| L(s) = 1 | + 1.32·2-s − 6.25·4-s − 12.9·5-s − 5.54·7-s − 18.8·8-s − 17.1·10-s + 0.841·11-s − 7.33·14-s + 25.1·16-s + 10.1·17-s + 73.0·19-s + 81.0·20-s + 1.11·22-s + 139.·23-s + 42.9·25-s + 34.6·28-s + 250.·29-s + 161.·31-s + 183.·32-s + 13.4·34-s + 71.8·35-s − 195.·37-s + 96.6·38-s + 244.·40-s − 183.·41-s + 115.·43-s − 5.26·44-s + ⋯ |
| L(s) = 1 | + 0.467·2-s − 0.781·4-s − 1.15·5-s − 0.299·7-s − 0.832·8-s − 0.541·10-s + 0.0230·11-s − 0.139·14-s + 0.392·16-s + 0.145·17-s + 0.882·19-s + 0.905·20-s + 0.0107·22-s + 1.26·23-s + 0.343·25-s + 0.233·28-s + 1.60·29-s + 0.935·31-s + 1.01·32-s + 0.0679·34-s + 0.347·35-s − 0.870·37-s + 0.412·38-s + 0.965·40-s − 0.699·41-s + 0.408·43-s − 0.0180·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.32T + 8T^{2} \) |
| 5 | \( 1 + 12.9T + 125T^{2} \) |
| 7 | \( 1 + 5.54T + 343T^{2} \) |
| 11 | \( 1 - 0.841T + 1.33e3T^{2} \) |
| 17 | \( 1 - 10.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 73.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 139.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 250.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 161.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 195.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 183.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 115.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 551.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 324.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 521.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 444.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 55.0T + 3.00e5T^{2} \) |
| 71 | \( 1 + 279.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 908.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 941.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 946.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 32.9T + 7.04e5T^{2} \) |
| 97 | \( 1 - 68.2T + 9.12e5T^{2} \) |
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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
−8.563050505981261130662808505413, −8.047302075267213315184283506073, −7.07424903932400770380906313609, −6.22775823231725285900345444244, −5.03762268417425135426783053270, −4.59434493319926792421787438844, −3.48428016512730317322997004749, −3.03002731637897479847319776591, −1.05395959182500777650808121561, 0,
1.05395959182500777650808121561, 3.03002731637897479847319776591, 3.48428016512730317322997004749, 4.59434493319926792421787438844, 5.03762268417425135426783053270, 6.22775823231725285900345444244, 7.07424903932400770380906313609, 8.047302075267213315184283506073, 8.563050505981261130662808505413
