| L(s) = 1 | + 2.88·2-s − 23.7·4-s − 16.0·5-s + 243.·7-s − 160.·8-s − 46.1·10-s − 571.·11-s − 491.·13-s + 700.·14-s + 296.·16-s + 870.·17-s + 67.4·19-s + 379.·20-s − 1.64e3·22-s + 4.37e3·23-s − 2.86e3·25-s − 1.41e3·26-s − 5.76e3·28-s − 4.11e3·29-s − 1.71e3·31-s + 5.98e3·32-s + 2.50e3·34-s − 3.89e3·35-s − 6.22e3·37-s + 194.·38-s + 2.57e3·40-s + 7.96e3·41-s + ⋯ |
| L(s) = 1 | + 0.509·2-s − 0.740·4-s − 0.286·5-s + 1.87·7-s − 0.886·8-s − 0.145·10-s − 1.42·11-s − 0.807·13-s + 0.955·14-s + 0.289·16-s + 0.730·17-s + 0.0428·19-s + 0.212·20-s − 0.725·22-s + 1.72·23-s − 0.917·25-s − 0.411·26-s − 1.38·28-s − 0.909·29-s − 0.321·31-s + 1.03·32-s + 0.372·34-s − 0.537·35-s − 0.748·37-s + 0.0218·38-s + 0.254·40-s + 0.739·41-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
| \(L(3)\) |
\(\approx\) |
\(2.051479745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.051479745\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 2.88T + 32T^{2} \) |
| 5 | \( 1 + 16.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 243.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 571.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 491.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 870.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 67.4T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.37e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.71e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.22e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.96e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.38e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 909.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.25e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.70e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.04e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.49e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.61e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.81e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.09e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.14e5T + 8.58e9T^{2} \) |
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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
−9.615351578677452663326678576912, −8.597274397193709930976988346436, −7.907839454443137711875879556689, −7.30991011307863636319017643465, −5.46042171467489326804952142404, −5.18846188105030868085972999791, −4.40024300999191344186918363428, −3.20711821276203771356806636940, −1.98968947584369685411626203406, −0.61805123184293225673673586069,
0.61805123184293225673673586069, 1.98968947584369685411626203406, 3.20711821276203771356806636940, 4.40024300999191344186918363428, 5.18846188105030868085972999791, 5.46042171467489326804952142404, 7.30991011307863636319017643465, 7.907839454443137711875879556689, 8.597274397193709930976988346436, 9.615351578677452663326678576912
