| L(s) = 1 | − 9.79·2-s + 20.6·3-s + 63.9·4-s − 25·5-s − 202.·6-s + 61.8·7-s − 313.·8-s + 185.·9-s + 244.·10-s + 636.·11-s + 1.32e3·12-s − 14.1·13-s − 605.·14-s − 517.·15-s + 1.01e3·16-s + 1.05e3·17-s − 1.81e3·18-s − 2.38e3·19-s − 1.59e3·20-s + 1.27e3·21-s − 6.23e3·22-s + 529·23-s − 6.47e3·24-s + 625·25-s + 138.·26-s − 1.19e3·27-s + 3.95e3·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 1.32·3-s + 1.99·4-s − 0.447·5-s − 2.29·6-s + 0.477·7-s − 1.72·8-s + 0.761·9-s + 0.774·10-s + 1.58·11-s + 2.65·12-s − 0.0231·13-s − 0.826·14-s − 0.593·15-s + 0.995·16-s + 0.886·17-s − 1.31·18-s − 1.51·19-s − 0.893·20-s + 0.633·21-s − 2.74·22-s + 0.208·23-s − 2.29·24-s + 0.200·25-s + 0.0401·26-s − 0.316·27-s + 0.953·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.380390539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.380390539\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 + 9.79T + 32T^{2} \) |
| 3 | \( 1 - 20.6T + 243T^{2} \) |
| 7 | \( 1 - 61.8T + 1.68e4T^{2} \) |
| 11 | \( 1 - 636.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 14.1T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.05e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.38e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 2.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.12e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.15e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.75e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.32e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.65e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.93e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 377.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.06e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.61e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.40e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.23e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.18e5T + 8.58e9T^{2} \) |
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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
−12.29907580421913329475273077869, −11.28137240074761892540608186842, −10.13540195812454718363126645294, −9.068133589759071087496195401863, −8.525155456727522574676969073463, −7.69388971620210917828158724287, −6.58067595971239528257181366930, −3.94969256054595103028631608388, −2.36840226193348725904225118056, −1.06270293561828920338506828092,
1.06270293561828920338506828092, 2.36840226193348725904225118056, 3.94969256054595103028631608388, 6.58067595971239528257181366930, 7.69388971620210917828158724287, 8.525155456727522574676969073463, 9.068133589759071087496195401863, 10.13540195812454718363126645294, 11.28137240074761892540608186842, 12.29907580421913329475273077869
