| L(s) = 1 | + 8.93·2-s − 9·3-s + 47.8·4-s + 39.2·5-s − 80.4·6-s − 156.·7-s + 142.·8-s + 81·9-s + 350.·10-s − 13.1·11-s − 431.·12-s + 140.·13-s − 1.39e3·14-s − 352.·15-s − 262.·16-s + 713.·17-s + 724.·18-s − 12.9·19-s + 1.87e3·20-s + 1.40e3·21-s − 117.·22-s + 629.·23-s − 1.27e3·24-s − 1.58e3·25-s + 1.25e3·26-s − 729·27-s − 7.49e3·28-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.577·3-s + 1.49·4-s + 0.701·5-s − 0.912·6-s − 1.20·7-s + 0.784·8-s + 0.333·9-s + 1.10·10-s − 0.0327·11-s − 0.864·12-s + 0.230·13-s − 1.90·14-s − 0.404·15-s − 0.256·16-s + 0.598·17-s + 0.526·18-s − 0.00823·19-s + 1.04·20-s + 0.697·21-s − 0.0517·22-s + 0.248·23-s − 0.453·24-s − 0.508·25-s + 0.364·26-s − 0.192·27-s − 1.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 157 | \( 1 + 2.46e4T \) |
| good | 2 | \( 1 - 8.93T + 32T^{2} \) |
| 5 | \( 1 - 39.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 156.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 13.1T + 1.61e5T^{2} \) |
| 13 | \( 1 - 140.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 713.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 12.9T + 2.47e6T^{2} \) |
| 23 | \( 1 - 629.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.06e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.21e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.45e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.53e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.46e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.83e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.65e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.65e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 8.96e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.69e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.88e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.08e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.68e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.44e4T + 8.58e9T^{2} \) |
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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
−9.910401122543040133071694995002, −9.168038808162153128023264578403, −7.43326333677690596912071893656, −6.46901456239880192050408173023, −5.86272295758149033060564950406, −5.18308183653788195736512418411, −3.90839272610548554775365153113, −3.11320076684392116113878205954, −1.80925581664193570715972339843, 0,
1.80925581664193570715972339843, 3.11320076684392116113878205954, 3.90839272610548554775365153113, 5.18308183653788195736512418411, 5.86272295758149033060564950406, 6.46901456239880192050408173023, 7.43326333677690596912071893656, 9.168038808162153128023264578403, 9.910401122543040133071694995002
