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
−14.87335522888553030475303548955, −14.02868909576342240992633145893, −12.62656404965254033122017179548, −11.29210493499664423413833050861, −9.922869343941312586188026090341, −9.040147761282582615932088624294, −8.070299290907310401363555113007, −6.08335326072965825412400275890, −5.42628932110553112217658772082, −3.79344313556920506719072178367,
1.41869927989089725511735896628, 3.48468886026594374124899861375, 5.38884027613971922432085192774, 7.13570532252354771070861971241, 8.211283487273718451120327086168, 9.718287676634036635037085955054, 10.79442960775594879580257763366, 11.80730817069143060515163477959, 12.68615579482086776930035369851, 13.88588735327333810123918857677