L(s) = 1 | + (2.04 − 1.48i)2-s + (1.35 − 4.15i)4-s + (0.809 + 0.587i)5-s + (0.646 − 1.99i)7-s + (−1.85 − 5.69i)8-s + 2.52·10-s + (1.64 + 2.87i)11-s + (−1.04 + 0.757i)13-s + (−1.63 − 5.02i)14-s + (−5.16 − 3.74i)16-s + (−2.41 − 1.75i)17-s + (0.664 + 2.04i)19-s + (3.53 − 2.57i)20-s + (7.63 + 3.43i)22-s − 8.77·23-s + ⋯ |
L(s) = 1 | + (1.44 − 1.04i)2-s + (0.675 − 2.07i)4-s + (0.361 + 0.262i)5-s + (0.244 − 0.752i)7-s + (−0.654 − 2.01i)8-s + 0.798·10-s + (0.496 + 0.867i)11-s + (−0.289 + 0.210i)13-s + (−0.436 − 1.34i)14-s + (−1.29 − 0.937i)16-s + (−0.586 − 0.426i)17-s + (0.152 + 0.469i)19-s + (0.791 − 0.574i)20-s + (1.62 + 0.732i)22-s − 1.83·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.12133 - 2.42271i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.12133 - 2.42271i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (-1.64 - 2.87i)T \) |
good | 2 | \( 1 + (-2.04 + 1.48i)T + (0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 + (-0.646 + 1.99i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (1.04 - 0.757i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (2.41 + 1.75i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.664 - 2.04i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 8.77T + 23T^{2} \) |
| 29 | \( 1 + (-0.189 + 0.582i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.94 + 2.14i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.578 - 1.77i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.57 - 4.85i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 5.17T + 43T^{2} \) |
| 47 | \( 1 + (-2.25 - 6.94i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (2.38 - 1.72i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.00 + 6.17i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.406 - 0.295i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 7.80T + 67T^{2} \) |
| 71 | \( 1 + (9.14 + 6.64i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.43 - 10.5i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (4.33 - 3.14i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-8.77 - 6.37i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 4.32T + 89T^{2} \) |
| 97 | \( 1 + (-0.284 + 0.206i)T + (29.9 - 92.2i)T^{2} \) |
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
−10.89716338354318691682279544288, −10.10623818582571468559640744627, −9.493338854082436573093796813785, −7.77395994435933079558901390634, −6.66132871958880119241137970251, −5.80302539592572669617323876607, −4.53100671009751216785595535850, −4.05904487407791995470187751280, −2.64916746568538197057958707652, −1.58366093775201601225502476770,
2.36952804555738933025342172624, 3.69294958205671947613563480489, 4.69418115073864446081323110932, 5.73729358369997874071850514527, 6.15015264721593799004231856889, 7.28513260505348403934089490243, 8.333576061906870680998833637399, 9.001078336268335573407125032546, 10.43577322657034987081337664949, 11.75039611460540468033868826385