| L(s) = 1 | + (−1.75 − 0.488i)2-s + (1.13 + 0.684i)4-s + (−0.390 + 0.0151i)5-s + (−1.38 − 0.216i)7-s + (0.844 + 0.895i)8-s + (0.693 + 0.164i)10-s + (0.144 + 1.05i)11-s + (0.399 − 0.582i)13-s + (2.32 + 1.05i)14-s + (−2.28 − 4.33i)16-s + (1.16 − 0.585i)17-s + (−0.116 − 2.00i)19-s + (−0.453 − 0.250i)20-s + (0.263 − 1.92i)22-s + (−0.395 + 1.02i)23-s + ⋯ |
| L(s) = 1 | + (−1.24 − 0.345i)2-s + (0.567 + 0.342i)4-s + (−0.174 + 0.00677i)5-s + (−0.522 − 0.0817i)7-s + (0.298 + 0.316i)8-s + (0.219 + 0.0519i)10-s + (0.0435 + 0.318i)11-s + (0.110 − 0.161i)13-s + (0.621 + 0.282i)14-s + (−0.570 − 1.08i)16-s + (0.282 − 0.141i)17-s + (−0.0267 − 0.459i)19-s + (−0.101 − 0.0559i)20-s + (0.0561 − 0.410i)22-s + (−0.0825 + 0.213i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00578343 + 0.112712i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00578343 + 0.112712i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.75 + 0.488i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (0.390 - 0.0151i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (1.38 + 0.216i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (-0.144 - 1.05i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (-0.399 + 0.582i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-1.16 + 0.585i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (0.116 + 2.00i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (0.395 - 1.02i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (3.81 + 2.72i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (-1.34 + 1.54i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-2.79 - 3.76i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (0.191 - 0.743i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (3.68 + 3.34i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (4.92 + 5.63i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-0.355 + 2.01i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (13.4 - 5.49i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (9.43 - 5.69i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (-6.23 + 4.45i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (1.57 + 5.26i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (4.21 - 0.997i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (10.2 - 10.0i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (1.59 + 6.20i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (3.45 - 11.5i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (-2.31 - 0.0899i)T + (96.7 + 7.51i)T^{2} \) |
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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
−9.767735834998380869071582380240, −9.352251554729029960792008929327, −8.290067360893535506189821863009, −7.64661206910444358523525771613, −6.71055272012165805788184266604, −5.52039514834735294099290409742, −4.31617355838795910534942610319, −2.98987523395555423451614625954, −1.67044646886986546622727901662, −0.090329038704071426914012783958,
1.56805630182901902516689802959, 3.27417876410300908781703399342, 4.38029464139908270906644098268, 5.85456729089145680988439298557, 6.64049152824019566675761584315, 7.65596518636988086222855336557, 8.221429831874525044859503343786, 9.199478730196497258495943728876, 9.716368540807445916115421751724, 10.58861110218478341872506897262
