| L(s) = 1 | − 20.9·2-s + 310.·4-s + 149.·5-s + 1.20e3·7-s − 3.81e3·8-s − 3.13e3·10-s − 3.09e3·11-s − 9.76e3·13-s − 2.51e4·14-s + 4.01e4·16-s − 774.·17-s + 4.22e4·19-s + 4.64e4·20-s + 6.46e4·22-s − 1.92e4·23-s − 5.56e4·25-s + 2.04e5·26-s + 3.72e5·28-s + 1.48e5·29-s − 7.75e4·31-s − 3.51e5·32-s + 1.62e4·34-s + 1.80e5·35-s + 3.81e5·37-s − 8.85e5·38-s − 5.71e5·40-s − 3.13e5·41-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 2.42·4-s + 0.536·5-s + 1.32·7-s − 2.63·8-s − 0.992·10-s − 0.700·11-s − 1.23·13-s − 2.44·14-s + 2.44·16-s − 0.0382·17-s + 1.41·19-s + 1.29·20-s + 1.29·22-s − 0.330·23-s − 0.712·25-s + 2.28·26-s + 3.20·28-s + 1.12·29-s − 0.467·31-s − 1.89·32-s + 0.0707·34-s + 0.709·35-s + 1.23·37-s − 2.61·38-s − 1.41·40-s − 0.711·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 + 20.9T + 128T^{2} \) |
| 5 | \( 1 - 149.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.20e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.09e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 9.76e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 774.T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.22e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.92e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.48e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 7.75e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.81e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.13e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.68e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.09e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.35e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 1.17e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.20e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 6.43e4T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.88e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.67e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.61e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.26e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.21e6T + 8.07e13T^{2} \) |
| 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
−9.408829034346496048500861154319, −8.267912528807742111043780902431, −7.78195389184583460979734267803, −7.04578326254388953969138098543, −5.76879393967445654821986625252, −4.82131804003540636803118419175, −2.82071146240850958452357515029, −1.98943024694351746330997201387, −1.15191140831153648539350096702, 0,
1.15191140831153648539350096702, 1.98943024694351746330997201387, 2.82071146240850958452357515029, 4.82131804003540636803118419175, 5.76879393967445654821986625252, 7.04578326254388953969138098543, 7.78195389184583460979734267803, 8.267912528807742111043780902431, 9.408829034346496048500861154319
