| L(s) = 1 | + 4·2-s − 2.97·3-s + 16·4-s − 20.6·5-s − 11.8·6-s + 119.·7-s + 64·8-s − 234.·9-s − 82.7·10-s − 496.·11-s − 47.5·12-s − 1.06e3·13-s + 477.·14-s + 61.4·15-s + 256·16-s + 1.81e3·17-s − 936.·18-s + 1.79e3·19-s − 331.·20-s − 354.·21-s − 1.98e3·22-s − 2.25e3·23-s − 190.·24-s − 2.69e3·25-s − 4.25e3·26-s + 1.41e3·27-s + 1.91e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.190·3-s + 0.5·4-s − 0.370·5-s − 0.134·6-s + 0.920·7-s + 0.353·8-s − 0.963·9-s − 0.261·10-s − 1.23·11-s − 0.0953·12-s − 1.74·13-s + 0.651·14-s + 0.0705·15-s + 0.250·16-s + 1.52·17-s − 0.681·18-s + 1.13·19-s − 0.185·20-s − 0.175·21-s − 0.874·22-s − 0.888·23-s − 0.0674·24-s − 0.862·25-s − 1.23·26-s + 0.374·27-s + 0.460·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 79 | \( 1 - 6.24e3T \) |
| good | 3 | \( 1 + 2.97T + 243T^{2} \) |
| 5 | \( 1 + 20.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 119.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 496.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.06e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.81e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.79e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.25e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.13e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.28e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.37e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.23e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.90e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.14e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.74e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.56e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.69e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.93e4T + 2.07e9T^{2} \) |
| 83 | \( 1 - 1.11e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.46e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.57e4T + 8.58e9T^{2} \) |
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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
−11.79779294309739572075672301368, −10.75055034182570182280990327812, −9.638639914811876784220595023206, −7.83881707473907904915672791351, −7.56413167872320241707167938868, −5.47926122028392963744159289780, −5.19068134188141296154220918391, −3.47450991086011621828021691267, −2.13790119162261416652076629261, 0,
2.13790119162261416652076629261, 3.47450991086011621828021691267, 5.19068134188141296154220918391, 5.47926122028392963744159289780, 7.56413167872320241707167938868, 7.83881707473907904915672791351, 9.638639914811876784220595023206, 10.75055034182570182280990327812, 11.79779294309739572075672301368
