L(s) = 1 | + (−1.78 + 1.03i)2-s + (−1.08 + 1.35i)3-s + (1.12 − 1.95i)4-s + (0.5 + 0.866i)5-s + (0.543 − 3.53i)6-s + 0.527i·8-s + (−0.649 − 2.92i)9-s + (−1.78 − 1.03i)10-s + (4.06 + 2.34i)11-s + (1.41 + 3.64i)12-s − 0.638i·13-s + (−1.71 − 0.263i)15-s + (1.71 + 2.96i)16-s + (−2.07 + 3.59i)17-s + (4.18 + 4.56i)18-s + (0.776 − 0.448i)19-s + ⋯ |
L(s) = 1 | + (−1.26 + 0.729i)2-s + (−0.625 + 0.779i)3-s + (0.563 − 0.976i)4-s + (0.223 + 0.387i)5-s + (0.221 − 1.44i)6-s + 0.186i·8-s + (−0.216 − 0.976i)9-s + (−0.564 − 0.326i)10-s + (1.22 + 0.707i)11-s + (0.408 + 1.05i)12-s − 0.177i·13-s + (−0.442 − 0.0680i)15-s + (0.427 + 0.741i)16-s + (−0.503 + 0.871i)17-s + (0.985 + 1.07i)18-s + (0.178 − 0.102i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.302i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0709588 - 0.457888i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0709588 - 0.457888i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.08 - 1.35i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.78 - 1.03i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-4.06 - 2.34i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 0.638iT - 13T^{2} \) |
| 17 | \( 1 + (2.07 - 3.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.776 + 0.448i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.89 - 3.40i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.14iT - 29T^{2} \) |
| 31 | \( 1 + (-2.02 - 1.16i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.69 - 9.85i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 4.10T + 41T^{2} \) |
| 43 | \( 1 - 3.14T + 43T^{2} \) |
| 47 | \( 1 + (3.40 + 5.89i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.96 + 1.13i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.254 - 0.440i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.48 - 2.59i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.41 - 4.18i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 1.22iT - 71T^{2} \) |
| 73 | \( 1 + (12.5 + 7.22i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.54 + 7.86i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.76T + 83T^{2} \) |
| 89 | \( 1 + (6.90 + 11.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 12.9iT - 97T^{2} \) |
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
−10.34494798973475250325185245681, −9.975744228682386487986280032261, −9.207861975081687470925185859747, −8.462950181729917573035599157715, −7.35929026285637804711967063539, −6.46423064672026562299640384599, −5.99792244267821973731660472157, −4.56048038747747351703932390388, −3.55901376781911530314397102479, −1.49432980129507471105714550877,
0.42831850841198571723677883419, 1.52890066081788449923778387740, 2.61079952171386686701400025288, 4.32463207109042121002242461292, 5.66991431711035109860774654227, 6.48769961257876671150429076843, 7.55127037540905949877712103360, 8.365511457692565457717786605619, 9.130316019135914024974918719260, 9.841848147971877358561816238747