| L(s) = 1 | + (−1.82 − 0.0236i)2-s + (1.47 + 0.905i)3-s + (1.33 + 0.0344i)4-s + (2.72 + 0.833i)5-s + (−2.67 − 1.68i)6-s + (1.26 − 3.14i)7-s + (1.21 + 0.0472i)8-s + (1.35 + 2.67i)9-s + (−4.94 − 1.58i)10-s + (3.49 + 1.98i)11-s + (1.93 + 1.25i)12-s + (1.89 + 4.24i)13-s + (−2.37 + 5.71i)14-s + (3.26 + 3.69i)15-s + (−4.88 − 0.252i)16-s + (−4.20 − 1.91i)17-s + ⋯ |
| L(s) = 1 | + (−1.29 − 0.0166i)2-s + (0.852 + 0.522i)3-s + (0.666 + 0.0172i)4-s + (1.21 + 0.372i)5-s + (−1.09 − 0.689i)6-s + (0.476 − 1.18i)7-s + (0.430 + 0.0166i)8-s + (0.453 + 0.891i)9-s + (−1.56 − 0.501i)10-s + (1.05 + 0.599i)11-s + (0.558 + 0.363i)12-s + (0.526 + 1.17i)13-s + (−0.635 + 1.52i)14-s + (0.842 + 0.953i)15-s + (−1.22 − 0.0631i)16-s + (−1.01 − 0.463i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38281 + 0.388144i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38281 + 0.388144i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.47 - 0.905i)T \) |
| good | 2 | \( 1 + (1.82 + 0.0236i)T + (1.99 + 0.0517i)T^{2} \) |
| 5 | \( 1 + (-2.72 - 0.833i)T + (4.14 + 2.80i)T^{2} \) |
| 7 | \( 1 + (-1.26 + 3.14i)T + (-5.06 - 4.83i)T^{2} \) |
| 11 | \( 1 + (-3.49 - 1.98i)T + (5.62 + 9.45i)T^{2} \) |
| 13 | \( 1 + (-1.89 - 4.24i)T + (-8.67 + 9.68i)T^{2} \) |
| 17 | \( 1 + (4.20 + 1.91i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.127 + 0.185i)T + (-6.84 + 17.7i)T^{2} \) |
| 23 | \( 1 + (-2.01 + 6.13i)T + (-18.5 - 13.6i)T^{2} \) |
| 29 | \( 1 + (-4.61 + 2.94i)T + (12.1 - 26.3i)T^{2} \) |
| 31 | \( 1 + (0.792 + 2.92i)T + (-26.7 + 15.6i)T^{2} \) |
| 37 | \( 1 + (11.5 + 2.26i)T + (34.2 + 13.9i)T^{2} \) |
| 41 | \( 1 + (0.528 - 0.200i)T + (30.7 - 27.1i)T^{2} \) |
| 43 | \( 1 + (5.16 - 2.26i)T + (29.1 - 31.6i)T^{2} \) |
| 47 | \( 1 + (0.358 - 1.32i)T + (-40.5 - 23.7i)T^{2} \) |
| 53 | \( 1 + (-1.85 + 1.96i)T + (-3.08 - 52.9i)T^{2} \) |
| 59 | \( 1 + (-5.24 + 0.890i)T + (55.6 - 19.4i)T^{2} \) |
| 61 | \( 1 + (6.99 - 12.8i)T + (-33.1 - 51.1i)T^{2} \) |
| 67 | \( 1 + (6.25 - 3.24i)T + (38.6 - 54.7i)T^{2} \) |
| 71 | \( 1 + (5.07 + 3.93i)T + (17.7 + 68.7i)T^{2} \) |
| 73 | \( 1 + (-12.6 + 4.04i)T + (59.3 - 42.4i)T^{2} \) |
| 79 | \( 1 + (-1.91 + 10.0i)T + (-73.5 - 28.9i)T^{2} \) |
| 83 | \( 1 + (-2.55 - 15.6i)T + (-78.7 + 26.3i)T^{2} \) |
| 89 | \( 1 + (1.35 + 0.553i)T + (63.5 + 62.3i)T^{2} \) |
| 97 | \( 1 + (3.05 - 13.2i)T + (-87.2 - 42.4i)T^{2} \) |
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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
−10.40189561030590055182494131956, −9.388660585794255694527459771971, −9.075856141965710109275023766907, −8.199233122786012808335253912525, −6.98365673111750007543083520805, −6.68200006998033101942150322190, −4.69156308418908477131798945011, −4.08594686628529438978615804848, −2.27959891111963710849593955777, −1.47106982794630788901726828394,
1.33004254851605030368063110544, 1.96936066427705552130952915470, 3.35640464831955618508328685109, 5.12478558261028671985702620137, 6.11902867762160464181559490871, 7.00299243266003221395383641700, 8.313559148454548234133979138083, 8.670598665819418268428337453721, 9.133446415121263193326758017040, 9.941423347793370968448209494532
