| L(s) = 1 | − 0.0533·2-s − 3-s − 1.99·4-s − 0.797·5-s + 0.0533·6-s + 3.48·7-s + 0.213·8-s + 9-s + 0.0425·10-s − 3.03·11-s + 1.99·12-s + 0.795·13-s − 0.186·14-s + 0.797·15-s + 3.98·16-s + 17-s − 0.0533·18-s + 3.63·19-s + 1.59·20-s − 3.48·21-s + 0.161·22-s + 1.26·23-s − 0.213·24-s − 4.36·25-s − 0.0424·26-s − 27-s − 6.96·28-s + ⋯ |
| L(s) = 1 | − 0.0377·2-s − 0.577·3-s − 0.998·4-s − 0.356·5-s + 0.0217·6-s + 1.31·7-s + 0.0754·8-s + 0.333·9-s + 0.0134·10-s − 0.913·11-s + 0.576·12-s + 0.220·13-s − 0.0497·14-s + 0.206·15-s + 0.995·16-s + 0.242·17-s − 0.0125·18-s + 0.834·19-s + 0.356·20-s − 0.761·21-s + 0.0344·22-s + 0.264·23-s − 0.0435·24-s − 0.872·25-s − 0.00832·26-s − 0.192·27-s − 1.31·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 + 0.0533T + 2T^{2} \) |
| 5 | \( 1 + 0.797T + 5T^{2} \) |
| 7 | \( 1 - 3.48T + 7T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 - 0.795T + 13T^{2} \) |
| 19 | \( 1 - 3.63T + 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 + 2.04T + 29T^{2} \) |
| 31 | \( 1 + 3.15T + 31T^{2} \) |
| 37 | \( 1 + 7.53T + 37T^{2} \) |
| 41 | \( 1 + 2.37T + 41T^{2} \) |
| 43 | \( 1 + 3.88T + 43T^{2} \) |
| 47 | \( 1 + 0.578T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 - 1.61T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 2.73T + 67T^{2} \) |
| 71 | \( 1 - 5.40T + 71T^{2} \) |
| 73 | \( 1 + 9.51T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 5.66T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
| 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
−7.62215837907419616533794224497, −6.98681140863830013340259488444, −5.72020691613369008375253573142, −5.31187035849344700387092975044, −4.85167414604553818365663935585, −4.02398476234824074049198465342, −3.35117114057943912257578761229, −2.01702274824943821295345596439, −1.08939044347680917040293879943, 0,
1.08939044347680917040293879943, 2.01702274824943821295345596439, 3.35117114057943912257578761229, 4.02398476234824074049198465342, 4.85167414604553818365663935585, 5.31187035849344700387092975044, 5.72020691613369008375253573142, 6.98681140863830013340259488444, 7.62215837907419616533794224497
