| L(s) = 1 | − 0.481·2-s + 3-s − 1.76·4-s − 2.37·5-s − 0.481·6-s − 1.68·7-s + 1.81·8-s + 9-s + 1.14·10-s + 0.580·11-s − 1.76·12-s − 1.83·13-s + 0.811·14-s − 2.37·15-s + 2.66·16-s + 17-s − 0.481·18-s − 5.72·19-s + 4.19·20-s − 1.68·21-s − 0.279·22-s + 8.71·23-s + 1.81·24-s + 0.636·25-s + 0.885·26-s + 27-s + 2.97·28-s + ⋯ |
| L(s) = 1 | − 0.340·2-s + 0.577·3-s − 0.884·4-s − 1.06·5-s − 0.196·6-s − 0.636·7-s + 0.641·8-s + 0.333·9-s + 0.361·10-s + 0.174·11-s − 0.510·12-s − 0.509·13-s + 0.216·14-s − 0.612·15-s + 0.665·16-s + 0.242·17-s − 0.113·18-s − 1.31·19-s + 0.938·20-s − 0.367·21-s − 0.0595·22-s + 1.81·23-s + 0.370·24-s + 0.127·25-s + 0.173·26-s + 0.192·27-s + 0.562·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 0.481T + 2T^{2} \) |
| 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 + 1.68T + 7T^{2} \) |
| 11 | \( 1 - 0.580T + 11T^{2} \) |
| 13 | \( 1 + 1.83T + 13T^{2} \) |
| 19 | \( 1 + 5.72T + 19T^{2} \) |
| 23 | \( 1 - 8.71T + 23T^{2} \) |
| 29 | \( 1 + 3.19T + 29T^{2} \) |
| 31 | \( 1 - 3.03T + 31T^{2} \) |
| 37 | \( 1 + 6.02T + 37T^{2} \) |
| 41 | \( 1 - 4.69T + 41T^{2} \) |
| 43 | \( 1 - 7.91T + 43T^{2} \) |
| 47 | \( 1 - 0.490T + 47T^{2} \) |
| 53 | \( 1 + 3.47T + 53T^{2} \) |
| 59 | \( 1 - 6.19T + 59T^{2} \) |
| 61 | \( 1 - 6.44T + 61T^{2} \) |
| 67 | \( 1 + 3.79T + 67T^{2} \) |
| 71 | \( 1 + 1.53T + 71T^{2} \) |
| 73 | \( 1 - 0.320T + 73T^{2} \) |
| 79 | \( 1 - 9.23T + 79T^{2} \) |
| 83 | \( 1 - 9.23T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.58482728332991233444914076989, −7.08854958452595063286826346746, −6.24644924907581277502571487345, −5.16689569987434561699131486870, −4.52922566269732401217127496668, −3.83609426788868777685495509938, −3.30631707030067875712502998819, −2.30422169697606517099159892772, −0.980763858503870124607410682785, 0,
0.980763858503870124607410682785, 2.30422169697606517099159892772, 3.30631707030067875712502998819, 3.83609426788868777685495509938, 4.52922566269732401217127496668, 5.16689569987434561699131486870, 6.24644924907581277502571487345, 7.08854958452595063286826346746, 7.58482728332991233444914076989
