| L(s) = 1 | − 9.68·2-s + 27·3-s − 34.2·4-s − 195.·5-s − 261.·6-s + 946.·7-s + 1.57e3·8-s + 729·9-s + 1.88e3·10-s − 924.·12-s + 5.14e3·13-s − 9.16e3·14-s − 5.26e3·15-s − 1.08e4·16-s + 7.82e3·17-s − 7.05e3·18-s − 5.37e4·19-s + 6.68e3·20-s + 2.55e4·21-s − 1.05e5·23-s + 4.24e4·24-s − 4.00e4·25-s − 4.98e4·26-s + 1.96e4·27-s − 3.24e4·28-s + 9.19e4·29-s + 5.10e4·30-s + ⋯ |
| L(s) = 1 | − 0.855·2-s + 0.577·3-s − 0.267·4-s − 0.698·5-s − 0.494·6-s + 1.04·7-s + 1.08·8-s + 0.333·9-s + 0.597·10-s − 0.154·12-s + 0.649·13-s − 0.892·14-s − 0.403·15-s − 0.660·16-s + 0.386·17-s − 0.285·18-s − 1.79·19-s + 0.186·20-s + 0.602·21-s − 1.79·23-s + 0.626·24-s − 0.512·25-s − 0.556·26-s + 0.192·27-s − 0.278·28-s + 0.700·29-s + 0.344·30-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 | \( 1 + 9.68T + 128T^{2} \) |
| 5 | \( 1 + 195.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 946.T + 8.23e5T^{2} \) |
| 13 | \( 1 - 5.14e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 7.82e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.37e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.05e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.19e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.70e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.27e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.41e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.37e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.72e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.24e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.67e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.12e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 6.48e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.28e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.34e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.09e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.53e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.14e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.30e6T + 8.07e13T^{2} \) |
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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
−9.677685911759446599763835032528, −8.503789870916303676394737296959, −8.211452385019848401883852184206, −7.50410666354818036860054732028, −6.03842327588279138100563679479, −4.43288043856130993804030569501, −4.02558095443849284972817936342, −2.25023820043809453282514338294, −1.21009666982804796945949999569, 0,
1.21009666982804796945949999569, 2.25023820043809453282514338294, 4.02558095443849284972817936342, 4.43288043856130993804030569501, 6.03842327588279138100563679479, 7.50410666354818036860054732028, 8.211452385019848401883852184206, 8.503789870916303676394737296959, 9.677685911759446599763835032528
