| L(s) = 1 | − 1.91·2-s + 1.65·4-s − 3.25·5-s − 3.82·7-s + 0.656·8-s + 6.22·10-s − 1.31·11-s − 3.31·13-s + 7.31·14-s − 4.56·16-s − 5.13·17-s + 5.48·19-s − 5.39·20-s + 2.51·22-s + 4·23-s + 5.59·25-s + 6.33·26-s − 6.33·28-s − 4.59·29-s + 4·31-s + 7.42·32-s + 9.82·34-s + 12.4·35-s − 5.65·37-s − 10.4·38-s − 2.13·40-s + 9.64·41-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.828·4-s − 1.45·5-s − 1.44·7-s + 0.232·8-s + 1.96·10-s − 0.395·11-s − 0.918·13-s + 1.95·14-s − 1.14·16-s − 1.24·17-s + 1.25·19-s − 1.20·20-s + 0.535·22-s + 0.834·23-s + 1.11·25-s + 1.24·26-s − 1.19·28-s − 0.854·29-s + 0.718·31-s + 1.31·32-s + 1.68·34-s + 2.10·35-s − 0.929·37-s − 1.70·38-s − 0.337·40-s + 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2436583954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2436583954\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 + 1.91T + 2T^{2} \) |
| 5 | \( 1 + 3.25T + 5T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 11 | \( 1 + 1.31T + 11T^{2} \) |
| 13 | \( 1 + 3.31T + 13T^{2} \) |
| 17 | \( 1 + 5.13T + 17T^{2} \) |
| 19 | \( 1 - 5.48T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 4.59T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 5.65T + 37T^{2} \) |
| 41 | \( 1 - 9.64T + 41T^{2} \) |
| 43 | \( 1 - 1.43T + 43T^{2} \) |
| 47 | \( 1 + 7.13T + 47T^{2} \) |
| 53 | \( 1 - 8.62T + 53T^{2} \) |
| 59 | \( 1 - 4.51T + 59T^{2} \) |
| 61 | \( 1 + 3.64T + 61T^{2} \) |
| 67 | \( 1 + 5.31T + 67T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 - 5.08T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 3.22T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−10.42203499405892753343249522909, −9.506328799106680047107312272059, −9.010557068323536213873007641443, −7.927870934091078283875683935221, −7.31124187205206798321064111315, −6.64805170422585348217990203977, −4.98073530970822826741396555826, −3.80473500417618946989325561905, −2.65553038912858336648600127014, −0.49134239846547884008346021973,
0.49134239846547884008346021973, 2.65553038912858336648600127014, 3.80473500417618946989325561905, 4.98073530970822826741396555826, 6.64805170422585348217990203977, 7.31124187205206798321064111315, 7.927870934091078283875683935221, 9.010557068323536213873007641443, 9.506328799106680047107312272059, 10.42203499405892753343249522909
