| L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s + 7-s − 3·8-s + 9-s + 10-s − 11-s − 12-s + 6·13-s + 14-s + 15-s − 16-s + 2·17-s + 18-s + 4·19-s − 20-s + 21-s − 22-s − 4·23-s − 3·24-s + 25-s + 6·26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 1.66·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s − 0.213·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s + 1.17·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.824482658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.824482658\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−18.95846566513185, −18.55410690831292, −17.90482287072486, −17.33906983756548, −16.07820925104838, −15.72328021839119, −14.75121507171002, −14.10348112203972, −13.68753137609328, −13.11304189990101, −12.30169166835495, −11.48435481127724, −10.51328421423159, −9.698646968210728, −8.909303336785777, −8.304701526069343, −7.400205455970474, −6.054311972332621, −5.616192372994298, −4.468629861736962, −3.686973633746718, −2.757410293174219, −1.262564673649024,
1.262564673649024, 2.757410293174219, 3.686973633746718, 4.468629861736962, 5.616192372994298, 6.054311972332621, 7.400205455970474, 8.304701526069343, 8.909303336785777, 9.698646968210728, 10.51328421423159, 11.48435481127724, 12.30169166835495, 13.11304189990101, 13.68753137609328, 14.10348112203972, 14.75121507171002, 15.72328021839119, 16.07820925104838, 17.33906983756548, 17.90482287072486, 18.55410690831292, 18.95846566513185
