| L(s) = 1 | − 5.42·2-s − 30.3·3-s − 2.59·4-s − 25·5-s + 164.·6-s + 20.9·7-s + 187.·8-s + 677.·9-s + 135.·10-s + 121·11-s + 78.6·12-s + 1.15e3·13-s − 113.·14-s + 758.·15-s − 934.·16-s + 16.3·17-s − 3.67e3·18-s + 361·19-s + 64.8·20-s − 635.·21-s − 656.·22-s + 323.·23-s − 5.69e3·24-s + 625·25-s − 6.25e3·26-s − 1.31e4·27-s − 54.2·28-s + ⋯ |
| L(s) = 1 | − 0.958·2-s − 1.94·3-s − 0.0810·4-s − 0.447·5-s + 1.86·6-s + 0.161·7-s + 1.03·8-s + 2.78·9-s + 0.428·10-s + 0.301·11-s + 0.157·12-s + 1.89·13-s − 0.154·14-s + 0.870·15-s − 0.912·16-s + 0.0137·17-s − 2.67·18-s + 0.229·19-s + 0.0362·20-s − 0.314·21-s − 0.289·22-s + 0.127·23-s − 2.01·24-s + 0.200·25-s − 1.81·26-s − 3.47·27-s − 0.0130·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 5.42T + 32T^{2} \) |
| 3 | \( 1 + 30.3T + 243T^{2} \) |
| 7 | \( 1 - 20.9T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.15e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 16.3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 323.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 469.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.83e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.22e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.49e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.34e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.45e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.40e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.94e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.74e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.10e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.31e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.73e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.43e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.13e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.28e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.03e5T + 8.58e9T^{2} \) |
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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
−8.802676019259161991988545894179, −7.972736427783397541511820262440, −7.02585556000920386197909632633, −6.31038007203619174673074457340, −5.43649147890993525311056994751, −4.51431028219849094573287525504, −3.78581923252638624291420796290, −1.46477309621175464255320939013, −0.950975896093137423502927105893, 0,
0.950975896093137423502927105893, 1.46477309621175464255320939013, 3.78581923252638624291420796290, 4.51431028219849094573287525504, 5.43649147890993525311056994751, 6.31038007203619174673074457340, 7.02585556000920386197909632633, 7.972736427783397541511820262440, 8.802676019259161991988545894179
