| L(s) = 1 | + 2.59·2-s − 3-s + 4.75·4-s + 0.424·5-s − 2.59·6-s + 7-s + 7.15·8-s + 9-s + 1.10·10-s − 5.94·11-s − 4.75·12-s − 1.20·13-s + 2.59·14-s − 0.424·15-s + 9.07·16-s + 0.403·17-s + 2.59·18-s − 8.37·19-s + 2.01·20-s − 21-s − 15.4·22-s − 2.32·23-s − 7.15·24-s − 4.81·25-s − 3.12·26-s − 27-s + 4.75·28-s + ⋯ |
| L(s) = 1 | + 1.83·2-s − 0.577·3-s + 2.37·4-s + 0.189·5-s − 1.06·6-s + 0.377·7-s + 2.52·8-s + 0.333·9-s + 0.348·10-s − 1.79·11-s − 1.37·12-s − 0.334·13-s + 0.694·14-s − 0.109·15-s + 2.26·16-s + 0.0977·17-s + 0.612·18-s − 1.92·19-s + 0.450·20-s − 0.218·21-s − 3.29·22-s − 0.483·23-s − 1.45·24-s − 0.963·25-s − 0.613·26-s − 0.192·27-s + 0.898·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 2.59T + 2T^{2} \) |
| 5 | \( 1 - 0.424T + 5T^{2} \) |
| 11 | \( 1 + 5.94T + 11T^{2} \) |
| 13 | \( 1 + 1.20T + 13T^{2} \) |
| 17 | \( 1 - 0.403T + 17T^{2} \) |
| 19 | \( 1 + 8.37T + 19T^{2} \) |
| 23 | \( 1 + 2.32T + 23T^{2} \) |
| 29 | \( 1 - 3.39T + 29T^{2} \) |
| 31 | \( 1 - 4.27T + 31T^{2} \) |
| 37 | \( 1 + 8.53T + 37T^{2} \) |
| 41 | \( 1 + 1.39T + 41T^{2} \) |
| 43 | \( 1 + 7.70T + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 53 | \( 1 + 9.00T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 5.38T + 61T^{2} \) |
| 67 | \( 1 - 5.28T + 67T^{2} \) |
| 71 | \( 1 + 3.79T + 71T^{2} \) |
| 73 | \( 1 + 1.84T + 73T^{2} \) |
| 79 | \( 1 + 8.67T + 79T^{2} \) |
| 83 | \( 1 - 8.36T + 83T^{2} \) |
| 89 | \( 1 + 2.38T + 89T^{2} \) |
| 97 | \( 1 + 4.08T + 97T^{2} \) |
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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
−7.11781961980355576808524692538, −6.52020968153425634506048865731, −5.89195851494256784758673511628, −5.22769154658354695143035099124, −4.81028181139225050479748026272, −4.15575615797673367173755787470, −3.24873059841576553168701759367, −2.34772892624361206875324489802, −1.85690070325296963289947279104, 0,
1.85690070325296963289947279104, 2.34772892624361206875324489802, 3.24873059841576553168701759367, 4.15575615797673367173755787470, 4.81028181139225050479748026272, 5.22769154658354695143035099124, 5.89195851494256784758673511628, 6.52020968153425634506048865731, 7.11781961980355576808524692538
