| L(s) = 1 | − 0.635·2-s − 27.7·3-s − 31.5·4-s − 25·5-s + 17.6·6-s + 19.6·7-s + 40.4·8-s + 529.·9-s + 15.8·10-s + 121·11-s + 878.·12-s − 886.·13-s − 12.4·14-s + 694.·15-s + 985.·16-s − 2.21e3·17-s − 336.·18-s + 361·19-s + 789.·20-s − 545.·21-s − 76.8·22-s − 1.26e3·23-s − 1.12e3·24-s + 625·25-s + 563.·26-s − 7.95e3·27-s − 620.·28-s + ⋯ |
| L(s) = 1 | − 0.112·2-s − 1.78·3-s − 0.987·4-s − 0.447·5-s + 0.200·6-s + 0.151·7-s + 0.223·8-s + 2.17·9-s + 0.0502·10-s + 0.301·11-s + 1.76·12-s − 1.45·13-s − 0.0170·14-s + 0.797·15-s + 0.962·16-s − 1.86·17-s − 0.244·18-s + 0.229·19-s + 0.441·20-s − 0.269·21-s − 0.0338·22-s − 0.497·23-s − 0.398·24-s + 0.200·25-s + 0.163·26-s − 2.10·27-s − 0.149·28-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 | \( 1 + 0.635T + 32T^{2} \) |
| 3 | \( 1 + 27.7T + 243T^{2} \) |
| 7 | \( 1 - 19.6T + 1.68e4T^{2} \) |
| 13 | \( 1 + 886.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.21e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.26e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.24e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.37e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.49e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 639.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.01e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.96e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 318.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.36e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.06e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.32e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.16e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.69e4T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.069010246020895825733291303188, −7.66770450679970548739761403633, −7.13233823779565166897386680857, −6.03455587481448008090956717012, −5.27530886743148102824430972685, −4.51433173062679574737116522822, −3.97706091453534556119273728626, −2.03034890631683916546492372227, −0.63616452119864504507536498089, 0,
0.63616452119864504507536498089, 2.03034890631683916546492372227, 3.97706091453534556119273728626, 4.51433173062679574737116522822, 5.27530886743148102824430972685, 6.03455587481448008090956717012, 7.13233823779565166897386680857, 7.66770450679970548739761403633, 9.069010246020895825733291303188
