| L(s) = 1 | − 0.583·2-s − 1.65·4-s − 2.31·5-s − 3.54·7-s + 2.13·8-s + 1.35·10-s + 1.79·11-s − 3.62·13-s + 2.06·14-s + 2.07·16-s − 4.64·17-s + 2.93·19-s + 3.85·20-s − 1.04·22-s − 0.374·23-s + 0.379·25-s + 2.11·26-s + 5.88·28-s − 5.25·29-s − 5.47·32-s + 2.71·34-s + 8.22·35-s + 11.0·37-s − 1.70·38-s − 4.94·40-s − 9.94·41-s − 9.36·43-s + ⋯ |
| L(s) = 1 | − 0.412·2-s − 0.829·4-s − 1.03·5-s − 1.34·7-s + 0.754·8-s + 0.427·10-s + 0.540·11-s − 1.00·13-s + 0.552·14-s + 0.518·16-s − 1.12·17-s + 0.672·19-s + 0.860·20-s − 0.222·22-s − 0.0781·23-s + 0.0758·25-s + 0.414·26-s + 1.11·28-s − 0.975·29-s − 0.968·32-s + 0.464·34-s + 1.39·35-s + 1.81·37-s − 0.277·38-s − 0.782·40-s − 1.55·41-s − 1.42·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.001097513893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001097513893\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 0.583T + 2T^{2} \) |
| 5 | \( 1 + 2.31T + 5T^{2} \) |
| 7 | \( 1 + 3.54T + 7T^{2} \) |
| 11 | \( 1 - 1.79T + 11T^{2} \) |
| 13 | \( 1 + 3.62T + 13T^{2} \) |
| 17 | \( 1 + 4.64T + 17T^{2} \) |
| 19 | \( 1 - 2.93T + 19T^{2} \) |
| 23 | \( 1 + 0.374T + 23T^{2} \) |
| 29 | \( 1 + 5.25T + 29T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 9.94T + 41T^{2} \) |
| 43 | \( 1 + 9.36T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 7.84T + 59T^{2} \) |
| 61 | \( 1 - 4.14T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 1.45T + 71T^{2} \) |
| 73 | \( 1 - 9.35T + 73T^{2} \) |
| 79 | \( 1 + 5.91T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.972070472080527956855162516329, −7.00329905931133605817978844318, −6.73417190421673029856966454811, −5.67370654149784986435167080187, −4.82724062733178513759527722129, −4.18817096996081426417628446966, −3.56053925087998247851685496101, −2.84802266853537461265139154214, −1.52313601737931545758590440517, −0.01746395728790068105330523447,
0.01746395728790068105330523447, 1.52313601737931545758590440517, 2.84802266853537461265139154214, 3.56053925087998247851685496101, 4.18817096996081426417628446966, 4.82724062733178513759527722129, 5.67370654149784986435167080187, 6.73417190421673029856966454811, 7.00329905931133605817978844318, 7.972070472080527956855162516329
