| L(s) = 1 | − 2-s − 2.12·3-s + 4-s − 1.71·5-s + 2.12·6-s + 1.55·7-s − 8-s + 1.53·9-s + 1.71·10-s − 11-s − 2.12·12-s − 0.181·13-s − 1.55·14-s + 3.65·15-s + 16-s + 3.47·17-s − 1.53·18-s − 1.54·19-s − 1.71·20-s − 3.30·21-s + 22-s − 1.83·23-s + 2.12·24-s − 2.05·25-s + 0.181·26-s + 3.11·27-s + 1.55·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.22·3-s + 0.5·4-s − 0.767·5-s + 0.869·6-s + 0.586·7-s − 0.353·8-s + 0.511·9-s + 0.542·10-s − 0.301·11-s − 0.614·12-s − 0.0504·13-s − 0.414·14-s + 0.943·15-s + 0.250·16-s + 0.843·17-s − 0.361·18-s − 0.354·19-s − 0.383·20-s − 0.721·21-s + 0.213·22-s − 0.383·23-s + 0.434·24-s − 0.410·25-s + 0.0356·26-s + 0.600·27-s + 0.293·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 + 2.12T + 3T^{2} \) |
| 5 | \( 1 + 1.71T + 5T^{2} \) |
| 7 | \( 1 - 1.55T + 7T^{2} \) |
| 13 | \( 1 + 0.181T + 13T^{2} \) |
| 17 | \( 1 - 3.47T + 17T^{2} \) |
| 19 | \( 1 + 1.54T + 19T^{2} \) |
| 23 | \( 1 + 1.83T + 23T^{2} \) |
| 29 | \( 1 - 2.51T + 29T^{2} \) |
| 31 | \( 1 + 2.35T + 31T^{2} \) |
| 37 | \( 1 + 8.16T + 37T^{2} \) |
| 41 | \( 1 - 5.50T + 41T^{2} \) |
| 43 | \( 1 - 3.73T + 43T^{2} \) |
| 47 | \( 1 + 7.96T + 47T^{2} \) |
| 53 | \( 1 - 7.46T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 4.49T + 61T^{2} \) |
| 67 | \( 1 - 5.83T + 67T^{2} \) |
| 71 | \( 1 - 7.57T + 71T^{2} \) |
| 73 | \( 1 - 5.61T + 73T^{2} \) |
| 79 | \( 1 - 0.169T + 79T^{2} \) |
| 83 | \( 1 + 2.06T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 9.08T + 97T^{2} \) |
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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
−8.178967015789598344025870668897, −7.26545849213674004178612690297, −6.69372094212295890826659312925, −5.72920698407184925957399559313, −5.26244850247701490898578011288, −4.33042525438667020607576228686, −3.42224518486472476127010581360, −2.18042099190912884601958674593, −1.00023839347319069600799168408, 0,
1.00023839347319069600799168408, 2.18042099190912884601958674593, 3.42224518486472476127010581360, 4.33042525438667020607576228686, 5.26244850247701490898578011288, 5.72920698407184925957399559313, 6.69372094212295890826659312925, 7.26545849213674004178612690297, 8.178967015789598344025870668897
