| L(s) = 1 | + 3.24·2-s + 2.53·4-s + 16.0·5-s − 7·7-s − 17.7·8-s + 52.2·10-s + 11·11-s + 35.3·13-s − 22.7·14-s − 77.8·16-s − 40.4·17-s + 118.·19-s + 40.7·20-s + 35.7·22-s + 174.·23-s + 134.·25-s + 114.·26-s − 17.7·28-s + 262.·29-s − 36.1·31-s − 110.·32-s − 131.·34-s − 112.·35-s + 19.0·37-s + 383.·38-s − 285.·40-s − 156.·41-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.316·4-s + 1.43·5-s − 0.377·7-s − 0.784·8-s + 1.65·10-s + 0.301·11-s + 0.754·13-s − 0.433·14-s − 1.21·16-s − 0.577·17-s + 1.42·19-s + 0.455·20-s + 0.345·22-s + 1.58·23-s + 1.07·25-s + 0.865·26-s − 0.119·28-s + 1.68·29-s − 0.209·31-s − 0.611·32-s − 0.662·34-s − 0.544·35-s + 0.0846·37-s + 1.63·38-s − 1.12·40-s − 0.598·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.562993668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.562993668\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 3.24T + 8T^{2} \) |
| 5 | \( 1 - 16.0T + 125T^{2} \) |
| 13 | \( 1 - 35.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 40.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 118.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 174.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 262.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 36.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 19.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 156.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 287.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 397.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 272.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 507.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 35.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 979.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 750.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 395.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 736.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 582.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 806.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 957.T + 9.12e5T^{2} \) |
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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
−9.962223437802360562386329578037, −9.302529152621927091710348133463, −8.583271605804741534753721163594, −6.91086204534138667877711843395, −6.29312111275353564691331320819, −5.46916070647206661107948309512, −4.75082792786151851827272127204, −3.44565223634946139946960747399, −2.60661458855278224653023630310, −1.12830636890115657044299363123,
1.12830636890115657044299363123, 2.60661458855278224653023630310, 3.44565223634946139946960747399, 4.75082792786151851827272127204, 5.46916070647206661107948309512, 6.29312111275353564691331320819, 6.91086204534138667877711843395, 8.583271605804741534753721163594, 9.302529152621927091710348133463, 9.962223437802360562386329578037
