| L(s) = 1 | + (−1.97 − 0.289i)2-s + (−1.09 + 1.33i)3-s + (3.83 + 1.14i)4-s + (2.38 + 7.86i)5-s + (2.56 − 2.33i)6-s + (2.36 − 11.8i)7-s + (−7.25 − 3.37i)8-s + (−0.585 − 2.94i)9-s + (−2.44 − 16.2i)10-s + (0.949 + 0.0935i)11-s + (−5.74 + 3.87i)12-s + (−23.6 − 7.16i)13-s + (−8.10 + 22.8i)14-s + (−13.1 − 5.44i)15-s + (13.3 + 8.77i)16-s + (−5.56 − 13.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.989 − 0.144i)2-s + (−0.366 + 0.446i)3-s + (0.958 + 0.286i)4-s + (0.477 + 1.57i)5-s + (0.426 − 0.388i)6-s + (0.337 − 1.69i)7-s + (−0.906 − 0.421i)8-s + (−0.0650 − 0.326i)9-s + (−0.244 − 1.62i)10-s + (0.0863 + 0.00850i)11-s + (−0.478 + 0.322i)12-s + (−1.81 − 0.551i)13-s + (−0.578 + 1.62i)14-s + (−0.876 − 0.363i)15-s + (0.836 + 0.548i)16-s + (−0.327 − 0.790i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.450226 - 0.401866i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.450226 - 0.401866i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.97 + 0.289i)T \) |
| 3 | \( 1 + (1.09 - 1.33i)T \) |
| good | 5 | \( 1 + (-2.38 - 7.86i)T + (-20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (-2.36 + 11.8i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-0.949 - 0.0935i)T + (118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (23.6 + 7.16i)T + (140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (5.56 + 13.4i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (-16.7 + 8.96i)T + (200. - 300. i)T^{2} \) |
| 23 | \( 1 + (19.3 + 12.9i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (-43.5 + 4.29i)T + (824. - 164. i)T^{2} \) |
| 31 | \( 1 + (5.14 - 5.14i)T - 961iT^{2} \) |
| 37 | \( 1 + (39.8 + 21.3i)T + (760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (25.7 + 17.2i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (22.0 + 26.8i)T + (-360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (-6.51 - 15.7i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (-2.94 - 0.290i)T + (2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (12.8 + 42.3i)T + (-2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (-45.5 + 55.5i)T + (-725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (-46.8 - 38.4i)T + (875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (-15.3 - 3.04i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-123. + 24.4i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-36.1 + 87.2i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (-34.5 - 64.7i)T + (-3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (45.1 + 67.6i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (16.3 + 16.3i)T + 9.40e3iT^{2} \) |
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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
−10.55523278803127919877246663871, −10.18603372154523934669443023899, −9.580605870665370163231166454022, −7.901351725657855496569350742538, −7.05350882945685680876879106609, −6.71245634616351903103506236656, −5.04459607976086484181335774460, −3.49826711949904559665258553334, −2.39726353524972962398845552112, −0.37600483973988513097198591946,
1.48562840976827350455822790407, 2.34497052565570618593400519373, 4.99970525555033215160804880143, 5.52803462972049175925633509996, 6.59048111479873586174343764480, 7.985236016863544051645837816010, 8.554988624579717563004305422040, 9.418208333003246802184579960658, 10.01519511407864806592065333208, 11.67371970278745484558905108421
