| L(s) = 1 | − 0.696·2-s + 3-s − 1.51·4-s − 2.51·5-s − 0.696·6-s − 2.32·7-s + 2.44·8-s + 9-s + 1.75·10-s − 11-s − 1.51·12-s + 1.69·13-s + 1.61·14-s − 2.51·15-s + 1.32·16-s + 7.58·17-s − 0.696·18-s − 19-s + 3.81·20-s − 2.32·21-s + 0.696·22-s + 0.248·23-s + 2.44·24-s + 1.32·25-s − 1.18·26-s + 27-s + 3.52·28-s + ⋯ |
| L(s) = 1 | − 0.492·2-s + 0.577·3-s − 0.757·4-s − 1.12·5-s − 0.284·6-s − 0.878·7-s + 0.865·8-s + 0.333·9-s + 0.553·10-s − 0.301·11-s − 0.437·12-s + 0.470·13-s + 0.432·14-s − 0.649·15-s + 0.331·16-s + 1.83·17-s − 0.164·18-s − 0.229·19-s + 0.852·20-s − 0.507·21-s + 0.148·22-s + 0.0518·23-s + 0.499·24-s + 0.265·25-s − 0.231·26-s + 0.192·27-s + 0.665·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8537836998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8537836998\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.696T + 2T^{2} \) |
| 5 | \( 1 + 2.51T + 5T^{2} \) |
| 7 | \( 1 + 2.32T + 7T^{2} \) |
| 13 | \( 1 - 1.69T + 13T^{2} \) |
| 17 | \( 1 - 7.58T + 17T^{2} \) |
| 23 | \( 1 - 0.248T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 2.07T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 - 4.59T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 - 2.84T + 47T^{2} \) |
| 53 | \( 1 + 1.65T + 53T^{2} \) |
| 59 | \( 1 - 1.06T + 59T^{2} \) |
| 61 | \( 1 - 5.30T + 61T^{2} \) |
| 67 | \( 1 - 0.303T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 1.81T + 79T^{2} \) |
| 83 | \( 1 - 9.16T + 83T^{2} \) |
| 89 | \( 1 + 6.59T + 89T^{2} \) |
| 97 | \( 1 - 8.72T + 97T^{2} \) |
| 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
−10.14214238403987079433163187087, −9.855546284893027412325116176316, −8.704974591706763500573945728239, −8.079307424140824278202677101547, −7.49418691152323461639661794478, −6.22428184642417359821541244566, −4.83842175090924746385875659925, −3.83473079939382034289542235976, −3.05576065421609926345454169925, −0.860472693055647410967144995114,
0.860472693055647410967144995114, 3.05576065421609926345454169925, 3.83473079939382034289542235976, 4.83842175090924746385875659925, 6.22428184642417359821541244566, 7.49418691152323461639661794478, 8.079307424140824278202677101547, 8.704974591706763500573945728239, 9.855546284893027412325116176316, 10.14214238403987079433163187087
