| L(s) = 1 | − 7.02·2-s − 78.6·4-s + 266.·5-s + 665.·7-s + 1.45e3·8-s − 1.87e3·10-s − 2.99e3·11-s + 1.11e4·13-s − 4.67e3·14-s − 120.·16-s − 1.73e4·17-s − 7.56e3·19-s − 2.09e4·20-s + 2.10e4·22-s + 8.48e4·23-s − 6.95e3·25-s − 7.80e4·26-s − 5.23e4·28-s − 5.97e4·29-s − 5.62e4·31-s − 1.84e5·32-s + 1.21e5·34-s + 1.77e5·35-s − 3.75e5·37-s + 5.31e4·38-s + 3.87e5·40-s − 5.60e5·41-s + ⋯ |
| L(s) = 1 | − 0.620·2-s − 0.614·4-s + 0.954·5-s + 0.733·7-s + 1.00·8-s − 0.592·10-s − 0.678·11-s + 1.40·13-s − 0.455·14-s − 0.00733·16-s − 0.855·17-s − 0.253·19-s − 0.586·20-s + 0.421·22-s + 1.45·23-s − 0.0890·25-s − 0.870·26-s − 0.450·28-s − 0.454·29-s − 0.338·31-s − 0.997·32-s + 0.530·34-s + 0.700·35-s − 1.21·37-s + 0.157·38-s + 0.956·40-s − 1.26·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 + 7.02T + 128T^{2} \) |
| 5 | \( 1 - 266.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 665.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.99e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.11e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.73e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 7.56e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 8.48e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 5.97e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 5.62e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.75e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.60e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.99e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.08e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 9.51e4T + 1.17e12T^{2} \) |
| 61 | \( 1 - 1.08e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 7.31e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.30e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 8.04e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.53e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.87e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.83e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.47e7T + 8.07e13T^{2} \) |
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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
−8.974524062551969222155738155018, −8.717516449489611282296615315251, −7.69641009546881069485839533271, −6.57892450101585065582480243609, −5.44032190281632296829186463238, −4.76647308913372252908196180120, −3.50883841968051350769282064887, −1.98169019823110748198408245775, −1.26922795471849609815744761808, 0,
1.26922795471849609815744761808, 1.98169019823110748198408245775, 3.50883841968051350769282064887, 4.76647308913372252908196180120, 5.44032190281632296829186463238, 6.57892450101585065582480243609, 7.69641009546881069485839533271, 8.717516449489611282296615315251, 8.974524062551969222155738155018
