| L(s) = 1 | − 1.39·2-s − 7.96·3-s − 6.05·4-s + 5·5-s + 11.0·6-s − 16.3·7-s + 19.5·8-s + 36.3·9-s − 6.96·10-s − 11·11-s + 48.2·12-s − 57.2·13-s + 22.8·14-s − 39.8·15-s + 21.1·16-s + 87.7·17-s − 50.6·18-s + 19·19-s − 30.2·20-s + 130.·21-s + 15.3·22-s − 20.0·23-s − 155.·24-s + 25·25-s + 79.8·26-s − 74.5·27-s + 99.2·28-s + ⋯ |
| L(s) = 1 | − 0.492·2-s − 1.53·3-s − 0.757·4-s + 0.447·5-s + 0.754·6-s − 0.884·7-s + 0.865·8-s + 1.34·9-s − 0.220·10-s − 0.301·11-s + 1.16·12-s − 1.22·13-s + 0.435·14-s − 0.685·15-s + 0.330·16-s + 1.25·17-s − 0.663·18-s + 0.229·19-s − 0.338·20-s + 1.35·21-s + 0.148·22-s − 0.181·23-s − 1.32·24-s + 0.200·25-s + 0.602·26-s − 0.531·27-s + 0.669·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1673406535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1673406535\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 + 1.39T + 8T^{2} \) |
| 3 | \( 1 + 7.96T + 27T^{2} \) |
| 7 | \( 1 + 16.3T + 343T^{2} \) |
| 13 | \( 1 + 57.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 87.7T + 4.91e3T^{2} \) |
| 23 | \( 1 + 20.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 206.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 292.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 289.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 115.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 476.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 49.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 236.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 7.81T + 2.05e5T^{2} \) |
| 61 | \( 1 + 810.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 286.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 405.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 72.8T + 4.93e5T^{2} \) |
| 83 | \( 1 - 386.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 789.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 848.T + 9.12e5T^{2} \) |
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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
−9.656563893219060492728706751718, −9.082595507586659518936258344522, −7.62985998074377300808696571503, −7.12039942075174988615029825912, −5.82643330206816449665336466301, −5.46657598032004496752835488915, −4.57038448027827117743166320336, −3.32070073401375536287205693951, −1.58421380906745969511427100588, −0.25860445397092356424000860138,
0.25860445397092356424000860138, 1.58421380906745969511427100588, 3.32070073401375536287205693951, 4.57038448027827117743166320336, 5.46657598032004496752835488915, 5.82643330206816449665336466301, 7.12039942075174988615029825912, 7.62985998074377300808696571503, 9.082595507586659518936258344522, 9.656563893219060492728706751718
