| L(s) = 1 | + (−2.31 − 0.408i)2-s + (−1.62 − 2.52i)3-s + (1.43 + 0.523i)4-s + (−3.71 − 4.42i)5-s + (2.73 + 6.50i)6-s + (4.57 − 1.66i)7-s + (5.02 + 2.90i)8-s + (−3.71 + 8.19i)9-s + (6.79 + 11.7i)10-s + (1.90 − 2.27i)11-s + (−1.01 − 4.47i)12-s + (−3.38 − 19.1i)13-s + (−11.2 + 1.98i)14-s + (−5.13 + 16.5i)15-s + (−15.1 − 12.7i)16-s + (21.7 − 12.5i)17-s + ⋯ |
| L(s) = 1 | + (−1.15 − 0.204i)2-s + (−0.541 − 0.840i)3-s + (0.359 + 0.130i)4-s + (−0.743 − 0.885i)5-s + (0.455 + 1.08i)6-s + (0.653 − 0.237i)7-s + (0.628 + 0.362i)8-s + (−0.413 + 0.910i)9-s + (0.679 + 1.17i)10-s + (0.173 − 0.206i)11-s + (−0.0847 − 0.373i)12-s + (−0.260 − 1.47i)13-s + (−0.805 + 0.142i)14-s + (−0.342 + 1.10i)15-s + (−0.947 − 0.794i)16-s + (1.27 − 0.737i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.184971 - 0.354970i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.184971 - 0.354970i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.62 + 2.52i)T \) |
| good | 2 | \( 1 + (2.31 + 0.408i)T + (3.75 + 1.36i)T^{2} \) |
| 5 | \( 1 + (3.71 + 4.42i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-4.57 + 1.66i)T + (37.5 - 31.4i)T^{2} \) |
| 11 | \( 1 + (-1.90 + 2.27i)T + (-21.0 - 119. i)T^{2} \) |
| 13 | \( 1 + (3.38 + 19.1i)T + (-158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (-21.7 + 12.5i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (8.92 - 15.4i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (6.02 - 16.5i)T + (-405. - 340. i)T^{2} \) |
| 29 | \( 1 + (3.42 + 0.603i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (-47.5 - 17.3i)T + (736. + 617. i)T^{2} \) |
| 37 | \( 1 + (11.1 + 19.3i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (2.61 - 0.461i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-19.4 - 16.3i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (7.07 + 19.4i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + 12.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (13.4 + 15.9i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-86.5 + 31.5i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-9.10 - 51.6i)T + (-4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-77.7 + 44.8i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (6.60 - 11.4i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (1.53 - 8.71i)T + (-5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (33.4 + 5.89i)T + (6.47e3 + 2.35e3i)T^{2} \) |
| 89 | \( 1 + (12.6 + 7.31i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (82.6 + 69.3i)T + (1.63e3 + 9.26e3i)T^{2} \) |
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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
−17.11070055368052523723551674342, −16.08862873497479888234789122497, −14.10062453054003640397244647913, −12.57548595615208537070857083974, −11.53037678242504968638465848353, −10.18303030812079517634092736261, −8.260714467635490451076849914759, −7.73483422594157646365233409597, −5.18252863897306006704747229543, −0.903157666080147258568918987037,
4.30569106328665436014640329538, 6.78581847380761053552446675223, 8.342993829594852539683117240493, 9.737652849953661838549476580150, 10.89071979984473912256815613756, 11.89835750788639806886832318409, 14.39365394100025247834966867336, 15.39156771345773883845176312648, 16.62608514511436396129378296299, 17.42894210950092938442927946649
