| L(s) = 1 | − 1.38·2-s − 3-s − 0.0885·4-s + 2.12·5-s + 1.38·6-s − 0.203·7-s + 2.88·8-s + 9-s − 2.94·10-s − 1.88·11-s + 0.0885·12-s + 3.56·13-s + 0.281·14-s − 2.12·15-s − 3.81·16-s − 17-s − 1.38·18-s − 0.0124·19-s − 0.188·20-s + 0.203·21-s + 2.60·22-s + 6.58·23-s − 2.88·24-s − 0.474·25-s − 4.92·26-s − 27-s + 0.0180·28-s + ⋯ |
| L(s) = 1 | − 0.977·2-s − 0.577·3-s − 0.0442·4-s + 0.951·5-s + 0.564·6-s − 0.0770·7-s + 1.02·8-s + 0.333·9-s − 0.930·10-s − 0.567·11-s + 0.0255·12-s + 0.988·13-s + 0.0753·14-s − 0.549·15-s − 0.953·16-s − 0.242·17-s − 0.325·18-s − 0.00285·19-s − 0.0421·20-s + 0.0445·21-s + 0.554·22-s + 1.37·23-s − 0.589·24-s − 0.0949·25-s − 0.966·26-s − 0.192·27-s + 0.00341·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 1.38T + 2T^{2} \) |
| 5 | \( 1 - 2.12T + 5T^{2} \) |
| 7 | \( 1 + 0.203T + 7T^{2} \) |
| 11 | \( 1 + 1.88T + 11T^{2} \) |
| 13 | \( 1 - 3.56T + 13T^{2} \) |
| 19 | \( 1 + 0.0124T + 19T^{2} \) |
| 23 | \( 1 - 6.58T + 23T^{2} \) |
| 29 | \( 1 - 0.647T + 29T^{2} \) |
| 31 | \( 1 - 0.398T + 31T^{2} \) |
| 37 | \( 1 - 4.09T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 8.80T + 43T^{2} \) |
| 47 | \( 1 - 5.83T + 47T^{2} \) |
| 53 | \( 1 + 1.05T + 53T^{2} \) |
| 59 | \( 1 + 14.7T + 59T^{2} \) |
| 61 | \( 1 - 2.57T + 61T^{2} \) |
| 67 | \( 1 - 6.56T + 67T^{2} \) |
| 71 | \( 1 - 3.69T + 71T^{2} \) |
| 73 | \( 1 - 0.453T + 73T^{2} \) |
| 79 | \( 1 + 2.56T + 79T^{2} \) |
| 83 | \( 1 - 8.30T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−7.61213374044503571159433720619, −6.72456859312871463530861832028, −6.30513885119871287464516041503, −5.30285609899960454169427781793, −4.96873725462056531094340324339, −3.96009890727752934381373248827, −2.93709976748810573873609819724, −1.80684749592890611441395034733, −1.17387196158058177258123325999, 0,
1.17387196158058177258123325999, 1.80684749592890611441395034733, 2.93709976748810573873609819724, 3.96009890727752934381373248827, 4.96873725462056531094340324339, 5.30285609899960454169427781793, 6.30513885119871287464516041503, 6.72456859312871463530861832028, 7.61213374044503571159433720619
