| L(s) = 1 | + 2·3-s − 2·5-s + 7-s + 9-s + 2·13-s − 4·15-s + 6·17-s + 19-s + 2·21-s − 25-s − 4·27-s + 8·29-s − 2·35-s − 2·37-s + 4·39-s + 2·41-s − 4·43-s − 2·45-s + 47-s + 49-s + 12·51-s − 6·53-s + 2·57-s − 14·59-s − 6·61-s + 63-s − 4·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 1.03·15-s + 1.45·17-s + 0.229·19-s + 0.436·21-s − 1/5·25-s − 0.769·27-s + 1.48·29-s − 0.338·35-s − 0.328·37-s + 0.640·39-s + 0.312·41-s − 0.609·43-s − 0.298·45-s + 0.145·47-s + 1/7·49-s + 1.68·51-s − 0.824·53-s + 0.264·57-s − 1.82·59-s − 0.768·61-s + 0.125·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.73630246484840, −14.28517323819813, −13.73056699075890, −13.60742151988755, −12.59908604502950, −12.26318783371799, −11.76868668756320, −11.21328477555269, −10.65650597655435, −10.03069928048874, −9.495363046662879, −8.945409752708462, −8.264771655827864, −8.094288668379158, −7.621607733259406, −7.051388425604411, −6.247990565840620, −5.649184605760326, −4.937410070995354, −4.244441381961511, −3.738510808245489, −3.069875154172941, −2.809110892980071, −1.700965293077333, −1.153678665092452, 0,
1.153678665092452, 1.700965293077333, 2.809110892980071, 3.069875154172941, 3.738510808245489, 4.244441381961511, 4.937410070995354, 5.649184605760326, 6.247990565840620, 7.051388425604411, 7.621607733259406, 8.094288668379158, 8.264771655827864, 8.945409752708462, 9.495363046662879, 10.03069928048874, 10.65650597655435, 11.21328477555269, 11.76868668756320, 12.26318783371799, 12.59908604502950, 13.60742151988755, 13.73056699075890, 14.28517323819813, 14.73630246484840
