| 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
−11.19606764847308579593486744415, −10.04503240597234401750929711367, −9.144559072814020836721679892351, −8.515281293940267027194450830073, −7.82128403720502112187578864191, −6.58700555851749536584353849899, −5.94182536583533272227603258761, −5.04310419720541950665144636645, −3.60650968377835172554712129354, −2.07469954812116890611637222881,
0.29170233864630120088462572004, 2.43111299029325920593385527486, 2.90793815937401898692840323342, 4.21388022297434717842421852117, 5.88988018334051520368278684852, 6.71196380622787504471564487944, 7.38762182404967037331698041126, 8.939508201824492717435058380202, 9.650420620694325152991266509490, 10.59700387774312193484693326482
