L(s) = 1 | + 2-s + 4-s − 2·5-s + 7-s + 8-s − 2·10-s + 2·13-s + 14-s + 16-s − 17-s + 8·19-s − 2·20-s − 8·23-s − 25-s + 2·26-s + 28-s − 2·29-s − 4·31-s + 32-s − 34-s − 2·35-s + 6·37-s + 8·38-s − 2·40-s − 10·41-s + 4·43-s − 8·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 0.632·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.83·19-s − 0.447·20-s − 1.66·23-s − 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.371·29-s − 0.718·31-s + 0.176·32-s − 0.171·34-s − 0.338·35-s + 0.986·37-s + 1.29·38-s − 0.316·40-s − 1.56·41-s + 0.609·43-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.27101188749962, −12.40107240193375, −12.05993941435635, −11.74723300832036, −11.44215134983126, −10.80516195941669, −10.51716001938913, −9.778852912774661, −9.428161000070805, −8.790310429930686, −8.159652598453441, −7.905681841271143, −7.281668460722490, −7.146478729406596, −6.276695124106920, −5.757125415745089, −5.510571517963980, −4.804811704454718, −4.250368942927149, −3.837583173703587, −3.473610319648311, −2.817230901372638, −2.149660400386703, −1.532341618639478, −0.8535583170031585, 0,
0.8535583170031585, 1.532341618639478, 2.149660400386703, 2.817230901372638, 3.473610319648311, 3.837583173703587, 4.250368942927149, 4.804811704454718, 5.510571517963980, 5.757125415745089, 6.276695124106920, 7.146478729406596, 7.281668460722490, 7.905681841271143, 8.159652598453441, 8.790310429930686, 9.428161000070805, 9.778852912774661, 10.51716001938913, 10.80516195941669, 11.44215134983126, 11.74723300832036, 12.05993941435635, 12.40107240193375, 13.27101188749962