| L(s) = 1 | + 2-s − 0.120·3-s + 4-s + 3.54·5-s − 0.120·6-s + 7-s + 8-s − 2.98·9-s + 3.54·10-s − 2.55·11-s − 0.120·12-s + 4.07·13-s + 14-s − 0.429·15-s + 16-s − 2.26·17-s − 2.98·18-s + 3.11·19-s + 3.54·20-s − 0.120·21-s − 2.55·22-s + 3.66·23-s − 0.120·24-s + 7.59·25-s + 4.07·26-s + 0.724·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0698·3-s + 0.5·4-s + 1.58·5-s − 0.0493·6-s + 0.377·7-s + 0.353·8-s − 0.995·9-s + 1.12·10-s − 0.771·11-s − 0.0349·12-s + 1.13·13-s + 0.267·14-s − 0.110·15-s + 0.250·16-s − 0.549·17-s − 0.703·18-s + 0.714·19-s + 0.793·20-s − 0.0263·21-s − 0.545·22-s + 0.763·23-s − 0.0246·24-s + 1.51·25-s + 0.799·26-s + 0.139·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.419794978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.419794978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 0.120T + 3T^{2} \) |
| 5 | \( 1 - 3.54T + 5T^{2} \) |
| 11 | \( 1 + 2.55T + 11T^{2} \) |
| 13 | \( 1 - 4.07T + 13T^{2} \) |
| 17 | \( 1 + 2.26T + 17T^{2} \) |
| 19 | \( 1 - 3.11T + 19T^{2} \) |
| 23 | \( 1 - 3.66T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 - 3.09T + 31T^{2} \) |
| 37 | \( 1 + 2.36T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 7.11T + 43T^{2} \) |
| 47 | \( 1 - 3.96T + 47T^{2} \) |
| 53 | \( 1 + 0.0703T + 53T^{2} \) |
| 59 | \( 1 + 8.20T + 59T^{2} \) |
| 61 | \( 1 - 7.87T + 61T^{2} \) |
| 67 | \( 1 - 5.07T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 4.93T + 73T^{2} \) |
| 79 | \( 1 + 1.75T + 79T^{2} \) |
| 83 | \( 1 - 6.99T + 83T^{2} \) |
| 89 | \( 1 - 18.6T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.992401413414555803942420468944, −7.26934131891104755333746730438, −6.27644176435924115295483801910, −5.83402269072202676573298794017, −5.40343782231533716338111362074, −4.66346778494982967329016801556, −3.55466425080383783686098075727, −2.68489622858990448076115767745, −2.10805309701781971863329948420, −1.03710560726725679360921202869,
1.03710560726725679360921202869, 2.10805309701781971863329948420, 2.68489622858990448076115767745, 3.55466425080383783686098075727, 4.66346778494982967329016801556, 5.40343782231533716338111362074, 5.83402269072202676573298794017, 6.27644176435924115295483801910, 7.26934131891104755333746730438, 7.992401413414555803942420468944
