| L(s) = 1 | + 2-s + 0.618·3-s + 4-s + 5-s + 0.618·6-s − 5.23·7-s + 8-s − 2.61·9-s + 10-s − 1.76·11-s + 0.618·12-s − 5.47·13-s − 5.23·14-s + 0.618·15-s + 16-s + 5.61·17-s − 2.61·18-s + 4.61·19-s + 20-s − 3.23·21-s − 1.76·22-s − 2.76·23-s + 0.618·24-s − 4·25-s − 5.47·26-s − 3.47·27-s − 5.23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.356·3-s + 0.5·4-s + 0.447·5-s + 0.252·6-s − 1.97·7-s + 0.353·8-s − 0.872·9-s + 0.316·10-s − 0.531·11-s + 0.178·12-s − 1.51·13-s − 1.39·14-s + 0.159·15-s + 0.250·16-s + 1.36·17-s − 0.617·18-s + 1.05·19-s + 0.223·20-s − 0.706·21-s − 0.376·22-s − 0.576·23-s + 0.126·24-s − 0.800·25-s − 1.07·26-s − 0.668·27-s − 0.989·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 5.23T + 7T^{2} \) |
| 11 | \( 1 + 1.76T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 - 5.61T + 17T^{2} \) |
| 19 | \( 1 - 4.61T + 19T^{2} \) |
| 23 | \( 1 + 2.76T + 23T^{2} \) |
| 29 | \( 1 + 5.76T + 29T^{2} \) |
| 31 | \( 1 - 4.70T + 31T^{2} \) |
| 37 | \( 1 + 8.85T + 37T^{2} \) |
| 41 | \( 1 + 2.70T + 41T^{2} \) |
| 43 | \( 1 + 8.23T + 43T^{2} \) |
| 47 | \( 1 + 4.85T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 1.23T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 - 6.23T + 71T^{2} \) |
| 73 | \( 1 - 8.61T + 73T^{2} \) |
| 79 | \( 1 + 7.38T + 79T^{2} \) |
| 83 | \( 1 + 1.47T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−9.474273155281802719738019381045, −8.126490677066377407169094439015, −7.35271424119939252870992267135, −6.51847578805526254439374360720, −5.65591276353402806588494664944, −5.16864859358646200569749557077, −3.51655297847361956486936928176, −3.15132089542222290942687491943, −2.20893035615662922830076136590, 0,
2.20893035615662922830076136590, 3.15132089542222290942687491943, 3.51655297847361956486936928176, 5.16864859358646200569749557077, 5.65591276353402806588494664944, 6.51847578805526254439374360720, 7.35271424119939252870992267135, 8.126490677066377407169094439015, 9.474273155281802719738019381045
