| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 11-s − 12-s + 13-s − 15-s + 16-s + 17-s + 18-s − 19-s + 20-s − 22-s − 24-s + 25-s + 26-s − 27-s − 8·29-s − 30-s − 5·31-s + 32-s + 33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.213·22-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 1.48·29-s − 0.182·30-s − 0.898·31-s + 0.176·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.283204379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.283204379\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 11 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−12.74749140394655, −12.14056183544489, −11.71880177629329, −11.32608177156449, −10.92491489905161, −10.27957464606218, −10.15957263886031, −9.493701032570117, −8.981761429146479, −8.474202853459403, −7.879524979838981, −7.388971373143921, −6.904822403755674, −6.466912826615296, −5.978613218349128, −5.477926829006351, −5.111927853958157, −4.762760141693671, −3.838822730886797, −3.681925959423324, −3.040577683247121, −2.230143782203189, −1.848356389649970, −1.260767374135582, −0.3523556478233265,
0.3523556478233265, 1.260767374135582, 1.848356389649970, 2.230143782203189, 3.040577683247121, 3.681925959423324, 3.838822730886797, 4.762760141693671, 5.111927853958157, 5.477926829006351, 5.978613218349128, 6.466912826615296, 6.904822403755674, 7.388971373143921, 7.879524979838981, 8.474202853459403, 8.981761429146479, 9.493701032570117, 10.15957263886031, 10.27957464606218, 10.92491489905161, 11.32608177156449, 11.71880177629329, 12.14056183544489, 12.74749140394655
