| L(s) = 1 | − 1.77·2-s − 3.40·3-s + 1.14·4-s − 3.60·5-s + 6.03·6-s + 0.417·7-s + 1.52·8-s + 8.60·9-s + 6.38·10-s + 11-s − 3.89·12-s − 4.37·13-s − 0.739·14-s + 12.2·15-s − 4.97·16-s + 0.423·17-s − 15.2·18-s + 1.68·19-s − 4.11·20-s − 1.42·21-s − 1.77·22-s − 3.07·23-s − 5.17·24-s + 7.98·25-s + 7.75·26-s − 19.0·27-s + 0.476·28-s + ⋯ |
| L(s) = 1 | − 1.25·2-s − 1.96·3-s + 0.571·4-s − 1.61·5-s + 2.46·6-s + 0.157·7-s + 0.537·8-s + 2.86·9-s + 2.01·10-s + 0.301·11-s − 1.12·12-s − 1.21·13-s − 0.197·14-s + 3.16·15-s − 1.24·16-s + 0.102·17-s − 3.59·18-s + 0.386·19-s − 0.920·20-s − 0.309·21-s − 0.377·22-s − 0.640·23-s − 1.05·24-s + 1.59·25-s + 1.52·26-s − 3.67·27-s + 0.0900·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08789844724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08789844724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 3 | \( 1 + 3.40T + 3T^{2} \) |
| 5 | \( 1 + 3.60T + 5T^{2} \) |
| 7 | \( 1 - 0.417T + 7T^{2} \) |
| 13 | \( 1 + 4.37T + 13T^{2} \) |
| 17 | \( 1 - 0.423T + 17T^{2} \) |
| 19 | \( 1 - 1.68T + 19T^{2} \) |
| 23 | \( 1 + 3.07T + 23T^{2} \) |
| 29 | \( 1 - 5.87T + 29T^{2} \) |
| 31 | \( 1 + 7.64T + 31T^{2} \) |
| 37 | \( 1 + 9.81T + 37T^{2} \) |
| 41 | \( 1 + 7.64T + 41T^{2} \) |
| 43 | \( 1 + 9.71T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 4.51T + 53T^{2} \) |
| 59 | \( 1 + 6.56T + 59T^{2} \) |
| 61 | \( 1 - 3.04T + 61T^{2} \) |
| 67 | \( 1 - 6.16T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 1.92T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 7.15T + 89T^{2} \) |
| 97 | \( 1 - 2.96T + 97T^{2} \) |
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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
−9.818102093473677280173413325679, −8.684868999369537863007131013273, −7.74403530237982791643999083302, −7.20777890597808188390414540234, −6.65118261233994420991102522438, −5.23780822877071149648386923633, −4.68144990182682518939562033895, −3.77684262006355554878339918701, −1.55687676074105032825732241968, −0.29498918348361619142796068272,
0.29498918348361619142796068272, 1.55687676074105032825732241968, 3.77684262006355554878339918701, 4.68144990182682518939562033895, 5.23780822877071149648386923633, 6.65118261233994420991102522438, 7.20777890597808188390414540234, 7.74403530237982791643999083302, 8.684868999369537863007131013273, 9.818102093473677280173413325679
