| L(s) = 1 | + 2-s − 0.0630·3-s + 4-s − 5-s − 0.0630·6-s − 2.56·7-s + 8-s − 2.99·9-s − 10-s + 11-s − 0.0630·12-s + 2.53·13-s − 2.56·14-s + 0.0630·15-s + 16-s + 7.70·17-s − 2.99·18-s − 5.99·19-s − 20-s + 0.161·21-s + 22-s − 6.26·23-s − 0.0630·24-s + 25-s + 2.53·26-s + 0.377·27-s − 2.56·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0363·3-s + 0.5·4-s − 0.447·5-s − 0.0257·6-s − 0.968·7-s + 0.353·8-s − 0.998·9-s − 0.316·10-s + 0.301·11-s − 0.0181·12-s + 0.703·13-s − 0.684·14-s + 0.0162·15-s + 0.250·16-s + 1.86·17-s − 0.706·18-s − 1.37·19-s − 0.223·20-s + 0.0352·21-s + 0.213·22-s − 1.30·23-s − 0.0128·24-s + 0.200·25-s + 0.497·26-s + 0.0727·27-s − 0.484·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 + 0.0630T + 3T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 13 | \( 1 - 2.53T + 13T^{2} \) |
| 17 | \( 1 - 7.70T + 17T^{2} \) |
| 19 | \( 1 + 5.99T + 19T^{2} \) |
| 23 | \( 1 + 6.26T + 23T^{2} \) |
| 29 | \( 1 - 2.04T + 29T^{2} \) |
| 31 | \( 1 - 5.03T + 31T^{2} \) |
| 37 | \( 1 - 0.296T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 1.91T + 43T^{2} \) |
| 47 | \( 1 + 2.78T + 47T^{2} \) |
| 53 | \( 1 - 1.03T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 2.65T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 - 5.26T + 71T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 + 1.69T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−7.50485411795597806833530158014, −6.44651876640958968446977206294, −6.10085874389935696091150597819, −5.61253734279676864367701319294, −4.49316945167251250099451401410, −3.87833820035552797383646116047, −3.17604303090631890574967582915, −2.59975860241770046505031756019, −1.28581116565976922865393402638, 0,
1.28581116565976922865393402638, 2.59975860241770046505031756019, 3.17604303090631890574967582915, 3.87833820035552797383646116047, 4.49316945167251250099451401410, 5.61253734279676864367701319294, 6.10085874389935696091150597819, 6.44651876640958968446977206294, 7.50485411795597806833530158014
