| L(s) = 1 | − 3.56·2-s + 4.68·4-s + 5.68·5-s + 7·7-s + 11.8·8-s − 20.2·10-s + 11·11-s − 39.3·13-s − 24.9·14-s − 79.5·16-s + 64.2·17-s − 82.9·19-s + 26.6·20-s − 39.1·22-s + 11.5·23-s − 92.6·25-s + 139.·26-s + 32.7·28-s − 128.·29-s − 101.·31-s + 188.·32-s − 228.·34-s + 39.7·35-s + 188.·37-s + 295.·38-s + 67.1·40-s − 198.·41-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 0.585·4-s + 0.508·5-s + 0.377·7-s + 0.521·8-s − 0.640·10-s + 0.301·11-s − 0.838·13-s − 0.475·14-s − 1.24·16-s + 0.916·17-s − 1.00·19-s + 0.297·20-s − 0.379·22-s + 0.104·23-s − 0.741·25-s + 1.05·26-s + 0.221·28-s − 0.822·29-s − 0.587·31-s + 1.04·32-s − 1.15·34-s + 0.192·35-s + 0.835·37-s + 1.26·38-s + 0.265·40-s − 0.755·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 3.56T + 8T^{2} \) |
| 5 | \( 1 - 5.68T + 125T^{2} \) |
| 13 | \( 1 + 39.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 64.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 82.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 11.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 128.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 101.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 188.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 231.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 285.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 202.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 103.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 382.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 259.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 352.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 490.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 141.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 192.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 423.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.09e3T + 9.12e5T^{2} \) |
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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
−9.531035240868883419639633291960, −8.968459757617524801808675741112, −7.920439516533049504713021588117, −7.38875864869095962304574483612, −6.22981283637926550830252299727, −5.15031104586740018230284077222, −4.03362215639515154882147189413, −2.35792862686512919898333997571, −1.38836932986094169123160992142, 0,
1.38836932986094169123160992142, 2.35792862686512919898333997571, 4.03362215639515154882147189413, 5.15031104586740018230284077222, 6.22981283637926550830252299727, 7.38875864869095962304574483612, 7.920439516533049504713021588117, 8.968459757617524801808675741112, 9.531035240868883419639633291960
