| L(s) = 1 | + (−0.223 − 1.26i)2-s + (0.326 − 0.118i)4-s + (−0.342 − 0.286i)5-s + (3.31 + 1.20i)7-s + (−1.50 − 2.61i)8-s + (−0.286 + 0.497i)10-s + (2.12 − 1.78i)11-s + (0.571 − 3.24i)13-s + (0.788 − 4.47i)14-s + (−2.43 + 2.04i)16-s + (−3.51 + 6.09i)17-s + (2.59 + 4.49i)19-s + (−0.145 − 0.0530i)20-s + (−2.73 − 2.29i)22-s + (6.83 − 2.48i)23-s + ⋯ |
| L(s) = 1 | + (−0.157 − 0.895i)2-s + (0.163 − 0.0593i)4-s + (−0.152 − 0.128i)5-s + (1.25 + 0.456i)7-s + (−0.533 − 0.923i)8-s + (−0.0907 + 0.157i)10-s + (0.642 − 0.538i)11-s + (0.158 − 0.898i)13-s + (0.210 − 1.19i)14-s + (−0.609 + 0.511i)16-s + (−0.853 + 1.47i)17-s + (0.594 + 1.03i)19-s + (−0.0325 − 0.0118i)20-s + (−0.583 − 0.489i)22-s + (1.42 − 0.518i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17586 - 1.32131i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17586 - 1.32131i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.223 + 1.26i)T + (-1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (0.342 + 0.286i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-3.31 - 1.20i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-2.12 + 1.78i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.571 + 3.24i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (3.51 - 6.09i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.59 - 4.49i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.83 + 2.48i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.628 + 3.56i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.81 + 0.662i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-1.61 + 2.79i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.844 + 4.78i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (4.41 - 3.70i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-2.83 - 1.03i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 8.77T + 53T^{2} \) |
| 59 | \( 1 + (2.27 + 1.90i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (7.41 + 2.69i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.63 - 9.29i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.65 + 4.59i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.777 - 1.34i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.06 + 11.7i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.82 - 16.0i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-9.21 - 15.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.74 + 6.50i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.54434372821026241656967776520, −9.385037162840395375733615866549, −8.484901821605041078890384825059, −7.88574161353535375569251599763, −6.48319374421593855127228060325, −5.73440732552744479987290620475, −4.45598711844094275914958042865, −3.41123419895391201839411038381, −2.16737279158926257665121310871, −1.11119735248990956893665897811,
1.57945028510744542175089542199, 3.00686719217122568772108687612, 4.63184166556577512910198046248, 5.10640293565884219618409557187, 6.62736315680688159451719575350, 7.14494501248904269124009824410, 7.71102427361470472894637174435, 8.956861954297862600492063064956, 9.294772015554445938591103635210, 10.93635734603684646151072667286
