L(s) = 1 | + (1.09 + 0.796i)2-s + (−0.177 + 0.547i)3-s + (−0.0501 − 0.154i)4-s + (−0.631 + 0.458i)6-s + (1.12 + 3.47i)7-s + (0.905 − 2.78i)8-s + (2.15 + 1.56i)9-s + (0.490 + 3.28i)11-s + 0.0933·12-s + (−2.29 − 1.66i)13-s + (−1.52 + 4.70i)14-s + (2.95 − 2.14i)16-s + (2.98 − 2.17i)17-s + (1.11 + 3.44i)18-s + (−0.0293 + 0.0904i)19-s + ⋯ |
L(s) = 1 | + (0.775 + 0.563i)2-s + (−0.102 + 0.315i)3-s + (−0.0250 − 0.0771i)4-s + (−0.257 + 0.187i)6-s + (0.426 + 1.31i)7-s + (0.320 − 0.985i)8-s + (0.719 + 0.522i)9-s + (0.147 + 0.989i)11-s + 0.0269·12-s + (−0.635 − 0.461i)13-s + (−0.408 + 1.25i)14-s + (0.738 − 0.536i)16-s + (0.724 − 0.526i)17-s + (0.263 + 0.811i)18-s + (−0.00674 + 0.0207i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.59033 + 0.991121i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.59033 + 0.991121i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 + (-0.490 - 3.28i)T \) |
good | 2 | \( 1 + (-1.09 - 0.796i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.177 - 0.547i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + (-1.12 - 3.47i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (2.29 + 1.66i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.98 + 2.17i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.0293 - 0.0904i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 + (2.08 + 6.42i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (5.48 + 3.98i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (3.04 + 9.35i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.57 - 7.91i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 2.96T + 43T^{2} \) |
| 47 | \( 1 + (-0.687 + 2.11i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.42 - 1.75i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (2.62 + 8.09i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-6.86 + 4.98i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + (6.71 - 4.88i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.407 - 1.25i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-11.2 - 8.15i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (8.61 - 6.25i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + (3.50 + 2.54i)T + (29.9 + 92.2i)T^{2} \) |
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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
−12.38836071467941380466392405565, −11.23388094899694593617222065931, −9.904663210881224144876202571879, −9.515887235148610332305725503096, −7.919613458849685235502239636545, −7.05223653434846154336638277318, −5.64383470261916782754013713628, −5.12214293969337174337336525495, −4.06697958676918546708605482388, −2.13926329412048976717365163497,
1.47796327373117168298806617629, 3.41127725753430372802640010255, 4.16802929896670752819210137907, 5.36214654125187473310090821069, 6.86237776365364875183327742078, 7.66789367091162350496386936575, 8.787664258721151405176590817774, 10.20923906728179200229580819751, 10.94794730075471552947568815805, 11.92279024836021634243859758799