| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 2·7-s − 8-s − 2·9-s − 10-s + 12-s + 5·13-s − 2·14-s + 15-s + 16-s − 17-s + 2·18-s − 19-s + 20-s + 2·21-s + 6·23-s − 24-s + 25-s − 5·26-s − 5·27-s + 2·28-s − 9·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.288·12-s + 1.38·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.471·18-s − 0.229·19-s + 0.223·20-s + 0.436·21-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.980·26-s − 0.962·27-s + 0.377·28-s − 1.67·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.109702059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.109702059\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−12.88999568022307419894056096339, −11.31002289164133692017183026949, −10.94841428782720554477702180822, −9.477504028752912702082700733771, −8.698484614188417794114959369886, −7.958366536731621540507126793914, −6.57312904074948950648190208673, −5.32145208242139248244240386570, −3.40344370945167659881716597982, −1.79579054758907773709917280330,
1.79579054758907773709917280330, 3.40344370945167659881716597982, 5.32145208242139248244240386570, 6.57312904074948950648190208673, 7.958366536731621540507126793914, 8.698484614188417794114959369886, 9.477504028752912702082700733771, 10.94841428782720554477702180822, 11.31002289164133692017183026949, 12.88999568022307419894056096339
