| L(s) = 1 | − 2.02·2-s + 2.31·3-s + 2.08·4-s − 4.68·6-s + 7-s − 0.176·8-s + 2.36·9-s − 11-s + 4.83·12-s + 2.85·13-s − 2.02·14-s − 3.81·16-s − 7.19·17-s − 4.78·18-s + 6.14·19-s + 2.31·21-s + 2.02·22-s + 2.19·23-s − 0.408·24-s − 5.76·26-s − 1.47·27-s + 2.08·28-s + 3.27·29-s + 10.6·31-s + 8.07·32-s − 2.31·33-s + 14.5·34-s + ⋯ |
| L(s) = 1 | − 1.42·2-s + 1.33·3-s + 1.04·4-s − 1.91·6-s + 0.377·7-s − 0.0623·8-s + 0.788·9-s − 0.301·11-s + 1.39·12-s + 0.791·13-s − 0.540·14-s − 0.954·16-s − 1.74·17-s − 1.12·18-s + 1.40·19-s + 0.505·21-s + 0.431·22-s + 0.456·23-s − 0.0833·24-s − 1.13·26-s − 0.282·27-s + 0.394·28-s + 0.607·29-s + 1.92·31-s + 1.42·32-s − 0.403·33-s + 2.49·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.433811767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.433811767\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.02T + 2T^{2} \) |
| 3 | \( 1 - 2.31T + 3T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 + 7.19T + 17T^{2} \) |
| 19 | \( 1 - 6.14T + 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 - 3.27T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 - 2.60T + 37T^{2} \) |
| 41 | \( 1 + 1.33T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 - 4.68T + 47T^{2} \) |
| 53 | \( 1 - 8.02T + 53T^{2} \) |
| 59 | \( 1 - 3.51T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 6.03T + 67T^{2} \) |
| 71 | \( 1 - 5.45T + 71T^{2} \) |
| 73 | \( 1 + 5.24T + 73T^{2} \) |
| 79 | \( 1 - 2.43T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 5.58T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−8.939375862777339606505360375975, −8.504795220754578736793203548952, −8.079607306577947633077091721686, −7.19877860138826778646848127279, −6.51483067182927667062477294230, −5.05967740165244729964461765320, −4.06782777422899057745848181405, −2.89853299846346358017198392428, −2.11630885378681560227112434962, −0.974627384893865332800046341951,
0.974627384893865332800046341951, 2.11630885378681560227112434962, 2.89853299846346358017198392428, 4.06782777422899057745848181405, 5.05967740165244729964461765320, 6.51483067182927667062477294230, 7.19877860138826778646848127279, 8.079607306577947633077091721686, 8.504795220754578736793203548952, 8.939375862777339606505360375975
