T + 13T^{2} \) |
| 17 | \( 1 + 7.92T + 17T^{2} \) |
| 19 | \( 1 + 5.50T + 19T^{2} \) |
| 23 | \( 1 - 3.09T + 23T^{2} \) |
| 29 | \( 1 - 3.39T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 7.25T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 - 1.66T + 47T^{2} \) |
| 53 | \( 1 + 9.51T + 53T^{2} \) |
| 59 | \( 1 - 5.51T + 59T^{2} \) |
| 61 | \( 1 - 0.0570T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 7.73T + 71T^{2} \) |
| 73 | \( 1 - 1.79T + 73T^{2} \) |
| 79 | \( 1 - 1.81T + 79T^{2} \) |
| 83 | \( 1 + 6.90T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 - 4.44T + 97T^{2} \) |
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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
−8.116465760038070950827683694455, −7.22470166012811750262747706492, −6.27435493311400728242451003568, −6.10175363374678746047671575418, −4.56695027237760443302061867592, −4.19830252084406117248903890720, −3.99382864436602056496869112106, −2.71832711837020377185200953095, −1.41767343561810024295191800308, −0.69945726910162737877524927022,
0.69945726910162737877524927022, 1.41767343561810024295191800308, 2.71832711837020377185200953095, 3.99382864436602056496869112106, 4.19830252084406117248903890720, 4.56695027237760443302061867592, 6.10175363374678746047671575418, 6.27435493311400728242451003568, 7.22470166012811750262747706492, 8.116465760038070950827683694455
