| L(s) = 1 | + 2-s + 2·3-s − 4-s − 5-s + 2·6-s − 2·7-s − 3·8-s + 9-s − 10-s + 2·11-s − 2·12-s + 2·13-s − 2·14-s − 2·15-s − 16-s + 17-s + 18-s + 20-s − 4·21-s + 2·22-s + 6·23-s − 6·24-s + 25-s + 2·26-s − 4·27-s + 2·28-s − 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s − 0.577·12-s + 0.554·13-s − 0.534·14-s − 0.516·15-s − 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.223·20-s − 0.872·21-s + 0.426·22-s + 1.25·23-s − 1.22·24-s + 1/5·25-s + 0.392·26-s − 0.769·27-s + 0.377·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.397692218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.397692218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.32499099465889905092766203697, −13.23869509800644511047101744766, −12.61695801969404065473449905589, −11.16926979542542511914194656047, −9.342260097121774077335621975199, −8.918128413786971033454816799413, −7.46613087643891033346359456413, −5.83398257972728633601666952176, −4.03297779911247219461657332614, −3.13757525842324917581154074415,
3.13757525842324917581154074415, 4.03297779911247219461657332614, 5.83398257972728633601666952176, 7.46613087643891033346359456413, 8.918128413786971033454816799413, 9.342260097121774077335621975199, 11.16926979542542511914194656047, 12.61695801969404065473449905589, 13.23869509800644511047101744766, 14.32499099465889905092766203697
