| L(s) = 1 | − 4.55·2-s − 27·3-s − 107.·4-s − 540.·5-s + 123.·6-s − 1.23e3·7-s + 1.07e3·8-s + 729·9-s + 2.46e3·10-s + 3.52e3·11-s + 2.89e3·12-s − 4.83e3·13-s + 5.64e3·14-s + 1.45e4·15-s + 8.84e3·16-s + 2.24e3·17-s − 3.32e3·18-s + 5.31e4·19-s + 5.79e4·20-s + 3.34e4·21-s − 1.60e4·22-s − 5.63e4·23-s − 2.89e4·24-s + 2.13e5·25-s + 2.20e4·26-s − 1.96e4·27-s + 1.32e5·28-s + ⋯ |
| L(s) = 1 | − 0.402·2-s − 0.577·3-s − 0.837·4-s − 1.93·5-s + 0.232·6-s − 1.36·7-s + 0.740·8-s + 0.333·9-s + 0.778·10-s + 0.797·11-s + 0.483·12-s − 0.610·13-s + 0.549·14-s + 1.11·15-s + 0.539·16-s + 0.110·17-s − 0.134·18-s + 1.77·19-s + 1.61·20-s + 0.787·21-s − 0.321·22-s − 0.966·23-s − 0.427·24-s + 2.73·25-s + 0.245·26-s − 0.192·27-s + 1.14·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 27T \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 + 4.55T + 128T^{2} \) |
| 5 | \( 1 + 540.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.23e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.52e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 4.83e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.24e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.31e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.63e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.37e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.21e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.76e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.40e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.96e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.83e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.05e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 3.78e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.12e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.82e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.56e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.80e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.26e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.36e6T + 8.07e13T^{2} \) |
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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
−10.91879785768879829999743299612, −9.755414909826336575978606300606, −8.989440705573759226353044907497, −7.66212736081745067274006759829, −7.06003983732652805071821901495, −5.43208581459069187747024382233, −4.06798025285441737294379372128, −3.48148723352148063994612398685, −0.78777653555696522991821047370, 0,
0.78777653555696522991821047370, 3.48148723352148063994612398685, 4.06798025285441737294379372128, 5.43208581459069187747024382233, 7.06003983732652805071821901495, 7.66212736081745067274006759829, 8.989440705573759226353044907497, 9.755414909826336575978606300606, 10.91879785768879829999743299612
