| L(s) = 1 | + (−0.453 + 0.481i)2-s + (0.0908 + 1.55i)4-s + (−2.49 − 0.291i)5-s + (3.67 − 1.84i)7-s + (−1.80 − 1.51i)8-s + (1.27 − 1.06i)10-s + (−1.17 + 2.71i)11-s + (−3.15 + 0.747i)13-s + (−0.779 + 2.60i)14-s + (−1.55 + 0.181i)16-s + (0.572 + 3.24i)17-s + (−0.571 + 3.23i)19-s + (0.228 − 3.92i)20-s + (−0.775 − 1.79i)22-s + (−2.23 − 1.12i)23-s + ⋯ |
| L(s) = 1 | + (−0.320 + 0.340i)2-s + (0.0454 + 0.779i)4-s + (−1.11 − 0.130i)5-s + (1.38 − 0.697i)7-s + (−0.638 − 0.535i)8-s + (0.402 − 0.337i)10-s + (−0.353 + 0.819i)11-s + (−0.875 + 0.207i)13-s + (−0.208 + 0.696i)14-s + (−0.389 + 0.0454i)16-s + (0.138 + 0.787i)17-s + (−0.131 + 0.743i)19-s + (0.0510 − 0.876i)20-s + (−0.165 − 0.383i)22-s + (−0.465 − 0.233i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0125511 + 0.484235i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0125511 + 0.484235i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.453 - 0.481i)T + (-0.116 - 1.99i)T^{2} \) |
| 5 | \( 1 + (2.49 + 0.291i)T + (4.86 + 1.15i)T^{2} \) |
| 7 | \( 1 + (-3.67 + 1.84i)T + (4.18 - 5.61i)T^{2} \) |
| 11 | \( 1 + (1.17 - 2.71i)T + (-7.54 - 8.00i)T^{2} \) |
| 13 | \( 1 + (3.15 - 0.747i)T + (11.6 - 5.83i)T^{2} \) |
| 17 | \( 1 + (-0.572 - 3.24i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (0.571 - 3.23i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (2.23 + 1.12i)T + (13.7 + 18.4i)T^{2} \) |
| 29 | \( 1 + (-1.59 - 5.33i)T + (-24.2 + 15.9i)T^{2} \) |
| 31 | \( 1 + (5.36 - 3.52i)T + (12.2 - 28.4i)T^{2} \) |
| 37 | \( 1 + (2.56 + 0.935i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (6.20 + 6.58i)T + (-2.38 + 40.9i)T^{2} \) |
| 43 | \( 1 + (4.66 + 6.27i)T + (-12.3 + 41.1i)T^{2} \) |
| 47 | \( 1 + (8.45 + 5.56i)T + (18.6 + 43.1i)T^{2} \) |
| 53 | \( 1 + (0.00494 + 0.00856i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.50 - 10.4i)T + (-40.4 + 42.9i)T^{2} \) |
| 61 | \( 1 + (0.523 - 8.99i)T + (-60.5 - 7.08i)T^{2} \) |
| 67 | \( 1 + (1.67 - 5.59i)T + (-55.9 - 36.8i)T^{2} \) |
| 71 | \( 1 + (-1.81 + 1.52i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-3.61 - 3.03i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (6.17 - 6.54i)T + (-4.59 - 78.8i)T^{2} \) |
| 83 | \( 1 + (2.54 - 2.69i)T + (-4.82 - 82.8i)T^{2} \) |
| 89 | \( 1 + (-9.57 - 8.03i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-7.10 + 0.830i)T + (94.3 - 22.3i)T^{2} \) |
| 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
−10.76991073660420183286891946802, −10.05348482525379631224468462402, −8.588216980844779741150793157006, −8.248309529284372801617312818969, −7.32491794210681598488230565047, −7.07223499460260426459090093822, −5.24430966449234691893459303240, −4.29835240756846635109880145279, −3.61973145719317179296290120736, −1.89093294906283248700303716126,
0.26800452798213188437346128030, 1.94538978199236950741158896925, 3.09327956128281538959387557504, 4.71968267154454275850439788262, 5.21805543021896204456674651945, 6.38992547085823644138661745083, 7.75729695677646420301162087518, 8.140955244291453262578957592238, 9.149047498480448764676046795172, 10.00348441836382153316369131426
