| L(s) = 1 | + 1.40·2-s − 1.25·3-s − 0.0320·4-s − 0.712·5-s − 1.75·6-s − 1.84·7-s − 2.85·8-s − 1.43·9-s − 0.999·10-s + 5.75·11-s + 0.0400·12-s + 6.26·13-s − 2.59·14-s + 0.890·15-s − 3.93·16-s + 2.01·17-s − 2.01·18-s − 19-s + 0.0228·20-s + 2.30·21-s + 8.06·22-s − 4.33·23-s + 3.56·24-s − 4.49·25-s + 8.79·26-s + 5.54·27-s + 0.0591·28-s + ⋯ |
| L(s) = 1 | + 0.991·2-s − 0.721·3-s − 0.0160·4-s − 0.318·5-s − 0.716·6-s − 0.698·7-s − 1.00·8-s − 0.478·9-s − 0.316·10-s + 1.73·11-s + 0.0115·12-s + 1.73·13-s − 0.692·14-s + 0.230·15-s − 0.983·16-s + 0.487·17-s − 0.475·18-s − 0.229·19-s + 0.00510·20-s + 0.503·21-s + 1.72·22-s − 0.903·23-s + 0.727·24-s − 0.898·25-s + 1.72·26-s + 1.06·27-s + 0.0111·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.555242951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.555242951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + T \) |
| 317 | \( 1 - T \) |
| good | 2 | \( 1 - 1.40T + 2T^{2} \) |
| 3 | \( 1 + 1.25T + 3T^{2} \) |
| 5 | \( 1 + 0.712T + 5T^{2} \) |
| 7 | \( 1 + 1.84T + 7T^{2} \) |
| 11 | \( 1 - 5.75T + 11T^{2} \) |
| 13 | \( 1 - 6.26T + 13T^{2} \) |
| 17 | \( 1 - 2.01T + 17T^{2} \) |
| 23 | \( 1 + 4.33T + 23T^{2} \) |
| 29 | \( 1 + 6.17T + 29T^{2} \) |
| 31 | \( 1 + 6.86T + 31T^{2} \) |
| 37 | \( 1 + 12.0T + 37T^{2} \) |
| 41 | \( 1 - 7.18T + 41T^{2} \) |
| 43 | \( 1 + 2.35T + 43T^{2} \) |
| 47 | \( 1 - 9.47T + 47T^{2} \) |
| 53 | \( 1 + 0.965T + 53T^{2} \) |
| 59 | \( 1 - 6.10T + 59T^{2} \) |
| 61 | \( 1 - 8.64T + 61T^{2} \) |
| 67 | \( 1 - 9.94T + 67T^{2} \) |
| 71 | \( 1 - 1.79T + 71T^{2} \) |
| 73 | \( 1 + 2.91T + 73T^{2} \) |
| 79 | \( 1 + 1.96T + 79T^{2} \) |
| 83 | \( 1 + 7.46T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 8.73T + 97T^{2} \) |
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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
−8.142260282148330054873684087294, −6.94179335266198621534320414801, −6.40913663442038004789794406834, −5.74520667317975659018041674174, −5.52358855763463474500876834823, −4.19788378376832730407296541011, −3.71825014515191962161351071747, −3.39779794318057227061161106863, −1.86432122093770515445781435090, −0.58337497788521058268717157344,
0.58337497788521058268717157344, 1.86432122093770515445781435090, 3.39779794318057227061161106863, 3.71825014515191962161351071747, 4.19788378376832730407296541011, 5.52358855763463474500876834823, 5.74520667317975659018041674174, 6.40913663442038004789794406834, 6.94179335266198621534320414801, 8.142260282148330054873684087294
