| L(s) = 1 | − 2.74·2-s + 4.30·3-s − 0.453·4-s − 5·5-s − 11.8·6-s + 27.4·7-s + 23.2·8-s − 8.46·9-s + 13.7·10-s + 11·11-s − 1.95·12-s − 37.9·13-s − 75.5·14-s − 21.5·15-s − 60.1·16-s + 9.56·17-s + 23.2·18-s − 19·19-s + 2.26·20-s + 118.·21-s − 30.2·22-s + 60.7·23-s + 99.9·24-s + 25·25-s + 104.·26-s − 152.·27-s − 12.4·28-s + ⋯ |
| L(s) = 1 | − 0.971·2-s + 0.828·3-s − 0.0566·4-s − 0.447·5-s − 0.804·6-s + 1.48·7-s + 1.02·8-s − 0.313·9-s + 0.434·10-s + 0.301·11-s − 0.0469·12-s − 0.810·13-s − 1.44·14-s − 0.370·15-s − 0.940·16-s + 0.136·17-s + 0.304·18-s − 0.229·19-s + 0.0253·20-s + 1.22·21-s − 0.292·22-s + 0.550·23-s + 0.850·24-s + 0.200·25-s + 0.787·26-s − 1.08·27-s − 0.0840·28-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.74T + 8T^{2} \) |
| 3 | \( 1 - 4.30T + 27T^{2} \) |
| 7 | \( 1 - 27.4T + 343T^{2} \) |
| 13 | \( 1 + 37.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 9.56T + 4.91e3T^{2} \) |
| 23 | \( 1 - 60.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 62.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 270.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 389.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 356.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 196.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 286.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 383.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 75.5T + 2.05e5T^{2} \) |
| 61 | \( 1 - 610.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 59.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 882.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 771.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 708.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 125.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.59e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 163.T + 9.12e5T^{2} \) |
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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
−8.857092527011211356474478379216, −8.538431914825294773296850054414, −7.52774497872900502515543474593, −7.36924662217071776729896714005, −5.52611500521049442094723109691, −4.65851463261862325723387314738, −3.74210108501229945438872557577, −2.33873393760140362031789775724, −1.40113559758986855599942668267, 0,
1.40113559758986855599942668267, 2.33873393760140362031789775724, 3.74210108501229945438872557577, 4.65851463261862325723387314738, 5.52611500521049442094723109691, 7.36924662217071776729896714005, 7.52774497872900502515543474593, 8.538431914825294773296850054414, 8.857092527011211356474478379216
