| L(s) = 1 | − 1.03·2-s − 3.10·3-s − 0.923·4-s − 3.82·5-s + 3.21·6-s + 0.702·7-s + 3.03·8-s + 6.62·9-s + 3.97·10-s − 4.15·11-s + 2.86·12-s + 4.12·13-s − 0.728·14-s + 11.8·15-s − 1.30·16-s − 3.32·17-s − 6.87·18-s + 2.39·19-s + 3.53·20-s − 2.17·21-s + 4.30·22-s + 6.71·23-s − 9.40·24-s + 9.65·25-s − 4.27·26-s − 11.2·27-s − 0.648·28-s + ⋯ |
| L(s) = 1 | − 0.733·2-s − 1.79·3-s − 0.461·4-s − 1.71·5-s + 1.31·6-s + 0.265·7-s + 1.07·8-s + 2.20·9-s + 1.25·10-s − 1.25·11-s + 0.826·12-s + 1.14·13-s − 0.194·14-s + 3.06·15-s − 0.325·16-s − 0.807·17-s − 1.61·18-s + 0.549·19-s + 0.790·20-s − 0.475·21-s + 0.918·22-s + 1.39·23-s − 1.92·24-s + 1.93·25-s − 0.838·26-s − 2.16·27-s − 0.122·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 787 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 787 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 787 | \( 1 + T \) |
| good | 2 | \( 1 + 1.03T + 2T^{2} \) |
| 3 | \( 1 + 3.10T + 3T^{2} \) |
| 5 | \( 1 + 3.82T + 5T^{2} \) |
| 7 | \( 1 - 0.702T + 7T^{2} \) |
| 11 | \( 1 + 4.15T + 11T^{2} \) |
| 13 | \( 1 - 4.12T + 13T^{2} \) |
| 17 | \( 1 + 3.32T + 17T^{2} \) |
| 19 | \( 1 - 2.39T + 19T^{2} \) |
| 23 | \( 1 - 6.71T + 23T^{2} \) |
| 29 | \( 1 + 2.41T + 29T^{2} \) |
| 31 | \( 1 - 8.96T + 31T^{2} \) |
| 37 | \( 1 + 3.73T + 37T^{2} \) |
| 41 | \( 1 + 4.51T + 41T^{2} \) |
| 43 | \( 1 + 8.78T + 43T^{2} \) |
| 47 | \( 1 - 5.23T + 47T^{2} \) |
| 53 | \( 1 + 4.48T + 53T^{2} \) |
| 59 | \( 1 - 6.31T + 59T^{2} \) |
| 61 | \( 1 - 5.58T + 61T^{2} \) |
| 67 | \( 1 - 0.385T + 67T^{2} \) |
| 71 | \( 1 + 7.72T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−10.26954240304230645561213527633, −8.877329355668917773092535371558, −8.091949850606670229088041576870, −7.37905384611250542159360213674, −6.51450301670668152782654601425, −5.07008437937578139988599166540, −4.75400423201593163169978026228, −3.61355366001620931818502180205, −1.02451300589763919264305593358, 0,
1.02451300589763919264305593358, 3.61355366001620931818502180205, 4.75400423201593163169978026228, 5.07008437937578139988599166540, 6.51450301670668152782654601425, 7.37905384611250542159360213674, 8.091949850606670229088041576870, 8.877329355668917773092535371558, 10.26954240304230645561213527633
