| L(s) = 1 | + 4.85·2-s + 27·3-s − 104.·4-s + 273.·5-s + 131.·6-s − 474.·7-s − 1.12e3·8-s + 729·9-s + 1.33e3·10-s − 2.81e3·12-s − 731.·13-s − 2.30e3·14-s + 7.39e3·15-s + 7.87e3·16-s + 2.31e4·17-s + 3.54e3·18-s − 4.33e4·19-s − 2.85e4·20-s − 1.28e4·21-s + 3.77e4·23-s − 3.04e4·24-s − 3.17e3·25-s − 3.55e3·26-s + 1.96e4·27-s + 4.95e4·28-s − 1.22e5·29-s + 3.59e4·30-s + ⋯ |
| L(s) = 1 | + 0.429·2-s + 0.577·3-s − 0.815·4-s + 0.979·5-s + 0.247·6-s − 0.523·7-s − 0.779·8-s + 0.333·9-s + 0.420·10-s − 0.470·12-s − 0.0923·13-s − 0.224·14-s + 0.565·15-s + 0.480·16-s + 1.14·17-s + 0.143·18-s − 1.44·19-s − 0.798·20-s − 0.302·21-s + 0.646·23-s − 0.450·24-s − 0.0406·25-s − 0.0396·26-s + 0.192·27-s + 0.426·28-s − 0.929·29-s + 0.242·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 27T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 4.85T + 128T^{2} \) |
| 5 | \( 1 - 273.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 474.T + 8.23e5T^{2} \) |
| 13 | \( 1 + 731.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.31e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.33e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.77e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.22e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.90e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.01e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.97e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.74e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.17e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.63e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.52e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.88e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.57e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.16e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.88e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.29e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.10e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.61e6T + 8.07e13T^{2} \) |
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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
−9.842656190738131969637578612936, −8.904389954234165285354032112032, −8.149035237703486074845885883825, −6.71971737576270593315474220366, −5.81336091377889124872032765588, −4.85140718555611489956570381453, −3.70395500034139149958911118918, −2.76280944015801313301949505333, −1.47149781430235305181988039793, 0,
1.47149781430235305181988039793, 2.76280944015801313301949505333, 3.70395500034139149958911118918, 4.85140718555611489956570381453, 5.81336091377889124872032765588, 6.71971737576270593315474220366, 8.149035237703486074845885883825, 8.904389954234165285354032112032, 9.842656190738131969637578612936
