| L(s) = 1 | − 2·4-s + 5-s − 4·7-s − 3·11-s + 2·13-s + 4·16-s − 6·17-s − 2·20-s + 25-s + 8·28-s + 3·29-s − 7·31-s − 4·35-s + 8·37-s + 6·41-s − 4·43-s + 6·44-s − 6·47-s + 9·49-s − 4·52-s + 6·53-s − 3·55-s + 15·59-s + 5·61-s − 8·64-s + 2·65-s + 2·67-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 1.51·7-s − 0.904·11-s + 0.554·13-s + 16-s − 1.45·17-s − 0.447·20-s + 1/5·25-s + 1.51·28-s + 0.557·29-s − 1.25·31-s − 0.676·35-s + 1.31·37-s + 0.937·41-s − 0.609·43-s + 0.904·44-s − 0.875·47-s + 9/7·49-s − 0.554·52-s + 0.824·53-s − 0.404·55-s + 1.95·59-s + 0.640·61-s − 64-s + 0.248·65-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.30119689098478, −15.76343004709598, −15.10875206008760, −14.53473324374217, −13.80838973296866, −13.25813792310624, −12.98045060402339, −12.79172747321526, −11.82154761588390, −10.99027232124414, −10.53614897958416, −9.750125168316176, −9.585535982538966, −8.840256661932381, −8.421781329169140, −7.628873009475114, −6.771058136690333, −6.342456284513648, −5.606872878582650, −5.091029581597595, −4.190628396962405, −3.684027013863426, −2.851846828219073, −2.187955992227127, −0.8353148224984355, 0,
0.8353148224984355, 2.187955992227127, 2.851846828219073, 3.684027013863426, 4.190628396962405, 5.091029581597595, 5.606872878582650, 6.342456284513648, 6.771058136690333, 7.628873009475114, 8.421781329169140, 8.840256661932381, 9.585535982538966, 9.750125168316176, 10.53614897958416, 10.99027232124414, 11.82154761588390, 12.79172747321526, 12.98045060402339, 13.25813792310624, 13.80838973296866, 14.53473324374217, 15.10875206008760, 15.76343004709598, 16.30119689098478
