| L(s) = 1 | − 2-s − 0.326·3-s + 4-s + 0.597·5-s + 0.326·6-s + 7-s − 8-s − 2.89·9-s − 0.597·10-s − 6.29·11-s − 0.326·12-s + 2.92·13-s − 14-s − 0.195·15-s + 16-s − 4.67·17-s + 2.89·18-s + 6.26·19-s + 0.597·20-s − 0.326·21-s + 6.29·22-s + 3.24·23-s + 0.326·24-s − 4.64·25-s − 2.92·26-s + 1.92·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.188·3-s + 0.5·4-s + 0.267·5-s + 0.133·6-s + 0.377·7-s − 0.353·8-s − 0.964·9-s − 0.188·10-s − 1.89·11-s − 0.0942·12-s + 0.810·13-s − 0.267·14-s − 0.0503·15-s + 0.250·16-s − 1.13·17-s + 0.681·18-s + 1.43·19-s + 0.133·20-s − 0.0712·21-s + 1.34·22-s + 0.675·23-s + 0.0666·24-s − 0.928·25-s − 0.573·26-s + 0.370·27-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 0.326T + 3T^{2} \) |
| 5 | \( 1 - 0.597T + 5T^{2} \) |
| 11 | \( 1 + 6.29T + 11T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 17 | \( 1 + 4.67T + 17T^{2} \) |
| 19 | \( 1 - 6.26T + 19T^{2} \) |
| 23 | \( 1 - 3.24T + 23T^{2} \) |
| 29 | \( 1 - 1.70T + 29T^{2} \) |
| 31 | \( 1 + 3.86T + 31T^{2} \) |
| 37 | \( 1 - 4.62T + 37T^{2} \) |
| 41 | \( 1 - 9.20T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 - 3.88T + 47T^{2} \) |
| 53 | \( 1 + 6.94T + 53T^{2} \) |
| 59 | \( 1 + 1.43T + 59T^{2} \) |
| 61 | \( 1 + 1.56T + 61T^{2} \) |
| 67 | \( 1 + 5.47T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 7.01T + 73T^{2} \) |
| 79 | \( 1 + 1.13T + 79T^{2} \) |
| 83 | \( 1 + 2.18T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 8.33T + 97T^{2} \) |
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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
−7.87232509927956066901611390292, −7.24740006176863883064469092203, −6.22565515020334333150179763264, −5.60104131692329095517520899430, −5.14104707846357971009124756988, −4.01380729167982694158158451855, −2.78003668412882978921246539092, −2.46959228793119067685501823184, −1.14213338859200756952957641063, 0,
1.14213338859200756952957641063, 2.46959228793119067685501823184, 2.78003668412882978921246539092, 4.01380729167982694158158451855, 5.14104707846357971009124756988, 5.60104131692329095517520899430, 6.22565515020334333150179763264, 7.24740006176863883064469092203, 7.87232509927956066901611390292
