| L(s) = 1 | + (−1.82 − 0.285i)2-s + (1.35 + 0.432i)4-s + (0.984 + 0.447i)5-s + (0.0949 + 0.694i)7-s + (0.961 + 0.482i)8-s + (−1.66 − 1.09i)10-s + (0.605 − 0.0470i)11-s + (1.37 + 4.00i)13-s + (0.0251 − 1.29i)14-s + (−3.92 − 2.80i)16-s + (0.887 − 2.05i)17-s + (−2.74 + 3.68i)19-s + (1.13 + 1.03i)20-s + (−1.11 − 0.0869i)22-s + (2.68 + 2.08i)23-s + ⋯ |
| L(s) = 1 | + (−1.29 − 0.201i)2-s + (0.675 + 0.216i)4-s + (0.440 + 0.200i)5-s + (0.0358 + 0.262i)7-s + (0.340 + 0.170i)8-s + (−0.528 − 0.347i)10-s + (0.182 − 0.0141i)11-s + (0.380 + 1.11i)13-s + (0.00671 − 0.346i)14-s + (−0.981 − 0.701i)16-s + (0.215 − 0.499i)17-s + (−0.629 + 0.845i)19-s + (0.253 + 0.230i)20-s + (−0.238 − 0.0185i)22-s + (0.560 + 0.434i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.612345 + 0.382284i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.612345 + 0.382284i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.82 + 0.285i)T + (1.90 + 0.610i)T^{2} \) |
| 5 | \( 1 + (-0.984 - 0.447i)T + (3.28 + 3.76i)T^{2} \) |
| 7 | \( 1 + (-0.0949 - 0.694i)T + (-6.74 + 1.87i)T^{2} \) |
| 11 | \( 1 + (-0.605 + 0.0470i)T + (10.8 - 1.69i)T^{2} \) |
| 13 | \( 1 + (-1.37 - 4.00i)T + (-10.2 + 7.96i)T^{2} \) |
| 17 | \( 1 + (-0.887 + 2.05i)T + (-11.6 - 12.3i)T^{2} \) |
| 19 | \( 1 + (2.74 - 3.68i)T + (-5.44 - 18.2i)T^{2} \) |
| 23 | \( 1 + (-2.68 - 2.08i)T + (5.73 + 22.2i)T^{2} \) |
| 29 | \( 1 + (1.70 - 1.02i)T + (13.5 - 25.6i)T^{2} \) |
| 31 | \( 1 + (-2.00 + 0.0777i)T + (30.9 - 2.40i)T^{2} \) |
| 37 | \( 1 + (3.62 - 3.84i)T + (-2.15 - 36.9i)T^{2} \) |
| 41 | \( 1 + (1.82 - 4.72i)T + (-30.3 - 27.5i)T^{2} \) |
| 43 | \( 1 + (-0.202 - 0.787i)T + (-37.6 + 20.7i)T^{2} \) |
| 47 | \( 1 + (7.76 + 0.301i)T + (46.8 + 3.64i)T^{2} \) |
| 53 | \( 1 + (-2.73 + 0.996i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-4.04 - 8.44i)T + (-37.0 + 45.9i)T^{2} \) |
| 61 | \( 1 + (-7.31 + 2.34i)T + (49.6 - 35.4i)T^{2} \) |
| 67 | \( 1 + (-12.6 - 7.62i)T + (31.2 + 59.2i)T^{2} \) |
| 71 | \( 1 + (0.252 - 4.33i)T + (-70.5 - 8.24i)T^{2} \) |
| 73 | \( 1 + (6.65 - 4.37i)T + (28.9 - 67.0i)T^{2} \) |
| 79 | \( 1 + (9.22 - 11.4i)T + (-16.7 - 77.2i)T^{2} \) |
| 83 | \( 1 + (-4.63 - 12.0i)T + (-61.4 + 55.7i)T^{2} \) |
| 89 | \( 1 + (-0.360 - 6.18i)T + (-88.3 + 10.3i)T^{2} \) |
| 97 | \( 1 + (-10.2 + 4.65i)T + (63.7 - 73.0i)T^{2} \) |
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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
−10.14381589950480926303213433767, −9.807908432499797942330736177323, −8.827323282556415479616339975806, −8.318038676808959043807545805917, −7.19800888329568918086344388601, −6.41365169636570426044960623626, −5.23352092903163700186378038449, −3.99493226175352575785749171638, −2.41500878533328875547843439593, −1.37709518801858679629742757235,
0.62744519643688412847325934266, 1.98220759960074229448568799233, 3.60212922786616515841567712353, 4.89027788694518005843726110173, 6.02186934110780398042127314117, 6.99724044131721345791517563694, 7.83336366574656305008420641904, 8.651437174933965508863933469098, 9.231060392079235114636400616230, 10.23840873096634281308514237558
