| L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.0316 − 1.73i)3-s + (0.809 − 0.587i)4-s + (−0.866 + 0.5i)5-s + (0.505 + 1.65i)6-s + (−0.0802 − 0.763i)7-s + (−0.587 + 0.809i)8-s + (−2.99 − 0.109i)9-s + (0.669 − 0.743i)10-s + (−0.551 − 0.245i)11-s + (−0.992 − 1.41i)12-s + (1.24 + 5.85i)13-s + (0.312 + 0.701i)14-s + (0.838 + 1.51i)15-s + (0.309 − 0.951i)16-s + (−0.895 + 0.398i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (0.0182 − 0.999i)3-s + (0.404 − 0.293i)4-s + (−0.387 + 0.223i)5-s + (0.206 + 0.676i)6-s + (−0.0303 − 0.288i)7-s + (−0.207 + 0.286i)8-s + (−0.999 − 0.0364i)9-s + (0.211 − 0.235i)10-s + (−0.166 − 0.0740i)11-s + (−0.286 − 0.409i)12-s + (0.345 + 1.62i)13-s + (0.0834 + 0.187i)14-s + (0.216 + 0.391i)15-s + (0.0772 − 0.237i)16-s + (−0.217 + 0.0966i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.494985 + 0.363698i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.494985 + 0.363698i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (-0.0316 + 1.73i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 31 | \( 1 + (-1.97 - 5.20i)T \) |
| good | 7 | \( 1 + (0.0802 + 0.763i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (0.551 + 0.245i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-1.24 - 5.85i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (0.895 - 0.398i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (4.05 + 0.861i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-0.492 - 0.357i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.63 - 8.11i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (9.50 + 5.48i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.20 - 7.38i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-0.784 + 3.69i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (9.44 + 3.07i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.409 - 3.89i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-7.27 + 6.54i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 11.7iT - 61T^{2} \) |
| 67 | \( 1 + (0.364 + 0.630i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.35 - 0.667i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (5.73 - 12.8i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-2.71 - 6.09i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-4.71 + 5.24i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-4.80 + 3.49i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-7.85 + 5.70i)T + (29.9 - 92.2i)T^{2} \) |
| 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
−10.34845249372009785120269990396, −8.933315320709948796135602588508, −8.705624183286340083098013148940, −7.63232191703897071923074614998, −6.78367995663747474208558265371, −6.52674428632309438429452593361, −5.15807413318413765169562764470, −3.82034986664959901185654149061, −2.45467265582844882083477369365, −1.33112273317405327454656260947,
0.37547657057347697524428022520, 2.43995726779555032386767894391, 3.43859428126690824990324697182, 4.44627725051430524662470903450, 5.51482453581324424915619948470, 6.37944203793653677145406315235, 7.86892587762655840405964634057, 8.274820616500979042905921116781, 9.106318093072895976117242202524, 9.992722499849843992337580207829
