L(s) = 1 | + (1.68 + 0.404i)3-s + (−2.10 − 1.60i)7-s + (2.67 + 1.36i)9-s + (−2.05 − 1.18i)11-s + 0.748·13-s + (6.53 + 3.77i)17-s + (6.11 − 3.53i)19-s + (−2.88 − 3.55i)21-s + (−1.63 − 2.83i)23-s + (3.95 + 3.37i)27-s + 2.48i·29-s + (−6.84 − 3.95i)31-s + (−2.98 − 2.83i)33-s + (−3.73 + 2.15i)37-s + (1.26 + 0.302i)39-s + ⋯ |
L(s) = 1 | + (0.972 + 0.233i)3-s + (−0.794 − 0.607i)7-s + (0.891 + 0.453i)9-s + (−0.620 − 0.358i)11-s + 0.207·13-s + (1.58 + 0.914i)17-s + (1.40 − 0.810i)19-s + (−0.630 − 0.776i)21-s + (−0.340 − 0.590i)23-s + (0.760 + 0.649i)27-s + 0.461i·29-s + (−1.22 − 0.709i)31-s + (−0.519 − 0.493i)33-s + (−0.614 + 0.354i)37-s + (0.201 + 0.0484i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.436843330\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.436843330\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.68 - 0.404i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.10 + 1.60i)T \) |
good | 11 | \( 1 + (2.05 + 1.18i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 0.748T + 13T^{2} \) |
| 17 | \( 1 + (-6.53 - 3.77i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.11 + 3.53i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.63 + 2.83i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.48iT - 29T^{2} \) |
| 31 | \( 1 + (6.84 + 3.95i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.73 - 2.15i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 3.03iT - 43T^{2} \) |
| 47 | \( 1 + (-5.59 + 3.22i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0540 + 0.0935i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.60 + 11.4i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.90 + 3.98i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.09 - 2.94i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 13.9iT - 71T^{2} \) |
| 73 | \( 1 + (-0.780 + 1.35i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.27 - 2.21i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.901iT - 83T^{2} \) |
| 89 | \( 1 + (-2.43 - 4.21i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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
−9.190531072652326928594585871666, −8.210317804281134241022657049288, −7.62507959795392829765274826565, −6.96652373652997447960227046963, −5.85412582634433697807235865266, −5.03539572539453645030957236844, −3.73950602498961974466863159441, −3.42478620250741970739163295015, −2.35664948476147957165732428740, −0.915661526217939627035710160558,
1.16658159031247204223694750075, 2.47123579275808581432478133905, 3.19603575960525338226532189573, 3.92469260836273127511200059077, 5.37232914128048974273762815012, 5.81705403739886034993053936194, 7.22671392315438251240599099095, 7.45680226754130088577051138806, 8.349092223204394681497374508250, 9.260312001639549376613949048941