| L(s) = 1 | + 1.66·2-s + 3-s + 0.759·4-s − 5-s + 1.66·6-s − 2.14·7-s − 2.06·8-s + 9-s − 1.66·10-s + 0.485·11-s + 0.759·12-s + 4.60·13-s − 3.56·14-s − 15-s − 4.94·16-s + 2.19·17-s + 1.66·18-s − 5.64·19-s − 0.759·20-s − 2.14·21-s + 0.805·22-s + 1.78·23-s − 2.06·24-s + 25-s + 7.64·26-s + 27-s − 1.62·28-s + ⋯ |
| L(s) = 1 | + 1.17·2-s + 0.577·3-s + 0.379·4-s − 0.447·5-s + 0.678·6-s − 0.810·7-s − 0.728·8-s + 0.333·9-s − 0.525·10-s + 0.146·11-s + 0.219·12-s + 1.27·13-s − 0.952·14-s − 0.258·15-s − 1.23·16-s + 0.531·17-s + 0.391·18-s − 1.29·19-s − 0.169·20-s − 0.468·21-s + 0.171·22-s + 0.371·23-s − 0.420·24-s + 0.200·25-s + 1.49·26-s + 0.192·27-s − 0.307·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 1.66T + 2T^{2} \) |
| 7 | \( 1 + 2.14T + 7T^{2} \) |
| 11 | \( 1 - 0.485T + 11T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 17 | \( 1 - 2.19T + 17T^{2} \) |
| 19 | \( 1 + 5.64T + 19T^{2} \) |
| 23 | \( 1 - 1.78T + 23T^{2} \) |
| 29 | \( 1 + 4.85T + 29T^{2} \) |
| 31 | \( 1 + 1.31T + 31T^{2} \) |
| 37 | \( 1 + 3.50T + 37T^{2} \) |
| 41 | \( 1 - 4.65T + 41T^{2} \) |
| 43 | \( 1 - 6.42T + 43T^{2} \) |
| 47 | \( 1 - 1.39T + 47T^{2} \) |
| 53 | \( 1 + 9.14T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 5.78T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 8.77T + 71T^{2} \) |
| 73 | \( 1 - 7.66T + 73T^{2} \) |
| 79 | \( 1 + 7.56T + 79T^{2} \) |
| 83 | \( 1 + 0.691T + 83T^{2} \) |
| 89 | \( 1 + 9.79T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−7.66917706246260668289466323843, −6.80999410420031145044121882586, −6.14905137821018322181605582189, −5.65981546979263136340607185046, −4.50856159705638181320268011393, −4.04984041224048216789678273269, −3.33113954951673031438636632154, −2.84443089316374841437626030468, −1.56419486972949319416557392905, 0,
1.56419486972949319416557392905, 2.84443089316374841437626030468, 3.33113954951673031438636632154, 4.04984041224048216789678273269, 4.50856159705638181320268011393, 5.65981546979263136340607185046, 6.14905137821018322181605582189, 6.80999410420031145044121882586, 7.66917706246260668289466323843
