| L(s) = 1 | + (−1.44 − 0.226i)2-s + (0.139 + 0.0448i)4-s + (−3.00 − 1.36i)5-s + (−0.603 − 4.41i)7-s + (2.42 + 1.21i)8-s + (4.04 + 2.65i)10-s + (4.13 − 0.321i)11-s + (−0.395 − 1.15i)13-s + (−0.126 + 6.53i)14-s + (−3.47 − 2.48i)16-s + (2.88 − 6.69i)17-s + (0.997 − 1.33i)19-s + (−0.358 − 0.325i)20-s + (−6.05 − 0.470i)22-s + (−0.314 − 0.243i)23-s + ⋯ |
| L(s) = 1 | + (−1.02 − 0.160i)2-s + (0.0698 + 0.0224i)4-s + (−1.34 − 0.610i)5-s + (−0.228 − 1.67i)7-s + (0.857 + 0.430i)8-s + (1.27 + 0.840i)10-s + (1.24 − 0.0968i)11-s + (−0.109 − 0.320i)13-s + (−0.0338 + 1.74i)14-s + (−0.868 − 0.621i)16-s + (0.700 − 1.62i)17-s + (0.228 − 0.307i)19-s + (−0.0802 − 0.0727i)20-s + (−1.29 − 0.100i)22-s + (−0.0654 − 0.0507i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0449423 + 0.387480i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0449423 + 0.387480i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.44 + 0.226i)T + (1.90 + 0.610i)T^{2} \) |
| 5 | \( 1 + (3.00 + 1.36i)T + (3.28 + 3.76i)T^{2} \) |
| 7 | \( 1 + (0.603 + 4.41i)T + (-6.74 + 1.87i)T^{2} \) |
| 11 | \( 1 + (-4.13 + 0.321i)T + (10.8 - 1.69i)T^{2} \) |
| 13 | \( 1 + (0.395 + 1.15i)T + (-10.2 + 7.96i)T^{2} \) |
| 17 | \( 1 + (-2.88 + 6.69i)T + (-11.6 - 12.3i)T^{2} \) |
| 19 | \( 1 + (-0.997 + 1.33i)T + (-5.44 - 18.2i)T^{2} \) |
| 23 | \( 1 + (0.314 + 0.243i)T + (5.73 + 22.2i)T^{2} \) |
| 29 | \( 1 + (8.76 - 5.29i)T + (13.5 - 25.6i)T^{2} \) |
| 31 | \( 1 + (4.29 - 0.166i)T + (30.9 - 2.40i)T^{2} \) |
| 37 | \( 1 + (-2.55 + 2.70i)T + (-2.15 - 36.9i)T^{2} \) |
| 41 | \( 1 + (-0.944 + 2.44i)T + (-30.3 - 27.5i)T^{2} \) |
| 43 | \( 1 + (0.0744 + 0.289i)T + (-37.6 + 20.7i)T^{2} \) |
| 47 | \( 1 + (-3.26 - 0.126i)T + (46.8 + 3.64i)T^{2} \) |
| 53 | \( 1 + (7.96 - 2.89i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (3.24 + 6.78i)T + (-37.0 + 45.9i)T^{2} \) |
| 61 | \( 1 + (-1.62 + 0.521i)T + (49.6 - 35.4i)T^{2} \) |
| 67 | \( 1 + (-7.10 - 4.28i)T + (31.2 + 59.2i)T^{2} \) |
| 71 | \( 1 + (0.445 - 7.65i)T + (-70.5 - 8.24i)T^{2} \) |
| 73 | \( 1 + (-4.00 + 2.63i)T + (28.9 - 67.0i)T^{2} \) |
| 79 | \( 1 + (0.497 - 0.616i)T + (-16.7 - 77.2i)T^{2} \) |
| 83 | \( 1 + (-3.45 - 8.95i)T + (-61.4 + 55.7i)T^{2} \) |
| 89 | \( 1 + (0.648 + 11.1i)T + (-88.3 + 10.3i)T^{2} \) |
| 97 | \( 1 + (13.9 - 6.33i)T + (63.7 - 73.0i)T^{2} \) |
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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
−9.625761841496320732063871757414, −9.279729190251189999471726420184, −8.195237552590806833658281385179, −7.36858570725190828019483217250, −7.11488377807614904659207860621, −5.17436326632395707406869726708, −4.21300011523042479476413948585, −3.52706050755915812435179190983, −1.17400349581524467671276267908, −0.35157009762581705115843753086,
1.78169385087781222864465891979, 3.46687169651245334887262942486, 4.18836580463965341875003677099, 5.77525126612080164720112235732, 6.67033626727719682130358184594, 7.74424149405394048548889731503, 8.236030023786773229647801681420, 9.150053835788677316612804338658, 9.647404596073713020459191189274, 10.84137041126405236633989898991
