| L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s − 7-s − 3·8-s + 9-s + 10-s − 12-s + 2·13-s − 14-s + 15-s − 16-s − 17-s + 18-s + 4·19-s − 20-s − 21-s − 8·23-s − 3·24-s + 25-s + 2·26-s + 27-s + 28-s − 6·29-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 − 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s − 1.66·23-s − 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215985 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215985 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.913515659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.913515659\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.22061327658806, −12.54182044482978, −12.16297458559742, −11.95190617313334, −10.97022276827727, −10.80338128710207, −9.913893236607481, −9.755244802203533, −9.157090771324675, −8.940058855097284, −8.224075870882046, −7.892583201114819, −7.215120954112684, −6.706849625055799, −5.961194060871498, −5.855587154827128, −5.195030135176506, −4.685749152484572, −3.968512487477837, −3.663227562655562, −3.263844299584498, −2.466682336509135, −2.033609462889634, −1.235981921191888, −0.4135416222009074,
0.4135416222009074, 1.235981921191888, 2.033609462889634, 2.466682336509135, 3.263844299584498, 3.663227562655562, 3.968512487477837, 4.685749152484572, 5.195030135176506, 5.855587154827128, 5.961194060871498, 6.706849625055799, 7.215120954112684, 7.892583201114819, 8.224075870882046, 8.940058855097284, 9.157090771324675, 9.755244802203533, 9.913893236607481, 10.80338128710207, 10.97022276827727, 11.95190617313334, 12.16297458559742, 12.54182044482978, 13.22061327658806
