| L(s) = 1 | − 14.2·2-s + 74.5·4-s + 290.·5-s + 1.11e3·7-s + 760.·8-s − 4.13e3·10-s + 4.49e3·11-s + 2.43e3·13-s − 1.58e4·14-s − 2.03e4·16-s + 1.59e4·17-s − 4.99e4·19-s + 2.16e4·20-s − 6.39e4·22-s + 6.93e4·23-s + 6.32e3·25-s − 3.46e4·26-s + 8.28e4·28-s − 9.40e4·29-s − 1.99e4·31-s + 1.92e5·32-s − 2.26e5·34-s + 3.23e5·35-s + 3.31e5·37-s + 7.10e5·38-s + 2.21e5·40-s + 2.42e5·41-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 0.582·4-s + 1.03·5-s + 1.22·7-s + 0.525·8-s − 1.30·10-s + 1.01·11-s + 0.307·13-s − 1.54·14-s − 1.24·16-s + 0.785·17-s − 1.67·19-s + 0.605·20-s − 1.27·22-s + 1.18·23-s + 0.0809·25-s − 0.386·26-s + 0.713·28-s − 0.716·29-s − 0.120·31-s + 1.03·32-s − 0.987·34-s + 1.27·35-s + 1.07·37-s + 2.10·38-s + 0.546·40-s + 0.548·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.496097973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496097973\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 14.2T + 128T^{2} \) |
| 5 | \( 1 - 290.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.11e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.49e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 2.43e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.59e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.99e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.93e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 9.40e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.99e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.31e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.42e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.31e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.60e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.11e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 3.12e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.74e4T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.10e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.03e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 8.23e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 9.78e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.70e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.09e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.50e6T + 8.07e13T^{2} \) |
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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
−12.96375728042968142918505499274, −11.36210084138746734075159962240, −10.55952030852226740476443738743, −9.389222683591918371319411715694, −8.654287309981505640931922828213, −7.47554848879276967017564152923, −6.02264590651411064791226156746, −4.45604385211629073373731706009, −1.99515232277572454129991145907, −1.06858299542411536497273128326,
1.06858299542411536497273128326, 1.99515232277572454129991145907, 4.45604385211629073373731706009, 6.02264590651411064791226156746, 7.47554848879276967017564152923, 8.654287309981505640931922828213, 9.389222683591918371319411715694, 10.55952030852226740476443738743, 11.36210084138746734075159962240, 12.96375728042968142918505499274
