| L(s) = 1 | + 27·3-s + 432.·5-s − 113.·7-s + 729·9-s − 4.05e3·11-s + 1.27e4·13-s + 1.16e4·15-s − 3.71e4·17-s − 4.36e4·19-s − 3.06e3·21-s − 1.06e5·23-s + 1.08e5·25-s + 1.96e4·27-s − 374.·29-s − 1.76e5·31-s − 1.09e5·33-s − 4.91e4·35-s + 2.56e5·37-s + 3.43e5·39-s − 8.53e5·41-s + 1.37e5·43-s + 3.15e5·45-s − 1.05e6·47-s − 8.10e5·49-s − 1.00e6·51-s − 1.02e6·53-s − 1.75e6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.54·5-s − 0.125·7-s + 0.333·9-s − 0.918·11-s + 1.60·13-s + 0.893·15-s − 1.83·17-s − 1.46·19-s − 0.0722·21-s − 1.81·23-s + 1.39·25-s + 0.192·27-s − 0.00285·29-s − 1.06·31-s − 0.530·33-s − 0.193·35-s + 0.832·37-s + 0.928·39-s − 1.93·41-s + 0.264·43-s + 0.515·45-s − 1.48·47-s − 0.984·49-s − 1.05·51-s − 0.942·53-s − 1.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - 27T \) |
| good | 5 | \( 1 - 432.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 113.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.05e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.27e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.71e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.36e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.06e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + 374.T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.76e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.56e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.53e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.37e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.05e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.02e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 6.39e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.60e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.63e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.31e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.28e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.00e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.53e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.52e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.71e6T + 8.07e13T^{2} \) |
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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
−9.684287979301767124486976324301, −8.760757544890736434981197404224, −8.146309134564481345146719329862, −6.49862442679794338152802829259, −6.14294161918630615937799561146, −4.83582716942884055068904074964, −3.61885037787874930740571916117, −2.20095749563206351032223108134, −1.80755641176517936723678368218, 0,
1.80755641176517936723678368218, 2.20095749563206351032223108134, 3.61885037787874930740571916117, 4.83582716942884055068904074964, 6.14294161918630615937799561146, 6.49862442679794338152802829259, 8.146309134564481345146719329862, 8.760757544890736434981197404224, 9.684287979301767124486976324301
