| L(s) = 1 | + (2.26 − 1.89i)2-s + (5.05 − 1.18i)3-s + (0.123 − 0.703i)4-s + (−16.0 + 5.82i)5-s + (9.18 − 12.2i)6-s + (−2.22 − 12.6i)7-s + (10.7 + 18.6i)8-s + (24.1 − 12.0i)9-s + (−25.1 + 43.5i)10-s + (−48.2 − 17.5i)11-s + (−0.207 − 3.70i)12-s + (42.4 + 35.6i)13-s + (−29.0 − 24.3i)14-s + (−74.0 + 48.4i)15-s + (65.0 + 23.6i)16-s + (18.6 − 32.2i)17-s + ⋯ |
| L(s) = 1 | + (0.799 − 0.670i)2-s + (0.973 − 0.228i)3-s + (0.0154 − 0.0879i)4-s + (−1.43 + 0.521i)5-s + (0.625 − 0.835i)6-s + (−0.120 − 0.682i)7-s + (0.475 + 0.823i)8-s + (0.895 − 0.444i)9-s + (−0.795 + 1.37i)10-s + (−1.32 − 0.481i)11-s + (−0.00499 − 0.0891i)12-s + (0.906 + 0.760i)13-s + (−0.553 − 0.464i)14-s + (−1.27 + 0.834i)15-s + (1.01 + 0.369i)16-s + (0.265 − 0.460i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.70057 - 0.569866i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70057 - 0.569866i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-5.05 + 1.18i)T \) |
| good | 2 | \( 1 + (-2.26 + 1.89i)T + (1.38 - 7.87i)T^{2} \) |
| 5 | \( 1 + (16.0 - 5.82i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (2.22 + 12.6i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (48.2 + 17.5i)T + (1.01e3 + 855. i)T^{2} \) |
| 13 | \( 1 + (-42.4 - 35.6i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-18.6 + 32.2i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (3.43 + 5.94i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-5.91 + 33.5i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (78.5 - 65.8i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-38.1 + 216. i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (159. - 275. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (77.7 + 65.2i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (97.2 + 35.3i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-85.0 - 482. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 - 136.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-195. + 71.1i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-103. - 587. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (336. + 282. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (69.9 - 121. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (213. + 370. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (555. - 466. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-589. + 494. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-371. - 642. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.28e3 + 466. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−16.35212924636990458857818222690, −15.23801066179166500263777283372, −13.93191131333126109834920879659, −13.13960456117214986645442527731, −11.75042050531947149249127609003, −10.65197439708208584226565117417, −8.339547171840579891679976874018, −7.36517375460138514240794208246, −4.17218920404259020432204081166, −3.07458366110846014994426482639,
3.68381314538710594896729655739, 5.21378327244305811766685806340, 7.50690021053379772851534056314, 8.542448590438372136752126157050, 10.40892788285278227939266349365, 12.46839986442950781521692037345, 13.32861838424103976944873382124, 14.89561990588879921356640289779, 15.61422307208800208590287224928, 16.02450731723506500100630738115
