| L(s) = 1 | + (1.39 + 0.387i)2-s + (0.0768 + 0.0463i)4-s + (−1.01 + 0.0395i)5-s + (−0.161 − 0.0252i)7-s + (−1.89 − 2.00i)8-s + (−1.43 − 0.340i)10-s + (−0.568 − 4.16i)11-s + (1.68 − 2.45i)13-s + (−0.215 − 0.0978i)14-s + (−1.94 − 3.69i)16-s + (6.87 − 3.45i)17-s + (0.307 + 5.27i)19-s + (−0.0802 − 0.0442i)20-s + (0.821 − 6.01i)22-s + (2.81 − 7.29i)23-s + ⋯ |
| L(s) = 1 | + (0.984 + 0.274i)2-s + (0.0384 + 0.0231i)4-s + (−0.456 + 0.0176i)5-s + (−0.0611 − 0.00955i)7-s + (−0.669 − 0.710i)8-s + (−0.453 − 0.107i)10-s + (−0.171 − 1.25i)11-s + (0.467 − 0.682i)13-s + (−0.0575 − 0.0261i)14-s + (−0.486 − 0.922i)16-s + (1.66 − 0.837i)17-s + (0.0705 + 1.21i)19-s + (−0.0179 − 0.00989i)20-s + (0.175 − 1.28i)22-s + (0.587 − 1.52i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.52152 - 1.01659i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52152 - 1.01659i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.39 - 0.387i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (1.01 - 0.0395i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (0.161 + 0.0252i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (0.568 + 4.16i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (-1.68 + 2.45i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-6.87 + 3.45i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (-0.307 - 5.27i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (-2.81 + 7.29i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (7.18 + 5.13i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (-1.77 + 2.03i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-2.96 - 3.98i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (2.73 - 10.6i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (-0.972 - 0.882i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-2.91 - 3.34i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-1.30 + 7.41i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (9.14 - 3.73i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (1.32 - 0.801i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (-5.87 + 4.20i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (-1.31 - 4.37i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (-3.87 + 0.917i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (-7.95 + 7.79i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (1.44 + 5.60i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (1.18 - 3.94i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (-5.41 - 0.210i)T + (96.7 + 7.51i)T^{2} \) |
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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
−10.19910913804457871739731669769, −9.487058809234202230725161627078, −8.233597689359697838193374363911, −7.72673872844008847060560377860, −6.25635289252315335435782096163, −5.79616263188837531912883929683, −4.82951488644132654615014055388, −3.65481000968587599641851496279, −3.08088886309032962873278489311, −0.71874771632796074597455271387,
1.85376810605837827297434089825, 3.33699177958311045636544308563, 3.99251382032091112328097746318, 5.03868904857739685652059526244, 5.76460891890493635958046350864, 7.10378057057733846017757743154, 7.79553169119261579375753691252, 8.974816958701404967053824421163, 9.623410104325637713886003450827, 10.84200833677158942803713159316
