L(s) = 1 | + (0.753 − 0.435i)3-s + (0.866 − 1.5i)5-s + (−0.615 + 2.57i)7-s + (−1.12 + 1.94i)9-s + (1.81 + 3.15i)11-s − 1.01·13-s − 1.50i·15-s + (−0.621 + 0.358i)17-s + (5.45 + 3.15i)19-s + (0.655 + 2.20i)21-s + (3.15 + 1.81i)23-s + (1 + 1.73i)25-s + 4.56i·27-s + 4.58i·29-s + (−3.76 − 6.52i)31-s + ⋯ |
L(s) = 1 | + (0.435 − 0.251i)3-s + (0.387 − 0.670i)5-s + (−0.232 + 0.972i)7-s + (−0.373 + 0.647i)9-s + (0.548 + 0.950i)11-s − 0.281·13-s − 0.389i·15-s + (−0.150 + 0.0870i)17-s + (1.25 + 0.723i)19-s + (0.143 + 0.481i)21-s + (0.657 + 0.379i)23-s + (0.200 + 0.346i)25-s + 0.878i·27-s + 0.851i·29-s + (−0.676 − 1.17i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.72262 + 0.614893i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.72262 + 0.614893i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.615 - 2.57i)T \) |
good | 3 | \( 1 + (-0.753 + 0.435i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.866 + 1.5i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.81 - 3.15i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.01T + 13T^{2} \) |
| 17 | \( 1 + (0.621 - 0.358i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.45 - 3.15i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.15 - 1.81i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4.58iT - 29T^{2} \) |
| 31 | \( 1 + (3.76 + 6.52i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.27 + 3.62i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 7.34iT - 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + (-1.30 + 2.26i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.82 + 2.20i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.753 + 0.435i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.27 + 10.8i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.88 - 4.99i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.5iT - 71T^{2} \) |
| 73 | \( 1 + (3.25 - 1.88i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.76 + 3.32i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.48iT - 83T^{2} \) |
| 89 | \( 1 + (6.98 + 4.03i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 1.01iT - 97T^{2} \) |
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
−9.940013883958860282595843525594, −9.186479026852673897811751131810, −8.780322967490597704308602260375, −7.66251294384061837945509305427, −6.96930415796978943463711528438, −5.47132203494787440701641149587, −5.30331760618033595886019168614, −3.79144895605246729641669213897, −2.51167313261814997622220579580, −1.60473482649529668352624186058,
0.888763603801423088561723942793, 2.82837637774494790128462221030, 3.40034897027623583444560926837, 4.51542602738627574862995292654, 5.82153577298314063432987855722, 6.68140877019741189690626356730, 7.31653592006193153603258576465, 8.513917889901105166229381539199, 9.246252713950065974317916297611, 9.968376512718168825712598991200