| L(s) = 1 | + 27·3-s − 445.·5-s + 729·9-s − 3.23e3·11-s + 1.65e3·13-s − 1.20e4·15-s − 1.69e4·17-s + 2.79e4·19-s + 6.94e4·23-s + 1.20e5·25-s + 1.96e4·27-s + 6.97e4·29-s + 1.20e5·31-s − 8.74e4·33-s − 4.53e5·37-s + 4.46e4·39-s + 6.90e5·41-s − 7.52e5·43-s − 3.24e5·45-s + 1.12e5·47-s − 4.56e5·51-s + 6.78e5·53-s + 1.44e6·55-s + 7.53e5·57-s + 2.11e6·59-s − 1.45e6·61-s − 7.36e5·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.59·5-s + 0.333·9-s − 0.733·11-s + 0.208·13-s − 0.920·15-s − 0.835·17-s + 0.933·19-s + 1.19·23-s + 1.54·25-s + 0.192·27-s + 0.530·29-s + 0.728·31-s − 0.423·33-s − 1.47·37-s + 0.120·39-s + 1.56·41-s − 1.44·43-s − 0.531·45-s + 0.158·47-s − 0.482·51-s + 0.625·53-s + 1.16·55-s + 0.539·57-s + 1.34·59-s − 0.822·61-s − 0.332·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 27T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 445.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 3.23e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.65e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.69e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.79e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.94e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.97e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.20e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.53e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.90e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.52e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.12e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.78e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.11e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.45e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 9.86e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.31e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 8.11e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.58e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.96e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.84e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.02e7T + 8.07e13T^{2} \) |
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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
−8.829194391899188599079020832024, −8.293349538646276484559300443674, −7.43539526116620686423394564579, −6.80875807994771310536420809108, −5.22528727393304311646607662293, −4.34637821189083153009995395528, −3.42218078673459530523953764861, −2.62593994327689382564445830086, −1.06380777419738511475784647893, 0,
1.06380777419738511475784647893, 2.62593994327689382564445830086, 3.42218078673459530523953764861, 4.34637821189083153009995395528, 5.22528727393304311646607662293, 6.80875807994771310536420809108, 7.43539526116620686423394564579, 8.293349538646276484559300443674, 8.829194391899188599079020832024
