| L(s) = 1 | + (−0.880 − 2.04i)2-s + (−2.01 + 2.13i)4-s + (0.171 − 2.94i)5-s + (2.87 − 0.681i)7-s + (1.96 + 0.714i)8-s + (−6.15 + 2.23i)10-s + (4.43 − 2.91i)11-s + (0.730 − 0.0853i)13-s + (−3.92 − 5.26i)14-s + (0.0720 + 1.23i)16-s + (1.11 + 0.933i)17-s + (4.21 − 3.53i)19-s + (5.94 + 6.29i)20-s + (−9.85 − 6.48i)22-s + (−0.438 − 0.103i)23-s + ⋯ |
| L(s) = 1 | + (−0.622 − 1.44i)2-s + (−1.00 + 1.06i)4-s + (0.0765 − 1.31i)5-s + (1.08 − 0.257i)7-s + (0.693 + 0.252i)8-s + (−1.94 + 0.708i)10-s + (1.33 − 0.879i)11-s + (0.202 − 0.0236i)13-s + (−1.04 − 1.40i)14-s + (0.0180 + 0.309i)16-s + (0.269 + 0.226i)17-s + (0.966 − 0.810i)19-s + (1.32 + 1.40i)20-s + (−2.10 − 1.38i)22-s + (−0.0914 − 0.0216i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0613i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0394365 - 1.28504i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0394365 - 1.28504i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.880 + 2.04i)T + (-1.37 + 1.45i)T^{2} \) |
| 5 | \( 1 + (-0.171 + 2.94i)T + (-4.96 - 0.580i)T^{2} \) |
| 7 | \( 1 + (-2.87 + 0.681i)T + (6.25 - 3.14i)T^{2} \) |
| 11 | \( 1 + (-4.43 + 2.91i)T + (4.35 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-0.730 + 0.0853i)T + (12.6 - 2.99i)T^{2} \) |
| 17 | \( 1 + (-1.11 - 0.933i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-4.21 + 3.53i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (0.438 + 0.103i)T + (20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (-2.12 + 2.85i)T + (-8.31 - 27.7i)T^{2} \) |
| 31 | \( 1 + (-0.319 - 1.06i)T + (-25.9 + 17.0i)T^{2} \) |
| 37 | \( 1 + (0.643 - 3.64i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (4.05 - 9.38i)T + (-28.1 - 29.8i)T^{2} \) |
| 43 | \( 1 + (7.08 + 3.55i)T + (25.6 + 34.4i)T^{2} \) |
| 47 | \( 1 + (1.58 - 5.29i)T + (-39.2 - 25.8i)T^{2} \) |
| 53 | \( 1 + (1.83 - 3.17i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.98 + 3.93i)T + (23.3 + 54.1i)T^{2} \) |
| 61 | \( 1 + (2.98 + 3.16i)T + (-3.54 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-6.75 - 9.07i)T + (-19.2 + 64.1i)T^{2} \) |
| 71 | \( 1 + (12.6 - 4.60i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-3.72 - 1.35i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.12 - 2.61i)T + (-54.2 + 57.4i)T^{2} \) |
| 83 | \( 1 + (-0.139 - 0.323i)T + (-56.9 + 60.3i)T^{2} \) |
| 89 | \( 1 + (-5.34 - 1.94i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.409 - 7.03i)T + (-96.3 + 11.2i)T^{2} \) |
| 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
−9.920550216268107466355433411393, −9.190364082358578569068132505816, −8.559978823355172931886224061253, −7.973541472845460694754491677604, −6.38578377828716027087756220032, −5.07806968042640228260925264678, −4.24937650798810599315203017920, −3.16585885389045233439465858786, −1.50166744937843276642666865480, −0.997777402832915122052052988521,
1.71997295530448616917943575954, 3.40781573630554051711551298566, 4.80875562640251683153786322180, 5.80191874198925807991981215181, 6.70356636858261267858334408563, 7.25116114875964801518140625936, 8.001501781756751856025735031414, 8.932259149505946580040662285718, 9.740416792835433107294419601676, 10.54213517551659822238488161989
