| L(s) = 1 | + (−0.492 − 1.35i)2-s + (−0.0577 + 0.0484i)4-s + (0.440 + 2.49i)5-s + (−2.34 − 1.22i)7-s + (−2.40 − 1.38i)8-s + (3.16 − 1.82i)10-s + (−2.06 − 0.364i)11-s + (−0.0331 + 0.0909i)13-s + (−0.494 + 3.77i)14-s + (−0.719 + 4.08i)16-s + (−3.94 − 6.82i)17-s + (−1.53 − 0.887i)19-s + (−0.146 − 0.122i)20-s + (0.525 + 2.98i)22-s + (−3.82 − 4.56i)23-s + ⋯ |
| L(s) = 1 | + (−0.348 − 0.957i)2-s + (−0.0288 + 0.0242i)4-s + (0.196 + 1.11i)5-s + (−0.887 − 0.461i)7-s + (−0.848 − 0.490i)8-s + (1.00 − 0.577i)10-s + (−0.623 − 0.110i)11-s + (−0.00918 + 0.0252i)13-s + (−0.132 + 1.01i)14-s + (−0.179 + 1.02i)16-s + (−0.956 − 1.65i)17-s + (−0.352 − 0.203i)19-s + (−0.0327 − 0.0274i)20-s + (0.112 + 0.635i)22-s + (−0.798 − 0.951i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0573879 + 0.504725i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0573879 + 0.504725i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.34 + 1.22i)T \) |
| good | 2 | \( 1 + (0.492 + 1.35i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.440 - 2.49i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (2.06 + 0.364i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (0.0331 - 0.0909i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (3.94 + 6.82i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.53 + 0.887i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.82 + 4.56i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.573 + 1.57i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (0.916 + 1.09i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (1.88 + 3.26i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.60 - 0.946i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.03 + 5.87i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.23 - 2.71i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 9.25iT - 53T^{2} \) |
| 59 | \( 1 + (-1.28 - 7.27i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (7.80 - 9.30i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (10.6 + 3.86i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-11.9 + 6.88i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-12.0 - 6.94i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.82 - 2.11i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (8.67 - 3.15i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-6.28 + 10.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-15.0 - 2.65i)T + (91.1 + 33.1i)T^{2} \) |
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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
−10.59700387774312193484693326482, −9.650420620694325152991266509490, −8.939508201824492717435058380202, −7.38762182404967037331698041126, −6.71196380622787504471564487944, −5.88988018334051520368278684852, −4.21388022297434717842421852117, −2.90793815937401898692840323342, −2.43111299029325920593385527486, −0.29170233864630120088462572004,
2.07469954812116890611637222881, 3.60650968377835172554712129354, 5.04310419720541950665144636645, 5.94182536583533272227603258761, 6.58700555851749536584353849899, 7.82128403720502112187578864191, 8.515281293940267027194450830073, 9.144559072814020836721679892351, 10.04503240597234401750929711367, 11.19606764847308579593486744415
