| L(s) = 1 | + (2.58 − 0.454i)2-s + (−2.92 + 0.680i)3-s + (2.69 − 0.979i)4-s + (0.519 − 0.618i)5-s + (−7.22 + 3.08i)6-s + (−5.56 − 2.02i)7-s + (−2.57 + 1.48i)8-s + (8.07 − 3.97i)9-s + (1.05 − 1.83i)10-s + (12.1 + 14.5i)11-s + (−7.19 + 4.69i)12-s + (3.42 − 19.4i)13-s + (−15.2 − 2.69i)14-s + (−1.09 + 2.16i)15-s + (−14.7 + 12.3i)16-s + (4.04 + 2.33i)17-s + ⋯ |
| L(s) = 1 | + (1.29 − 0.227i)2-s + (−0.973 + 0.226i)3-s + (0.672 − 0.244i)4-s + (0.103 − 0.123i)5-s + (−1.20 + 0.514i)6-s + (−0.794 − 0.289i)7-s + (−0.322 + 0.186i)8-s + (0.897 − 0.441i)9-s + (0.105 − 0.183i)10-s + (1.10 + 1.32i)11-s + (−0.599 + 0.391i)12-s + (0.263 − 1.49i)13-s + (−1.09 − 0.192i)14-s + (−0.0730 + 0.144i)15-s + (−0.921 + 0.773i)16-s + (0.238 + 0.137i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.25457 - 0.0802728i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25457 - 0.0802728i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.92 - 0.680i)T \) |
| good | 2 | \( 1 + (-2.58 + 0.454i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-0.519 + 0.618i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (5.56 + 2.02i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (-12.1 - 14.5i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (-3.42 + 19.4i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (-4.04 - 2.33i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (4.96 + 8.59i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (4.36 + 11.9i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (5.94 - 1.04i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (34.4 - 12.5i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (-9.43 + 16.3i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-21.6 - 3.82i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (-26.6 + 22.3i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (9.90 - 27.2i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + 19.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-38.4 + 45.8i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-13.9 - 5.07i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-10.1 + 57.5i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-72.4 - 41.8i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-57.2 - 99.1i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-4.96 - 28.1i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (22.9 - 4.04i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (121. - 70.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-50.5 + 42.4i)T + (1.63e3 - 9.26e3i)T^{2} \) |
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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
−17.07312998879302964186673782432, −15.64271140399436967237381423273, −14.65074758205668703049558938767, −12.88231333713587201843512371796, −12.52400162602157180867810680896, −11.02628664095643699320985552905, −9.576286139460278884636756244083, −6.77294171203726658654750551908, −5.39448064111390287488015047932, −3.87609201833165429165325121441,
3.97194568714181189231229484010, 5.88480441734308616487697379630, 6.61187295482400601111181777503, 9.341736928578118106503052373211, 11.33886366763022294457441456388, 12.22848708472081394520725260115, 13.46053641514415603783922654912, 14.41325372510188678970162134257, 16.08480922080659380963090721445, 16.65593977035768622346111244815
