| L(s) = 1 | + 2-s + 4-s + 2.92·5-s + 3.47·7-s + 8-s + 2.92·10-s − 3.75·11-s + 7.10·13-s + 3.47·14-s + 16-s + 5.91·17-s − 1.62·19-s + 2.92·20-s − 3.75·22-s − 1.54·23-s + 3.56·25-s + 7.10·26-s + 3.47·28-s + 5.22·29-s − 6.74·31-s + 32-s + 5.91·34-s + 10.1·35-s − 3.91·37-s − 1.62·38-s + 2.92·40-s − 8.17·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.30·5-s + 1.31·7-s + 0.353·8-s + 0.925·10-s − 1.13·11-s + 1.96·13-s + 0.929·14-s + 0.250·16-s + 1.43·17-s − 0.372·19-s + 0.654·20-s − 0.799·22-s − 0.321·23-s + 0.712·25-s + 1.39·26-s + 0.656·28-s + 0.970·29-s − 1.21·31-s + 0.176·32-s + 1.01·34-s + 1.71·35-s − 0.644·37-s − 0.263·38-s + 0.462·40-s − 1.27·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.008944520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.008944520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 5 | \( 1 - 2.92T + 5T^{2} \) |
| 7 | \( 1 - 3.47T + 7T^{2} \) |
| 11 | \( 1 + 3.75T + 11T^{2} \) |
| 13 | \( 1 - 7.10T + 13T^{2} \) |
| 17 | \( 1 - 5.91T + 17T^{2} \) |
| 19 | \( 1 + 1.62T + 19T^{2} \) |
| 23 | \( 1 + 1.54T + 23T^{2} \) |
| 29 | \( 1 - 5.22T + 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 + 3.91T + 37T^{2} \) |
| 41 | \( 1 + 8.17T + 41T^{2} \) |
| 43 | \( 1 - 6.00T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 4.86T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 4.70T + 61T^{2} \) |
| 67 | \( 1 + 2.36T + 67T^{2} \) |
| 71 | \( 1 + 9.47T + 71T^{2} \) |
| 73 | \( 1 - 7.98T + 73T^{2} \) |
| 79 | \( 1 - 3.75T + 79T^{2} \) |
| 83 | \( 1 + 4.41T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 + 8.52T + 97T^{2} \) |
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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
−8.218829426840996330388692984210, −7.78712739989471327353417829660, −6.66891953116158864034550357203, −5.92000577912675308401903480731, −5.43429949207744051799447530505, −4.88995540165860205527213507685, −3.80331145926147481885727764936, −2.94892764203252726203754431168, −1.84772986138652466546077392090, −1.35484446575397674918582154273,
1.35484446575397674918582154273, 1.84772986138652466546077392090, 2.94892764203252726203754431168, 3.80331145926147481885727764936, 4.88995540165860205527213507685, 5.43429949207744051799447530505, 5.92000577912675308401903480731, 6.66891953116158864034550357203, 7.78712739989471327353417829660, 8.218829426840996330388692984210
