| L(s) = 1 | − 10.1·2-s + 9·3-s + 71.2·4-s + 25·5-s − 91.4·6-s + 177.·7-s − 399.·8-s + 81·9-s − 254.·10-s − 242.·11-s + 641.·12-s − 169·13-s − 1.80e3·14-s + 225·15-s + 1.77e3·16-s − 2.15e3·17-s − 823.·18-s − 2.44e3·19-s + 1.78e3·20-s + 1.59e3·21-s + 2.46e3·22-s − 2.31e3·23-s − 3.59e3·24-s + 625·25-s + 1.71e3·26-s + 729·27-s + 1.26e4·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.577·3-s + 2.22·4-s + 0.447·5-s − 1.03·6-s + 1.36·7-s − 2.20·8-s + 0.333·9-s − 0.803·10-s − 0.604·11-s + 1.28·12-s − 0.277·13-s − 2.45·14-s + 0.258·15-s + 1.73·16-s − 1.80·17-s − 0.598·18-s − 1.55·19-s + 0.996·20-s + 0.789·21-s + 1.08·22-s − 0.913·23-s − 1.27·24-s + 0.200·25-s + 0.498·26-s + 0.192·27-s + 3.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 9T \) |
| 5 | \( 1 - 25T \) |
| 13 | \( 1 + 169T \) |
| good | 2 | \( 1 + 10.1T + 32T^{2} \) |
| 7 | \( 1 - 177.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 242.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 2.15e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.44e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.31e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 550.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.96e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.86e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.15e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.98e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.57e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.03e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.08e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.64e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.42e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.42e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.94e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.32e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.37e5T + 8.58e9T^{2} \) |
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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
−10.76019327914351845726276339339, −10.09476970092954004458432378741, −8.808389236938870029138926531982, −8.442528139930943302911173233562, −7.47097419004719226147916187528, −6.37714676921522357119539044271, −4.61264535085130128486391072591, −2.33886415484967366231048390978, −1.75760940851970976289839247184, 0,
1.75760940851970976289839247184, 2.33886415484967366231048390978, 4.61264535085130128486391072591, 6.37714676921522357119539044271, 7.47097419004719226147916187528, 8.442528139930943302911173233562, 8.808389236938870029138926531982, 10.09476970092954004458432378741, 10.76019327914351845726276339339
