| L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s + 4·11-s + 2·13-s + 15-s − 2·17-s − 4·21-s + 8·23-s + 25-s + 27-s − 6·29-s + 4·31-s + 4·33-s − 4·35-s + 10·37-s + 2·39-s + 2·41-s − 12·43-s + 45-s + 9·49-s − 2·51-s − 6·53-s + 4·55-s − 10·61-s − 4·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.258·15-s − 0.485·17-s − 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.696·33-s − 0.676·35-s + 1.64·37-s + 0.320·39-s + 0.312·41-s − 1.82·43-s + 0.149·45-s + 9/7·49-s − 0.280·51-s − 0.824·53-s + 0.539·55-s − 1.28·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.10128868891077, −13.43270199026698, −13.22878650629845, −12.97403963313806, −12.22136708813274, −11.79894526380340, −10.94715223839450, −10.86664589965615, −9.864670003962275, −9.545491161176579, −9.339400870117523, −8.709813077596463, −8.317822576099138, −7.403452001300069, −6.968875364116418, −6.468059210229109, −6.135846736287056, −5.522913356778663, −4.587738573237262, −4.206674854382593, −3.344461437391882, −3.161831589982657, −2.482231243979510, −1.576606014766127, −1.051796046352562, 0,
1.051796046352562, 1.576606014766127, 2.482231243979510, 3.161831589982657, 3.344461437391882, 4.206674854382593, 4.587738573237262, 5.522913356778663, 6.135846736287056, 6.468059210229109, 6.968875364116418, 7.403452001300069, 8.317822576099138, 8.709813077596463, 9.339400870117523, 9.545491161176579, 9.864670003962275, 10.86664589965615, 10.94715223839450, 11.79894526380340, 12.22136708813274, 12.97403963313806, 13.22878650629845, 13.43270199026698, 14.10128868891077
