L(s) = 1 | + (−0.866 + 0.5i)3-s + (−1.25 + 2.16i)5-s + (1.36 − 2.26i)7-s + (0.499 − 0.866i)9-s + (−2.83 − 4.91i)11-s + 5.31·13-s − 2.50i·15-s + (−0.393 + 0.227i)17-s + (3.19 + 1.84i)19-s + (−0.0468 + 2.64i)21-s + (4.43 + 2.56i)23-s + (−0.632 − 1.09i)25-s + 0.999i·27-s − 2.57i·29-s + (3.00 + 5.20i)31-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.288i)3-s + (−0.559 + 0.969i)5-s + (0.515 − 0.857i)7-s + (0.166 − 0.288i)9-s + (−0.855 − 1.48i)11-s + 1.47·13-s − 0.646i·15-s + (−0.0955 + 0.0551i)17-s + (0.733 + 0.423i)19-s + (−0.0102 + 0.577i)21-s + (0.924 + 0.533i)23-s + (−0.126 − 0.219i)25-s + 0.192i·27-s − 0.479i·29-s + (0.539 + 0.934i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23557 + 0.0526740i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23557 + 0.0526740i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (-1.36 + 2.26i)T \) |
good | 5 | \( 1 + (1.25 - 2.16i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.83 + 4.91i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.31T + 13T^{2} \) |
| 17 | \( 1 + (0.393 - 0.227i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.19 - 1.84i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.43 - 2.56i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2.57iT - 29T^{2} \) |
| 31 | \( 1 + (-3.00 - 5.20i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.80 - 4.50i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 4.65iT - 41T^{2} \) |
| 43 | \( 1 + 3.66T + 43T^{2} \) |
| 47 | \( 1 + (0.478 - 0.829i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.41 + 3.12i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-8.76 + 5.06i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.50 + 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.65 + 8.05i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.35iT - 71T^{2} \) |
| 73 | \( 1 + (-5.93 + 3.42i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.71 + 4.45i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.96iT - 83T^{2} \) |
| 89 | \( 1 + (-5.91 - 3.41i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 3.71iT - 97T^{2} \) |
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
−10.84262325878523313603190180080, −9.980711354897185226883939831017, −8.584931901718786062554235032102, −7.912652389338098737893894036067, −6.95850627518495225431340738809, −6.06107351034229718975870565343, −5.09337839731599577886741349094, −3.74980827537897326510941309881, −3.17021867981647526253875238018, −0.959301639495827290297095503968,
1.10517337310326101468422538666, 2.52625875633902065791827761660, 4.29897132600367747786453686223, 4.99195798711470733651584884096, 5.82764968656480868861412717771, 7.04794214230497536941842312139, 7.972919362640111598203508865294, 8.647261371417984709675699177388, 9.520600035751402965016118875157, 10.65962525564882485776453509917