| L(s) = 1 | − 16.0·2-s + 27·3-s + 128.·4-s + 211.·5-s − 432.·6-s − 7.07·8-s + 729·9-s − 3.38e3·10-s − 954.·11-s + 3.46e3·12-s + 1.30e3·13-s + 5.70e3·15-s − 1.63e4·16-s − 3.03e4·17-s − 1.16e4·18-s + 9.69e3·19-s + 2.71e4·20-s + 1.52e4·22-s − 8.86e4·23-s − 191.·24-s − 3.34e4·25-s − 2.09e4·26-s + 1.96e4·27-s + 9.71e4·29-s − 9.13e4·30-s + 1.36e5·31-s + 2.62e5·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 1.00·4-s + 0.756·5-s − 0.817·6-s − 0.00488·8-s + 0.333·9-s − 1.07·10-s − 0.216·11-s + 0.579·12-s + 0.165·13-s + 0.436·15-s − 0.996·16-s − 1.49·17-s − 0.471·18-s + 0.324·19-s + 0.758·20-s + 0.306·22-s − 1.51·23-s − 0.00282·24-s − 0.428·25-s − 0.233·26-s + 0.192·27-s + 0.739·29-s − 0.617·30-s + 0.820·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 27T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 16.0T + 128T^{2} \) |
| 5 | \( 1 - 211.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 954.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.30e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.03e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 9.69e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.86e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.71e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.36e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.48e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.97e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.91e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.69e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.23e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.08e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.43e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.42e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.80e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.81e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.94e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.40e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.15e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.80e6T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.69616803117664377812804893609, −10.00063185928293416759461557115, −9.099320106872103845267485591117, −8.364192346475987549672672974279, −7.30032175236316333265622404811, −6.12443361380815810382015305652, −4.38107566889524032456741847490, −2.49292197254431025626536931959, −1.54160882605527365066264957284, 0,
1.54160882605527365066264957284, 2.49292197254431025626536931959, 4.38107566889524032456741847490, 6.12443361380815810382015305652, 7.30032175236316333265622404811, 8.364192346475987549672672974279, 9.099320106872103845267485591117, 10.00063185928293416759461557115, 10.69616803117664377812804893609
