| L(s) = 1 | + (2.50 − 0.696i)2-s + (4.05 − 2.44i)4-s + (−0.716 − 0.0278i)5-s + (2.72 − 0.426i)7-s + (4.87 − 5.17i)8-s + (−1.81 + 0.429i)10-s + (−0.742 + 5.43i)11-s + (−1.59 − 2.32i)13-s + (6.52 − 2.96i)14-s + (4.18 − 7.94i)16-s + (1.84 + 0.928i)17-s + (0.137 − 2.36i)19-s + (−2.97 + 1.64i)20-s + (1.92 + 14.1i)22-s + (−2.77 − 7.18i)23-s + ⋯ |
| L(s) = 1 | + (1.76 − 0.492i)2-s + (2.02 − 1.22i)4-s + (−0.320 − 0.0124i)5-s + (1.03 − 0.161i)7-s + (1.72 − 1.82i)8-s + (−0.573 + 0.135i)10-s + (−0.223 + 1.63i)11-s + (−0.441 − 0.644i)13-s + (1.74 − 0.792i)14-s + (1.04 − 1.98i)16-s + (0.448 + 0.225i)17-s + (0.0316 − 0.542i)19-s + (−0.665 + 0.367i)20-s + (0.410 + 3.00i)22-s + (−0.578 − 1.49i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.93234 - 1.72459i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.93234 - 1.72459i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.50 + 0.696i)T + (1.71 - 1.03i)T^{2} \) |
| 5 | \( 1 + (0.716 + 0.0278i)T + (4.98 + 0.387i)T^{2} \) |
| 7 | \( 1 + (-2.72 + 0.426i)T + (6.66 - 2.13i)T^{2} \) |
| 11 | \( 1 + (0.742 - 5.43i)T + (-10.5 - 2.94i)T^{2} \) |
| 13 | \( 1 + (1.59 + 2.32i)T + (-4.68 + 12.1i)T^{2} \) |
| 17 | \( 1 + (-1.84 - 0.928i)T + (10.1 + 13.6i)T^{2} \) |
| 19 | \( 1 + (-0.137 + 2.36i)T + (-18.8 - 2.20i)T^{2} \) |
| 23 | \( 1 + (2.77 + 7.18i)T + (-17.0 + 15.4i)T^{2} \) |
| 29 | \( 1 + (2.18 - 1.56i)T + (9.38 - 27.4i)T^{2} \) |
| 31 | \( 1 + (-4.19 - 4.80i)T + (-4.19 + 30.7i)T^{2} \) |
| 37 | \( 1 + (3.11 - 4.19i)T + (-10.6 - 35.4i)T^{2} \) |
| 41 | \( 1 + (1.92 + 7.48i)T + (-35.8 + 19.8i)T^{2} \) |
| 43 | \( 1 + (5.74 - 5.21i)T + (4.16 - 42.7i)T^{2} \) |
| 47 | \( 1 + (1.85 - 2.12i)T + (-6.36 - 46.5i)T^{2} \) |
| 53 | \( 1 + (-0.472 - 2.67i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-13.5 - 5.54i)T + (42.1 + 41.3i)T^{2} \) |
| 61 | \( 1 + (3.82 + 2.30i)T + (28.4 + 53.9i)T^{2} \) |
| 67 | \( 1 + (-5.06 - 3.61i)T + (21.6 + 63.3i)T^{2} \) |
| 71 | \( 1 + (1.07 - 3.60i)T + (-59.3 - 39.0i)T^{2} \) |
| 73 | \( 1 + (-15.1 - 3.59i)T + (65.2 + 32.7i)T^{2} \) |
| 79 | \( 1 + (10.6 + 10.4i)T + (1.53 + 78.9i)T^{2} \) |
| 83 | \( 1 + (-1.27 + 4.95i)T + (-72.6 - 40.0i)T^{2} \) |
| 89 | \( 1 + (2.89 + 9.65i)T + (-74.3 + 48.9i)T^{2} \) |
| 97 | \( 1 + (5.23 - 0.203i)T + (96.7 - 7.51i)T^{2} \) |
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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
−10.45913937310695275640286580544, −10.00889075171452036795236781102, −8.302959531052696538049839609475, −7.39376400430707342548113030681, −6.58619090009368347894597204741, −5.29560039248061750918057622868, −4.75599472793735637722123444338, −4.01640884291222921562938627192, −2.69854611214415801265738587109, −1.72093120794329101901604271844,
2.00858719003842784105293939742, 3.37942039733364377769348059403, 4.06433601801901851508384723757, 5.26906566896295725202882348905, 5.68098179734486684179290433881, 6.74510049301886332243846317157, 7.87077977794558318738653939825, 8.181860065091021263689343166676, 9.737753855559106735802348215311, 11.17191187018958792412278729001
