| L(s) = 1 | + (−0.695 − 0.457i)2-s + (−0.517 − 1.19i)4-s + (0.827 + 0.877i)5-s + (1.30 − 0.152i)7-s + (−0.478 + 2.71i)8-s + (−0.174 − 0.989i)10-s + (−0.623 − 2.08i)11-s + (−0.264 − 4.54i)13-s + (−0.978 − 0.491i)14-s + (−0.218 + 0.231i)16-s + (3.54 + 1.28i)17-s + (−2.50 + 0.911i)19-s + (0.623 − 1.44i)20-s + (−0.519 + 1.73i)22-s + (5.99 + 0.700i)23-s + ⋯ |
| L(s) = 1 | + (−0.492 − 0.323i)2-s + (−0.258 − 0.599i)4-s + (0.370 + 0.392i)5-s + (0.493 − 0.0576i)7-s + (−0.169 + 0.958i)8-s + (−0.0551 − 0.312i)10-s + (−0.187 − 0.627i)11-s + (−0.0734 − 1.26i)13-s + (−0.261 − 0.131i)14-s + (−0.0545 + 0.0578i)16-s + (0.859 + 0.312i)17-s + (−0.574 + 0.209i)19-s + (0.139 − 0.323i)20-s + (−0.110 + 0.369i)22-s + (1.24 + 0.146i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0419 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0419 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.769692 - 0.802654i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.769692 - 0.802654i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (0.695 + 0.457i)T + (0.792 + 1.83i)T^{2} \) |
| 5 | \( 1 + (-0.827 - 0.877i)T + (-0.290 + 4.99i)T^{2} \) |
| 7 | \( 1 + (-1.30 + 0.152i)T + (6.81 - 1.61i)T^{2} \) |
| 11 | \( 1 + (0.623 + 2.08i)T + (-9.19 + 6.04i)T^{2} \) |
| 13 | \( 1 + (0.264 + 4.54i)T + (-12.9 + 1.50i)T^{2} \) |
| 17 | \( 1 + (-3.54 - 1.28i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (2.50 - 0.911i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-5.99 - 0.700i)T + (22.3 + 5.30i)T^{2} \) |
| 29 | \( 1 + (3.71 - 1.86i)T + (17.3 - 23.2i)T^{2} \) |
| 31 | \( 1 + (-4.45 + 5.98i)T + (-8.89 - 29.6i)T^{2} \) |
| 37 | \( 1 + (-7.47 + 6.27i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (4.83 - 3.18i)T + (16.2 - 37.6i)T^{2} \) |
| 43 | \( 1 + (4.91 + 1.16i)T + (38.4 + 19.2i)T^{2} \) |
| 47 | \( 1 + (1.79 + 2.40i)T + (-13.4 + 45.0i)T^{2} \) |
| 53 | \( 1 + (6.22 + 10.7i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.12 + 10.4i)T + (-49.2 - 32.4i)T^{2} \) |
| 61 | \( 1 + (-4.67 + 10.8i)T + (-41.8 - 44.3i)T^{2} \) |
| 67 | \( 1 + (0.741 + 0.372i)T + (40.0 + 53.7i)T^{2} \) |
| 71 | \( 1 + (-1.25 - 7.14i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.41 - 7.99i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-9.60 - 6.31i)T + (31.2 + 72.5i)T^{2} \) |
| 83 | \( 1 + (-3.98 - 2.62i)T + (32.8 + 76.2i)T^{2} \) |
| 89 | \( 1 + (-0.578 + 3.28i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (0.935 - 0.991i)T + (-5.64 - 96.8i)T^{2} \) |
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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
−10.09158284103367380661978582178, −9.615776064359737805950256525130, −8.333708364739405165206714982176, −8.002437336742166021128628054886, −6.51875519300748220679535576195, −5.64125106126854418519816065260, −4.93328146465046152282389508011, −3.36674470801726654692452556625, −2.16552236793342143707347403325, −0.74994051729090089979512670448,
1.44739878680172529949079065824, 2.99436095791601570557332654044, 4.37581457057526528957534418661, 5.04607756854869082952125339626, 6.47110917188335458401812339919, 7.25434241978419586457446543519, 8.063462442137681841571212293128, 8.999521938658439667137488230644, 9.428166395493983655475565059753, 10.39866860411277905028590782371
