L(s) = 1 | − 2-s + 1.63·3-s + 4-s + 1.33·5-s − 1.63·6-s − 2.68·7-s − 8-s − 0.316·9-s − 1.33·10-s − 0.0998·11-s + 1.63·12-s + 2.79·13-s + 2.68·14-s + 2.18·15-s + 16-s − 1.00·17-s + 0.316·18-s − 3.85·19-s + 1.33·20-s − 4.39·21-s + 0.0998·22-s + 4.15·23-s − 1.63·24-s − 3.21·25-s − 2.79·26-s − 5.43·27-s − 2.68·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.945·3-s + 0.5·4-s + 0.597·5-s − 0.668·6-s − 1.01·7-s − 0.353·8-s − 0.105·9-s − 0.422·10-s − 0.0301·11-s + 0.472·12-s + 0.775·13-s + 0.716·14-s + 0.565·15-s + 0.250·16-s − 0.242·17-s + 0.0746·18-s − 0.883·19-s + 0.298·20-s − 0.958·21-s + 0.0212·22-s + 0.866·23-s − 0.334·24-s − 0.642·25-s − 0.548·26-s − 1.04·27-s − 0.506·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
good | 3 | \( 1 - 1.63T + 3T^{2} \) |
| 5 | \( 1 - 1.33T + 5T^{2} \) |
| 7 | \( 1 + 2.68T + 7T^{2} \) |
| 11 | \( 1 + 0.0998T + 11T^{2} \) |
| 13 | \( 1 - 2.79T + 13T^{2} \) |
| 17 | \( 1 + 1.00T + 17T^{2} \) |
| 19 | \( 1 + 3.85T + 19T^{2} \) |
| 23 | \( 1 - 4.15T + 23T^{2} \) |
| 29 | \( 1 + 1.57T + 29T^{2} \) |
| 31 | \( 1 - 0.852T + 31T^{2} \) |
| 37 | \( 1 + 0.546T + 37T^{2} \) |
| 41 | \( 1 + 9.83T + 41T^{2} \) |
| 43 | \( 1 + 3.13T + 43T^{2} \) |
| 47 | \( 1 - 7.14T + 47T^{2} \) |
| 53 | \( 1 + 9.47T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 3.67T + 67T^{2} \) |
| 71 | \( 1 + 3.23T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 8.88T + 79T^{2} \) |
| 83 | \( 1 - 8.88T + 83T^{2} \) |
| 89 | \( 1 + 7.10T + 89T^{2} \) |
| 97 | \( 1 - 3.56T + 97T^{2} \) |
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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
−8.348732589216566506423158273711, −7.50186388059765615349784448745, −6.58060671396581286663829309652, −6.18202237349112132365047699718, −5.23869891292638001477798510959, −3.90165649306640179584898553626, −3.20563551145491891656013503673, −2.44509590814916244087774707899, −1.56299380438736006838142979642, 0,
1.56299380438736006838142979642, 2.44509590814916244087774707899, 3.20563551145491891656013503673, 3.90165649306640179584898553626, 5.23869891292638001477798510959, 6.18202237349112132365047699718, 6.58060671396581286663829309652, 7.50186388059765615349784448745, 8.348732589216566506423158273711