| L(s) = 1 | + (−0.971 + 2.25i)2-s + (−2.75 − 2.92i)4-s + (0.232 + 3.99i)5-s + (−1.28 − 0.303i)7-s + (4.64 − 1.69i)8-s + (−9.21 − 3.35i)10-s + (−2.04 − 1.34i)11-s + (−1.30 − 0.152i)13-s + (1.92 − 2.59i)14-s + (−0.237 + 4.08i)16-s + (0.206 − 0.173i)17-s + (1.02 + 0.862i)19-s + (11.0 − 11.6i)20-s + (5.02 − 3.30i)22-s + (−5.82 + 1.37i)23-s + ⋯ |
| L(s) = 1 | + (−0.686 + 1.59i)2-s + (−1.37 − 1.46i)4-s + (0.103 + 1.78i)5-s + (−0.484 − 0.114i)7-s + (1.64 − 0.597i)8-s + (−2.91 − 1.06i)10-s + (−0.617 − 0.406i)11-s + (−0.361 − 0.0422i)13-s + (0.515 − 0.692i)14-s + (−0.0594 + 1.02i)16-s + (0.0501 − 0.0420i)17-s + (0.235 + 0.197i)19-s + (2.46 − 2.61i)20-s + (1.07 − 0.704i)22-s + (−1.21 + 0.287i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.146720 - 0.0700348i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.146720 - 0.0700348i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.971 - 2.25i)T + (-1.37 - 1.45i)T^{2} \) |
| 5 | \( 1 + (-0.232 - 3.99i)T + (-4.96 + 0.580i)T^{2} \) |
| 7 | \( 1 + (1.28 + 0.303i)T + (6.25 + 3.14i)T^{2} \) |
| 11 | \( 1 + (2.04 + 1.34i)T + (4.35 + 10.1i)T^{2} \) |
| 13 | \( 1 + (1.30 + 0.152i)T + (12.6 + 2.99i)T^{2} \) |
| 17 | \( 1 + (-0.206 + 0.173i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-1.02 - 0.862i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (5.82 - 1.37i)T + (20.5 - 10.3i)T^{2} \) |
| 29 | \( 1 + (3.30 + 4.44i)T + (-8.31 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-0.369 + 1.23i)T + (-25.9 - 17.0i)T^{2} \) |
| 37 | \( 1 + (0.00841 + 0.0477i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (-1.64 - 3.80i)T + (-28.1 + 29.8i)T^{2} \) |
| 43 | \( 1 + (-5.59 + 2.80i)T + (25.6 - 34.4i)T^{2} \) |
| 47 | \( 1 + (-2.38 - 7.95i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (5.79 + 10.0i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-8.15 + 5.36i)T + (23.3 - 54.1i)T^{2} \) |
| 61 | \( 1 + (-2.93 + 3.10i)T + (-3.54 - 60.8i)T^{2} \) |
| 67 | \( 1 + (0.791 - 1.06i)T + (-19.2 - 64.1i)T^{2} \) |
| 71 | \( 1 + (-7.40 - 2.69i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (8.12 - 2.95i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.07 + 4.80i)T + (-54.2 - 57.4i)T^{2} \) |
| 83 | \( 1 + (2.24 - 5.21i)T + (-56.9 - 60.3i)T^{2} \) |
| 89 | \( 1 + (8.61 - 3.13i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (0.721 - 12.3i)T + (-96.3 - 11.2i)T^{2} \) |
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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
−10.76811644810527460139094407464, −9.906451945732342460386298808562, −9.556786310364658404131457648671, −8.076710396873248744557639901391, −7.66264752357498462314836637286, −6.76919109346512651050774497212, −6.18854626766393970398721693887, −5.42064841506694239121628508322, −3.79786733011712384701634333137, −2.54486833574031601228846913914,
0.10661665108787417220062582788, 1.43342675397763148518355420044, 2.51931276242227832229145971782, 3.89201845363056976404870654399, 4.74745291813144106471595006553, 5.75755122463835918971528152115, 7.51337731146621195835306873912, 8.418675378930964779452468775076, 9.048708929191847305081035765340, 9.705810560434402198838056978554
