| L(s) = 1 | − 8.67·2-s + 43.3·4-s − 26.8·5-s + 155.·7-s − 98.4·8-s + 233.·10-s + 447.·11-s − 776.·13-s − 1.34e3·14-s − 532.·16-s + 501.·17-s − 463.·19-s − 1.16e3·20-s − 3.88e3·22-s + 3.34e3·23-s − 2.40e3·25-s + 6.74e3·26-s + 6.72e3·28-s + 5.99e3·29-s + 6.16e3·31-s + 7.77e3·32-s − 4.34e3·34-s − 4.17e3·35-s − 1.07e4·37-s + 4.01e3·38-s + 2.64e3·40-s + 7.37e3·41-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 1.35·4-s − 0.481·5-s + 1.19·7-s − 0.543·8-s + 0.738·10-s + 1.11·11-s − 1.27·13-s − 1.83·14-s − 0.520·16-s + 0.420·17-s − 0.294·19-s − 0.651·20-s − 1.71·22-s + 1.31·23-s − 0.768·25-s + 1.95·26-s + 1.61·28-s + 1.32·29-s + 1.15·31-s + 1.34·32-s − 0.645·34-s − 0.575·35-s − 1.29·37-s + 0.451·38-s + 0.261·40-s + 0.685·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.092589224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.092589224\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 8.67T + 32T^{2} \) |
| 5 | \( 1 + 26.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 155.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 447.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 776.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 501.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 463.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.34e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.99e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.07e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.37e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.22e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.93e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 796.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.16e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 7.17e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.12e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.84e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.54e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.71e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.21e5T + 8.58e9T^{2} \) |
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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
−9.542934013679778476291878375814, −8.717683197511819167431480187242, −8.062394720751083184928897832373, −7.35105151757895619016819924056, −6.59397263435939523333354276822, −5.05690597143014196698117049445, −4.21922288207796720289118468643, −2.61429974608867500164374010214, −1.49295452356513735849598835442, −0.66903993034663422870104160026,
0.66903993034663422870104160026, 1.49295452356513735849598835442, 2.61429974608867500164374010214, 4.21922288207796720289118468643, 5.05690597143014196698117049445, 6.59397263435939523333354276822, 7.35105151757895619016819924056, 8.062394720751083184928897832373, 8.717683197511819167431480187242, 9.542934013679778476291878375814
