| L(s) = 1 | − 27·3-s + 451.·5-s + 729·9-s − 8.08e3·11-s + 3.48e3·13-s − 1.22e4·15-s + 1.54e4·17-s + 3.68e4·19-s − 6.12e4·23-s + 1.26e5·25-s − 1.96e4·27-s − 2.50e5·29-s + 1.22e5·31-s + 2.18e5·33-s + 1.20e5·37-s − 9.41e4·39-s − 3.95e5·41-s + 3.78e5·43-s + 3.29e5·45-s − 6.69e5·47-s − 4.16e5·51-s − 2.11e6·53-s − 3.65e6·55-s − 9.95e5·57-s + 9.87e5·59-s + 2.20e6·61-s + 1.57e6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.61·5-s + 0.333·9-s − 1.83·11-s + 0.440·13-s − 0.933·15-s + 0.761·17-s + 1.23·19-s − 1.04·23-s + 1.61·25-s − 0.192·27-s − 1.90·29-s + 0.738·31-s + 1.05·33-s + 0.392·37-s − 0.254·39-s − 0.895·41-s + 0.726·43-s + 0.539·45-s − 0.941·47-s − 0.439·51-s − 1.95·53-s − 2.96·55-s − 0.711·57-s + 0.625·59-s + 1.24·61-s + 0.712·65-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 | \( 1 - 451.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 8.08e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.48e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.54e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.68e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.12e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.50e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.22e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.20e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.95e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.78e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.69e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.11e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 9.87e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.20e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.69e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.16e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.33e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.57e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.56e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.36e6T + 8.07e13T^{2} \) |
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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
−9.518114339383784548436758276229, −8.161010165732066188001420038728, −7.33792246931507690562366674668, −6.05041047751528113534470826438, −5.61056536480869848629615357216, −4.90805177792251365895784673782, −3.27288383844648136253063286263, −2.21363699216053876942986421223, −1.29623333818925859312105027083, 0,
1.29623333818925859312105027083, 2.21363699216053876942986421223, 3.27288383844648136253063286263, 4.90805177792251365895784673782, 5.61056536480869848629615357216, 6.05041047751528113534470826438, 7.33792246931507690562366674668, 8.161010165732066188001420038728, 9.518114339383784548436758276229
