| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 3·7-s − 8-s − 2·9-s − 10-s − 2·11-s − 12-s + 3·13-s + 3·14-s − 15-s + 16-s − 7·17-s + 2·18-s − 19-s + 20-s + 3·21-s + 2·22-s − 4·23-s + 24-s + 25-s − 3·26-s + 5·27-s − 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.832·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 1.69·17-s + 0.471·18-s − 0.229·19-s + 0.223·20-s + 0.654·21-s + 0.426·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.588·26-s + 0.962·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.66481326545742, −14.75962590669363, −14.35639741778292, −13.53550344327397, −13.16117160463316, −12.76690321661302, −12.12968880455167, −11.42830136720553, −11.00012869821553, −10.64649236110427, −10.07299391950868, −9.420752126242826, −8.986261967087797, −8.560408007444302, −7.871674953510756, −7.159057713953136, −6.492131641276783, −6.197784288215762, −5.733007296782252, −5.040219306297649, −4.163769323461626, −3.482830549468046, −2.675540760304686, −2.228324885759866, −1.269057172612464, 0, 0,
1.269057172612464, 2.228324885759866, 2.675540760304686, 3.482830549468046, 4.163769323461626, 5.040219306297649, 5.733007296782252, 6.197784288215762, 6.492131641276783, 7.159057713953136, 7.871674953510756, 8.560408007444302, 8.986261967087797, 9.420752126242826, 10.07299391950868, 10.64649236110427, 11.00012869821553, 11.42830136720553, 12.12968880455167, 12.76690321661302, 13.16117160463316, 13.53550344327397, 14.35639741778292, 14.75962590669363, 15.66481326545742
