| L(s) = 1 | + 2-s − 2.07·3-s + 4-s − 3.66·5-s − 2.07·6-s − 3.45·7-s + 8-s + 1.31·9-s − 3.66·10-s − 0.0411·11-s − 2.07·12-s − 1.28·13-s − 3.45·14-s + 7.62·15-s + 16-s − 3.99·17-s + 1.31·18-s + 4.34·19-s − 3.66·20-s + 7.17·21-s − 0.0411·22-s − 2.81·23-s − 2.07·24-s + 8.46·25-s − 1.28·26-s + 3.49·27-s − 3.45·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.19·3-s + 0.5·4-s − 1.64·5-s − 0.848·6-s − 1.30·7-s + 0.353·8-s + 0.439·9-s − 1.16·10-s − 0.0124·11-s − 0.599·12-s − 0.357·13-s − 0.922·14-s + 1.96·15-s + 0.250·16-s − 0.968·17-s + 0.310·18-s + 0.996·19-s − 0.820·20-s + 1.56·21-s − 0.00877·22-s − 0.586·23-s − 0.424·24-s + 1.69·25-s − 0.252·26-s + 0.672·27-s − 0.652·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 4021 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.07T + 3T^{2} \) |
| 5 | \( 1 + 3.66T + 5T^{2} \) |
| 7 | \( 1 + 3.45T + 7T^{2} \) |
| 11 | \( 1 + 0.0411T + 11T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 17 | \( 1 + 3.99T + 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 + 2.81T + 23T^{2} \) |
| 29 | \( 1 + 2.72T + 29T^{2} \) |
| 31 | \( 1 - 8.98T + 31T^{2} \) |
| 37 | \( 1 - 8.77T + 37T^{2} \) |
| 41 | \( 1 - 0.445T + 41T^{2} \) |
| 43 | \( 1 + 9.80T + 43T^{2} \) |
| 47 | \( 1 - 5.77T + 47T^{2} \) |
| 53 | \( 1 - 7.93T + 53T^{2} \) |
| 59 | \( 1 + 4.48T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 7.58T + 71T^{2} \) |
| 73 | \( 1 + 3.83T + 73T^{2} \) |
| 79 | \( 1 + 1.20T + 79T^{2} \) |
| 83 | \( 1 - 4.24T + 83T^{2} \) |
| 89 | \( 1 - 7.73T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.26974264829885541666440934556, −6.61072368193119416853950219187, −6.19618942425838788492939993499, −5.34520887544570344141666211004, −4.60717573270248447297682542284, −4.05163778310350477540660985332, −3.28447252392450871916345122343, −2.58641894609559532713610394124, −0.828653576741461276515849808954, 0,
0.828653576741461276515849808954, 2.58641894609559532713610394124, 3.28447252392450871916345122343, 4.05163778310350477540660985332, 4.60717573270248447297682542284, 5.34520887544570344141666211004, 6.19618942425838788492939993499, 6.61072368193119416853950219187, 7.26974264829885541666440934556
