| L(s) = 1 | − 2.11·2-s + 2.48·4-s − 2.68·5-s + 0.972·7-s − 1.01·8-s + 5.67·10-s − 0.316·11-s − 1.51·13-s − 2.05·14-s − 2.80·16-s + 1.17·17-s + 6.22·19-s − 6.65·20-s + 0.670·22-s + 2.16·23-s + 2.19·25-s + 3.20·26-s + 2.41·28-s − 4.40·29-s − 8.67·31-s + 7.97·32-s − 2.48·34-s − 2.60·35-s − 4.46·37-s − 13.1·38-s + 2.72·40-s + 5.84·41-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.24·4-s − 1.19·5-s + 0.367·7-s − 0.359·8-s + 1.79·10-s − 0.0955·11-s − 0.419·13-s − 0.550·14-s − 0.702·16-s + 0.284·17-s + 1.42·19-s − 1.48·20-s + 0.143·22-s + 0.450·23-s + 0.439·25-s + 0.628·26-s + 0.455·28-s − 0.817·29-s − 1.55·31-s + 1.41·32-s − 0.426·34-s − 0.440·35-s − 0.734·37-s − 2.13·38-s + 0.431·40-s + 0.912·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 2.11T + 2T^{2} \) |
| 5 | \( 1 + 2.68T + 5T^{2} \) |
| 7 | \( 1 - 0.972T + 7T^{2} \) |
| 11 | \( 1 + 0.316T + 11T^{2} \) |
| 13 | \( 1 + 1.51T + 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 19 | \( 1 - 6.22T + 19T^{2} \) |
| 23 | \( 1 - 2.16T + 23T^{2} \) |
| 29 | \( 1 + 4.40T + 29T^{2} \) |
| 31 | \( 1 + 8.67T + 31T^{2} \) |
| 37 | \( 1 + 4.46T + 37T^{2} \) |
| 41 | \( 1 - 5.84T + 41T^{2} \) |
| 43 | \( 1 - 5.59T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 1.72T + 59T^{2} \) |
| 61 | \( 1 - 1.01T + 61T^{2} \) |
| 67 | \( 1 - 0.856T + 67T^{2} \) |
| 71 | \( 1 + 9.59T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 4.68T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 5.54T + 97T^{2} \) |
| 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
−9.739283871366419286669688923613, −9.139227218070714036981462880339, −8.190764938875812809500694971431, −7.51735644548264790221693832423, −7.14388856231166287824695764269, −5.53051695146413229094843262166, −4.35960612079513948442785350673, −3.11748094666312729388666379753, −1.49984527792155789246925648260, 0,
1.49984527792155789246925648260, 3.11748094666312729388666379753, 4.35960612079513948442785350673, 5.53051695146413229094843262166, 7.14388856231166287824695764269, 7.51735644548264790221693832423, 8.190764938875812809500694971431, 9.139227218070714036981462880339, 9.739283871366419286669688923613
