| L(s) = 1 | + 2.74·2-s + 5.55·4-s + 5-s + 0.361·7-s + 9.77·8-s + 2.74·10-s − 4.19·11-s + 13-s + 0.994·14-s + 15.7·16-s − 5.94·17-s + 2.86·19-s + 5.55·20-s − 11.5·22-s + 7.69·23-s + 25-s + 2.74·26-s + 2.01·28-s + 5.35·29-s + 5.91·31-s + 23.7·32-s − 16.3·34-s + 0.361·35-s − 1.10·37-s + 7.87·38-s + 9.77·40-s + 5.02·41-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 2.77·4-s + 0.447·5-s + 0.136·7-s + 3.45·8-s + 0.869·10-s − 1.26·11-s + 0.277·13-s + 0.265·14-s + 3.94·16-s − 1.44·17-s + 0.657·19-s + 1.24·20-s − 2.46·22-s + 1.60·23-s + 0.200·25-s + 0.539·26-s + 0.379·28-s + 0.994·29-s + 1.06·31-s + 4.20·32-s − 2.80·34-s + 0.0611·35-s − 0.181·37-s + 1.27·38-s + 1.54·40-s + 0.785·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.282199158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.282199158\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 7 | \( 1 - 0.361T + 7T^{2} \) |
| 11 | \( 1 + 4.19T + 11T^{2} \) |
| 17 | \( 1 + 5.94T + 17T^{2} \) |
| 19 | \( 1 - 2.86T + 19T^{2} \) |
| 23 | \( 1 - 7.69T + 23T^{2} \) |
| 29 | \( 1 - 5.35T + 29T^{2} \) |
| 31 | \( 1 - 5.91T + 31T^{2} \) |
| 37 | \( 1 + 1.10T + 37T^{2} \) |
| 41 | \( 1 - 5.02T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 9.60T + 53T^{2} \) |
| 59 | \( 1 - 6.22T + 59T^{2} \) |
| 61 | \( 1 + 1.38T + 61T^{2} \) |
| 67 | \( 1 - 4.84T + 67T^{2} \) |
| 71 | \( 1 - 0.306T + 71T^{2} \) |
| 73 | \( 1 - 5.22T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 2.65T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 6.20T + 97T^{2} \) |
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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
−7.961015645917219251854794838321, −6.97270053529076038724537698411, −6.70754592940285460678492141390, −5.77494175428546663164955472936, −5.13294015702507818447897027463, −4.72107721348089662066986054787, −3.83953450072875998287644504101, −2.72108221894098821011593675171, −2.57059210826059081979660222529, −1.27602644220480876942121008654,
1.27602644220480876942121008654, 2.57059210826059081979660222529, 2.72108221894098821011593675171, 3.83953450072875998287644504101, 4.72107721348089662066986054787, 5.13294015702507818447897027463, 5.77494175428546663164955472936, 6.70754592940285460678492141390, 6.97270053529076038724537698411, 7.961015645917219251854794838321
