L(s) = 1 | + (−0.162 − 1.40i)2-s + (−0.373 − 1.69i)3-s + (−1.94 + 0.455i)4-s + (0.432 + 0.249i)5-s + (−2.31 + 0.798i)6-s + (0.261 − 2.63i)7-s + (0.956 + 2.66i)8-s + (−2.72 + 1.26i)9-s + (0.280 − 0.648i)10-s + (0.695 + 1.20i)11-s + (1.49 + 3.12i)12-s + 2.75·13-s + (−3.74 + 0.0590i)14-s + (0.260 − 0.824i)15-s + (3.58 − 1.77i)16-s + (5.04 − 2.91i)17-s + ⋯ |
L(s) = 1 | + (−0.114 − 0.993i)2-s + (−0.215 − 0.976i)3-s + (−0.973 + 0.227i)4-s + (0.193 + 0.111i)5-s + (−0.945 + 0.326i)6-s + (0.0989 − 0.995i)7-s + (0.338 + 0.941i)8-s + (−0.907 + 0.420i)9-s + (0.0887 − 0.204i)10-s + (0.209 + 0.363i)11-s + (0.432 + 0.901i)12-s + 0.762·13-s + (−0.999 + 0.0157i)14-s + (0.0673 − 0.212i)15-s + (0.896 − 0.443i)16-s + (1.22 − 0.706i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.593 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.369347 - 0.731682i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.369347 - 0.731682i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.162 + 1.40i)T \) |
| 3 | \( 1 + (0.373 + 1.69i)T \) |
| 7 | \( 1 + (-0.261 + 2.63i)T \) |
good | 5 | \( 1 + (-0.432 - 0.249i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.695 - 1.20i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.75T + 13T^{2} \) |
| 17 | \( 1 + (-5.04 + 2.91i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.14 - 1.24i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.53 - 4.38i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6.32iT - 29T^{2} \) |
| 31 | \( 1 + (5.98 - 3.45i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.67 - 6.35i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 5.74iT - 41T^{2} \) |
| 43 | \( 1 - 3.52iT - 43T^{2} \) |
| 47 | \( 1 + (-2.53 + 4.38i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.54 - 2.62i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.57 - 2.72i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.45 + 4.25i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.30 + 5.37i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4.54T + 71T^{2} \) |
| 73 | \( 1 + (-4.98 - 8.63i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.84 + 1.06i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + (2.12 + 1.22i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 0.526T + 97T^{2} \) |
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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
−13.88588735327333810123918857677, −12.68615579482086776930035369851, −11.80730817069143060515163477959, −10.79442960775594879580257763366, −9.718287676634036635037085955054, −8.211283487273718451120327086168, −7.13570532252354771070861971241, −5.38884027613971922432085192774, −3.48468886026594374124899861375, −1.41869927989089725511735896628,
3.79344313556920506719072178367, 5.42628932110553112217658772082, 6.08335326072965825412400275890, 8.070299290907310401363555113007, 9.040147761282582615932088624294, 9.922869343941312586188026090341, 11.29210493499664423413833050861, 12.62656404965254033122017179548, 14.02868909576342240992633145893, 14.87335522888553030475303548955