L(s) = 1 | + 2-s − 4-s + 2·5-s − 4·7-s − 3·8-s + 2·10-s + 2·13-s − 4·14-s − 16-s − 2·17-s − 2·20-s − 8·23-s − 25-s + 2·26-s + 4·28-s − 6·29-s − 8·31-s + 5·32-s − 2·34-s − 8·35-s + 6·37-s − 6·40-s − 2·41-s − 8·46-s − 8·47-s + 9·49-s − 50-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.51·7-s − 1.06·8-s + 0.632·10-s + 0.554·13-s − 1.06·14-s − 1/4·16-s − 0.485·17-s − 0.447·20-s − 1.66·23-s − 1/5·25-s + 0.392·26-s + 0.755·28-s − 1.11·29-s − 1.43·31-s + 0.883·32-s − 0.342·34-s − 1.35·35-s + 0.986·37-s − 0.948·40-s − 0.312·41-s − 1.17·46-s − 1.16·47-s + 9/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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
−9.531112831864384789210057232688, −8.938244517857001424659052468572, −7.77744707656776841753907290662, −6.38866021531112361683475479825, −6.11982556539470572208453322290, −5.25909302550772251578024339366, −3.98994534926773298366607994601, −3.36662241945177810035682717763, −2.09666333780189433867543562159, 0,
2.09666333780189433867543562159, 3.36662241945177810035682717763, 3.98994534926773298366607994601, 5.25909302550772251578024339366, 6.11982556539470572208453322290, 6.38866021531112361683475479825, 7.77744707656776841753907290662, 8.938244517857001424659052468572, 9.531112831864384789210057232688