| L(s) = 1 | + 27·3-s − 223.·5-s + 1.52e3·7-s + 729·9-s − 1.60e3·11-s + 2.95e3·13-s − 6.02e3·15-s − 2.01e4·17-s − 7.74e3·19-s + 4.12e4·21-s − 7.14e4·23-s − 2.83e4·25-s + 1.96e4·27-s + 2.51e4·29-s − 2.70e5·31-s − 4.33e4·33-s − 3.40e5·35-s + 3.94e5·37-s + 7.99e4·39-s + 1.81e5·41-s + 5.55e5·43-s − 1.62e5·45-s − 6.69e5·47-s + 1.50e6·49-s − 5.43e5·51-s − 7.72e5·53-s + 3.57e5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.798·5-s + 1.68·7-s + 0.333·9-s − 0.363·11-s + 0.373·13-s − 0.460·15-s − 0.994·17-s − 0.259·19-s + 0.971·21-s − 1.22·23-s − 0.362·25-s + 0.192·27-s + 0.191·29-s − 1.63·31-s − 0.209·33-s − 1.34·35-s + 1.27·37-s + 0.215·39-s + 0.411·41-s + 1.06·43-s − 0.266·45-s − 0.941·47-s + 1.82·49-s − 0.574·51-s − 0.713·53-s + 0.290·55-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 | \( 1 + 223.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.52e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.60e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 2.95e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.01e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 7.74e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.14e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.51e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.70e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.94e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.81e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.55e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.69e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.72e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.30e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.68e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.08e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 8.59e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.61e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.33e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.43e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.18e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.05e6T + 8.07e13T^{2} \) |
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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
−9.574566503821871024791140618560, −8.440992438791823877197124688175, −8.029002267519586359721077940051, −7.21854907480454994814743640907, −5.75988846122063831806705948109, −4.52294405860063715929348984341, −3.93756601074717080092326459924, −2.41291158077584532720570255241, −1.48895696725030146810340124895, 0,
1.48895696725030146810340124895, 2.41291158077584532720570255241, 3.93756601074717080092326459924, 4.52294405860063715929348984341, 5.75988846122063831806705948109, 7.21854907480454994814743640907, 8.029002267519586359721077940051, 8.440992438791823877197124688175, 9.574566503821871024791140618560
