L(s) = 1 | + (−1.40 − 0.144i)2-s + (−0.466 + 1.74i)3-s + (1.95 + 0.406i)4-s + (−0.858 − 2.06i)5-s + (0.907 − 2.38i)6-s + (−2.69 − 0.854i)8-s + (−0.213 − 0.122i)9-s + (0.910 + 3.02i)10-s + (−2.91 + 1.68i)11-s + (−1.62 + 3.21i)12-s + (2.69 − 2.69i)13-s + (3.99 − 0.531i)15-s + (3.66 + 1.59i)16-s + (0.587 − 2.19i)17-s + (0.281 + 0.203i)18-s + (−0.601 + 1.04i)19-s + ⋯ |
L(s) = 1 | + (−0.994 − 0.102i)2-s + (−0.269 + 1.00i)3-s + (0.979 + 0.203i)4-s + (−0.384 − 0.923i)5-s + (0.370 − 0.972i)6-s + (−0.953 − 0.302i)8-s + (−0.0710 − 0.0409i)9-s + (0.287 + 0.957i)10-s + (−0.878 + 0.507i)11-s + (−0.467 + 0.929i)12-s + (0.746 − 0.746i)13-s + (1.03 − 0.137i)15-s + (0.917 + 0.397i)16-s + (0.142 − 0.531i)17-s + (0.0664 + 0.0480i)18-s + (−0.138 + 0.239i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00473074 + 0.187397i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00473074 + 0.187397i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 + 0.144i)T \) |
| 5 | \( 1 + (0.858 + 2.06i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.466 - 1.74i)T + (-2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (2.91 - 1.68i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.69 + 2.69i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.587 + 2.19i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.601 - 1.04i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (7.37 - 1.97i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 6.53iT - 29T^{2} \) |
| 31 | \( 1 + (-3.65 + 2.10i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.72 - 0.729i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 2.68T + 41T^{2} \) |
| 43 | \( 1 + (1.90 + 1.90i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.77 - 10.3i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (12.6 + 3.38i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (3.60 + 6.24i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.61 - 4.52i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.338 + 0.0907i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 11.5iT - 71T^{2} \) |
| 73 | \( 1 + (13.9 + 3.74i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.05 + 7.02i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (11.4 + 11.4i)T + 83iT^{2} \) |
| 89 | \( 1 + (6.54 + 3.77i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.0167 + 0.0167i)T + 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.25264199442523854033625413305, −9.687200255827739475792120524821, −8.869377072527852852919793471915, −7.998055198338730725691161354098, −7.48994301455520896279678335199, −6.02575654599507387551208826466, −5.20172856303263004538967911511, −4.22586027262484855683707693729, −3.14296134130816404795066413303, −1.54497109971324666020961275571,
0.12271201769942989679116029461, 1.69643325228264302451459988028, 2.71526014116417401188634567276, 4.01115115518874752459889619973, 5.92025178018079432232139981292, 6.34882580061184159606475627954, 7.13443662615558209259843765210, 7.969228814744705797860606614894, 8.390452791819684211878165792929, 9.700547957751395733164040218938