| L(s) = 1 | − 3.13·2-s + 27·3-s − 118.·4-s − 324.·5-s − 84.7·6-s − 1.65e3·7-s + 772.·8-s + 729·9-s + 1.02e3·10-s − 3.18e3·12-s − 5.92e3·13-s + 5.20e3·14-s − 8.77e3·15-s + 1.26e4·16-s + 2.92e4·17-s − 2.28e3·18-s + 3.24e4·19-s + 3.83e4·20-s − 4.47e4·21-s + 1.09e4·23-s + 2.08e4·24-s + 2.74e4·25-s + 1.86e4·26-s + 1.96e4·27-s + 1.95e5·28-s − 1.00e5·29-s + 2.75e4·30-s + ⋯ |
| L(s) = 1 | − 0.277·2-s + 0.577·3-s − 0.922·4-s − 1.16·5-s − 0.160·6-s − 1.82·7-s + 0.533·8-s + 0.333·9-s + 0.322·10-s − 0.532·12-s − 0.748·13-s + 0.506·14-s − 0.671·15-s + 0.774·16-s + 1.44·17-s − 0.0925·18-s + 1.08·19-s + 1.07·20-s − 1.05·21-s + 0.187·23-s + 0.308·24-s + 0.351·25-s + 0.207·26-s + 0.192·27-s + 1.68·28-s − 0.761·29-s + 0.186·30-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 | \( 1 + 3.13T + 128T^{2} \) |
| 5 | \( 1 + 324.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.65e3T + 8.23e5T^{2} \) |
| 13 | \( 1 + 5.92e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.92e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.24e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.09e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.00e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.14e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 8.52e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.47e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.92e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.53e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.33e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 5.69e4T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.82e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.57e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.98e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.30e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.14e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.57e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.21e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.72e7T + 8.07e13T^{2} \) |
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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
−9.688864712582977249300761618204, −9.010141705011446522924181005814, −7.78288840575256732522003170613, −7.39787378651935510430877197766, −5.94022584672486811188023189829, −4.60267682875578371908196992706, −3.53774388337659911978506700976, −3.03311935844154326011090550544, −0.906040062722935065392603651007, 0,
0.906040062722935065392603651007, 3.03311935844154326011090550544, 3.53774388337659911978506700976, 4.60267682875578371908196992706, 5.94022584672486811188023189829, 7.39787378651935510430877197766, 7.78288840575256732522003170613, 9.010141705011446522924181005814, 9.688864712582977249300761618204
