| L(s) = 1 | + 10.7·2-s + 27·3-s − 12.1·4-s + 451.·5-s + 290.·6-s − 504.·7-s − 1.50e3·8-s + 729·9-s + 4.86e3·10-s − 8.07e3·11-s − 328.·12-s − 4.67e3·13-s − 5.43e3·14-s + 1.22e4·15-s − 1.46e4·16-s − 1.34e4·17-s + 7.84e3·18-s − 1.13e4·19-s − 5.49e3·20-s − 1.36e4·21-s − 8.68e4·22-s − 6.64e4·23-s − 4.07e4·24-s + 1.26e5·25-s − 5.03e4·26-s + 1.96e4·27-s + 6.13e3·28-s + ⋯ |
| L(s) = 1 | + 0.951·2-s + 0.577·3-s − 0.0949·4-s + 1.61·5-s + 0.549·6-s − 0.556·7-s − 1.04·8-s + 0.333·9-s + 1.53·10-s − 1.82·11-s − 0.0548·12-s − 0.590·13-s − 0.529·14-s + 0.933·15-s − 0.895·16-s − 0.666·17-s + 0.317·18-s − 0.380·19-s − 0.153·20-s − 0.321·21-s − 1.73·22-s − 1.13·23-s − 0.601·24-s + 1.61·25-s − 0.561·26-s + 0.192·27-s + 0.0528·28-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 | \( 1 - 10.7T + 128T^{2} \) |
| 5 | \( 1 - 451.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 504.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 8.07e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 4.67e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.34e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.13e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.64e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.74e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.24e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.77e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.29e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.75e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.75e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.13e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 3.01e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 8.85e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.15e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.34e4T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.94e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.72e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.67e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.89e6T + 8.07e13T^{2} \) |
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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
−10.71943827056050149744978051259, −9.762663049357801183911139672402, −9.150154511026007440319140282806, −7.75953242891771597570719396529, −6.22215178088750118984607578552, −5.52560361283704725825879079218, −4.41860329384364271837411016948, −2.83207764001527264668292232126, −2.20751479641618996134483535616, 0,
2.20751479641618996134483535616, 2.83207764001527264668292232126, 4.41860329384364271837411016948, 5.52560361283704725825879079218, 6.22215178088750118984607578552, 7.75953242891771597570719396529, 9.150154511026007440319140282806, 9.762663049357801183911139672402, 10.71943827056050149744978051259
