| L(s) = 1 | − 0.618·2-s − 3-s − 1.61·4-s − 1.61·5-s + 0.618·6-s + 7-s + 2.23·8-s + 9-s + 1.00·10-s + 1.61·12-s − 4.23·13-s − 0.618·14-s + 1.61·15-s + 1.85·16-s + 3.47·17-s − 0.618·18-s − 6.47·19-s + 2.61·20-s − 21-s + 5.70·23-s − 2.23·24-s − 2.38·25-s + 2.61·26-s − 27-s − 1.61·28-s + 0.236·29-s − 1.00·30-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.577·3-s − 0.809·4-s − 0.723·5-s + 0.252·6-s + 0.377·7-s + 0.790·8-s + 0.333·9-s + 0.316·10-s + 0.467·12-s − 1.17·13-s − 0.165·14-s + 0.417·15-s + 0.463·16-s + 0.842·17-s − 0.145·18-s − 1.48·19-s + 0.585·20-s − 0.218·21-s + 1.19·23-s − 0.456·24-s − 0.476·25-s + 0.513·26-s − 0.192·27-s − 0.305·28-s + 0.0438·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 - 3.47T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 - 5.70T + 23T^{2} \) |
| 29 | \( 1 - 0.236T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 9.85T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 5.14T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 4.61T + 59T^{2} \) |
| 61 | \( 1 - 7.94T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 4.52T + 71T^{2} \) |
| 73 | \( 1 + 7.38T + 73T^{2} \) |
| 79 | \( 1 - 6.09T + 79T^{2} \) |
| 83 | \( 1 + 8.47T + 83T^{2} \) |
| 89 | \( 1 - 1.32T + 89T^{2} \) |
| 97 | \( 1 + 5.29T + 97T^{2} \) |
| 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
−8.409203886967204736861029989017, −7.81136190396394545685134643735, −7.26875613307984849339412334252, −6.18036841790961658661934997543, −5.22463236922030432296497100513, −4.53224967939523181058791409751, −3.97147462811103427462741779528, −2.59403428738265541736244246140, −1.11951291299399668742993901424, 0,
1.11951291299399668742993901424, 2.59403428738265541736244246140, 3.97147462811103427462741779528, 4.53224967939523181058791409751, 5.22463236922030432296497100513, 6.18036841790961658661934997543, 7.26875613307984849339412334252, 7.81136190396394545685134643735, 8.409203886967204736861029989017
