| L(s) = 1 | − 0.696·2-s + 0.135·3-s − 1.51·4-s − 0.0464·5-s − 0.0945·6-s + 3.33·7-s + 2.44·8-s − 2.98·9-s + 0.0323·10-s − 0.501·11-s − 0.205·12-s − 3.43·13-s − 2.31·14-s − 0.00630·15-s + 1.32·16-s + 2.08·17-s + 2.07·18-s + 5.70·19-s + 0.0703·20-s + 0.452·21-s + 0.349·22-s + 7.75·23-s + 0.332·24-s − 4.99·25-s + 2.39·26-s − 0.812·27-s − 5.05·28-s + ⋯ |
| L(s) = 1 | − 0.492·2-s + 0.0784·3-s − 0.757·4-s − 0.0207·5-s − 0.0386·6-s + 1.25·7-s + 0.865·8-s − 0.993·9-s + 0.0102·10-s − 0.151·11-s − 0.0594·12-s − 0.952·13-s − 0.619·14-s − 0.00162·15-s + 0.331·16-s + 0.505·17-s + 0.489·18-s + 1.30·19-s + 0.0157·20-s + 0.0988·21-s + 0.0744·22-s + 1.61·23-s + 0.0678·24-s − 0.999·25-s + 0.468·26-s − 0.156·27-s − 0.954·28-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
| \(L(1)\) |
\(\approx\) |
\(1.061476503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.061476503\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.696T + 2T^{2} \) |
| 3 | \( 1 - 0.135T + 3T^{2} \) |
| 5 | \( 1 + 0.0464T + 5T^{2} \) |
| 7 | \( 1 - 3.33T + 7T^{2} \) |
| 11 | \( 1 + 0.501T + 11T^{2} \) |
| 13 | \( 1 + 3.43T + 13T^{2} \) |
| 17 | \( 1 - 2.08T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 23 | \( 1 - 7.75T + 23T^{2} \) |
| 29 | \( 1 + 3.80T + 29T^{2} \) |
| 31 | \( 1 - 5.50T + 31T^{2} \) |
| 37 | \( 1 + 3.83T + 37T^{2} \) |
| 41 | \( 1 - 3.88T + 41T^{2} \) |
| 43 | \( 1 - 8.60T + 43T^{2} \) |
| 47 | \( 1 - 4.51T + 47T^{2} \) |
| 53 | \( 1 - 7.50T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 7.96T + 61T^{2} \) |
| 67 | \( 1 - 9.01T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 - 3.53T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 5.46T + 83T^{2} \) |
| 89 | \( 1 + 1.46T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−10.11564237339115359904292118071, −9.365680503496863187629580877493, −8.588128145121590290577396242240, −7.85210748243195540880131300189, −7.25029883035128964571559047941, −5.40987636408004744137426318076, −5.17617622802534000032796197346, −3.91792256911935125560291289834, −2.52351861587981377445662601293, −0.951559900702948005934061972520,
0.951559900702948005934061972520, 2.52351861587981377445662601293, 3.91792256911935125560291289834, 5.17617622802534000032796197346, 5.40987636408004744137426318076, 7.25029883035128964571559047941, 7.85210748243195540880131300189, 8.588128145121590290577396242240, 9.365680503496863187629580877493, 10.11564237339115359904292118071
