| L(s) = 1 | + 2.09·2-s + 0.301·3-s + 2.37·4-s − 5-s + 0.631·6-s + 7-s + 0.786·8-s − 2.90·9-s − 2.09·10-s + 3.01·11-s + 0.716·12-s + 3.92·13-s + 2.09·14-s − 0.301·15-s − 3.10·16-s − 0.774·17-s − 6.08·18-s − 4.72·19-s − 2.37·20-s + 0.301·21-s + 6.30·22-s − 6.84·23-s + 0.237·24-s + 25-s + 8.20·26-s − 1.78·27-s + 2.37·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 0.174·3-s + 1.18·4-s − 0.447·5-s + 0.257·6-s + 0.377·7-s + 0.278·8-s − 0.969·9-s − 0.661·10-s + 0.908·11-s + 0.206·12-s + 1.08·13-s + 0.559·14-s − 0.0778·15-s − 0.776·16-s − 0.187·17-s − 1.43·18-s − 1.08·19-s − 0.531·20-s + 0.0658·21-s + 1.34·22-s − 1.42·23-s + 0.0484·24-s + 0.200·25-s + 1.60·26-s − 0.343·27-s + 0.449·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 2.09T + 2T^{2} \) |
| 3 | \( 1 - 0.301T + 3T^{2} \) |
| 11 | \( 1 - 3.01T + 11T^{2} \) |
| 13 | \( 1 - 3.92T + 13T^{2} \) |
| 17 | \( 1 + 0.774T + 17T^{2} \) |
| 19 | \( 1 + 4.72T + 19T^{2} \) |
| 23 | \( 1 + 6.84T + 23T^{2} \) |
| 29 | \( 1 - 0.816T + 29T^{2} \) |
| 31 | \( 1 + 7.77T + 31T^{2} \) |
| 37 | \( 1 + 4.57T + 37T^{2} \) |
| 41 | \( 1 - 0.755T + 41T^{2} \) |
| 43 | \( 1 + 2.33T + 43T^{2} \) |
| 47 | \( 1 - 0.666T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 1.26T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 0.570T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 3.79T + 73T^{2} \) |
| 79 | \( 1 - 0.0156T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 7.49T + 89T^{2} \) |
| 97 | \( 1 + 5.32T + 97T^{2} \) |
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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
−7.28622993150173558518960162156, −6.47823980278467209233936458340, −5.98619860533153892057350795105, −5.42227379602762589891366485886, −4.47262700185122774261367946870, −3.88379416100146697279343031526, −3.49507671095900320138330105390, −2.47742911214062535427684909837, −1.64949446790465613714095700215, 0,
1.64949446790465613714095700215, 2.47742911214062535427684909837, 3.49507671095900320138330105390, 3.88379416100146697279343031526, 4.47262700185122774261367946870, 5.42227379602762589891366485886, 5.98619860533153892057350795105, 6.47823980278467209233936458340, 7.28622993150173558518960162156
