| L(s) = 1 | − 2.38·2-s + 0.0725·3-s + 3.68·4-s − 3.05·5-s − 0.172·6-s − 4.27·7-s − 4.01·8-s − 2.99·9-s + 7.29·10-s + 2.46·11-s + 0.267·12-s + 13-s + 10.1·14-s − 0.221·15-s + 2.19·16-s − 1.27·17-s + 7.13·18-s + 0.997·19-s − 11.2·20-s − 0.309·21-s − 5.88·22-s − 0.383·23-s − 0.291·24-s + 4.35·25-s − 2.38·26-s − 0.434·27-s − 15.7·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 0.0418·3-s + 1.84·4-s − 1.36·5-s − 0.0705·6-s − 1.61·7-s − 1.41·8-s − 0.998·9-s + 2.30·10-s + 0.743·11-s + 0.0771·12-s + 0.277·13-s + 2.72·14-s − 0.0572·15-s + 0.549·16-s − 0.308·17-s + 1.68·18-s + 0.228·19-s − 2.51·20-s − 0.0675·21-s − 1.25·22-s − 0.0799·23-s − 0.0594·24-s + 0.871·25-s − 0.467·26-s − 0.0836·27-s − 2.97·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 617 | \( 1 - T \) |
| good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 3 | \( 1 - 0.0725T + 3T^{2} \) |
| 5 | \( 1 + 3.05T + 5T^{2} \) |
| 7 | \( 1 + 4.27T + 7T^{2} \) |
| 11 | \( 1 - 2.46T + 11T^{2} \) |
| 17 | \( 1 + 1.27T + 17T^{2} \) |
| 19 | \( 1 - 0.997T + 19T^{2} \) |
| 23 | \( 1 + 0.383T + 23T^{2} \) |
| 29 | \( 1 + 4.72T + 29T^{2} \) |
| 31 | \( 1 + 3.53T + 31T^{2} \) |
| 37 | \( 1 - 6.07T + 37T^{2} \) |
| 41 | \( 1 + 1.30T + 41T^{2} \) |
| 43 | \( 1 + 9.71T + 43T^{2} \) |
| 47 | \( 1 - 9.52T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 1.91T + 59T^{2} \) |
| 61 | \( 1 + 6.27T + 61T^{2} \) |
| 67 | \( 1 + 1.81T + 67T^{2} \) |
| 71 | \( 1 + 0.711T + 71T^{2} \) |
| 73 | \( 1 - 0.825T + 73T^{2} \) |
| 79 | \( 1 - 4.55T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 0.955T + 89T^{2} \) |
| 97 | \( 1 + 1.49T + 97T^{2} \) |
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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
−7.64494769563137778118532580050, −7.04342911044428592822628984391, −6.40310881065905349660780312922, −5.84339387149340847730223127458, −4.47571944497340339061454657461, −3.48111840970206562960604542120, −3.16539901322779242377956081853, −2.02928276300796835202005747418, −0.69139683175836247634850169120, 0,
0.69139683175836247634850169120, 2.02928276300796835202005747418, 3.16539901322779242377956081853, 3.48111840970206562960604542120, 4.47571944497340339061454657461, 5.84339387149340847730223127458, 6.40310881065905349660780312922, 7.04342911044428592822628984391, 7.64494769563137778118532580050
