| L(s) = 1 | − 2-s − 0.0937·3-s + 4-s − 1.52·5-s + 0.0937·6-s − 2.85·7-s − 8-s − 2.99·9-s + 1.52·10-s − 11-s − 0.0937·12-s + 2.76·13-s + 2.85·14-s + 0.142·15-s + 16-s − 2.26·17-s + 2.99·18-s − 1.52·20-s + 0.267·21-s + 22-s + 6.69·23-s + 0.0937·24-s − 2.68·25-s − 2.76·26-s + 0.561·27-s − 2.85·28-s + 2.52·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0541·3-s + 0.5·4-s − 0.680·5-s + 0.0382·6-s − 1.07·7-s − 0.353·8-s − 0.997·9-s + 0.481·10-s − 0.301·11-s − 0.0270·12-s + 0.767·13-s + 0.763·14-s + 0.0368·15-s + 0.250·16-s − 0.549·17-s + 0.705·18-s − 0.340·20-s + 0.0584·21-s + 0.213·22-s + 1.39·23-s + 0.0191·24-s − 0.536·25-s − 0.542·26-s + 0.108·27-s − 0.539·28-s + 0.469·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.0937T + 3T^{2} \) |
| 5 | \( 1 + 1.52T + 5T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 13 | \( 1 - 2.76T + 13T^{2} \) |
| 17 | \( 1 + 2.26T + 17T^{2} \) |
| 23 | \( 1 - 6.69T + 23T^{2} \) |
| 29 | \( 1 - 2.52T + 29T^{2} \) |
| 31 | \( 1 - 7.90T + 31T^{2} \) |
| 37 | \( 1 + 7.30T + 37T^{2} \) |
| 41 | \( 1 + 1.66T + 41T^{2} \) |
| 43 | \( 1 + 0.0856T + 43T^{2} \) |
| 47 | \( 1 + 0.776T + 47T^{2} \) |
| 53 | \( 1 - 7.66T + 53T^{2} \) |
| 59 | \( 1 + 5.95T + 59T^{2} \) |
| 61 | \( 1 - 7.34T + 61T^{2} \) |
| 67 | \( 1 + 9.72T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 5.91T + 79T^{2} \) |
| 83 | \( 1 + 5.23T + 83T^{2} \) |
| 89 | \( 1 + 1.40T + 89T^{2} \) |
| 97 | \( 1 - 6.98T + 97T^{2} \) |
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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
−7.56481273119781579510071870046, −6.76862698275136670147330163686, −6.35495871119031902391209849660, −5.56682828791883970952505243358, −4.69297739008045224626596764412, −3.59677284600238126359310376928, −3.12436660832376087133631160341, −2.31126303203157070320027228346, −0.908699257618121201767637610829, 0,
0.908699257618121201767637610829, 2.31126303203157070320027228346, 3.12436660832376087133631160341, 3.59677284600238126359310376928, 4.69297739008045224626596764412, 5.56682828791883970952505243358, 6.35495871119031902391209849660, 6.76862698275136670147330163686, 7.56481273119781579510071870046
