| L(s) = 1 | + (−0.984 − 0.273i)2-s + (−0.818 − 0.494i)4-s + (−2.73 + 0.106i)5-s + (0.877 + 0.137i)7-s + (2.07 + 2.19i)8-s + (2.72 + 0.645i)10-s + (0.602 + 4.41i)11-s + (3.10 − 4.52i)13-s + (−0.825 − 0.375i)14-s + (−0.546 − 1.03i)16-s + (−1.67 + 0.838i)17-s + (0.0797 + 1.36i)19-s + (2.29 + 1.26i)20-s + (0.615 − 4.50i)22-s + (3.01 − 7.80i)23-s + ⋯ |
| L(s) = 1 | + (−0.695 − 0.193i)2-s + (−0.409 − 0.247i)4-s + (−1.22 + 0.0474i)5-s + (0.331 + 0.0518i)7-s + (0.732 + 0.776i)8-s + (0.860 + 0.204i)10-s + (0.181 + 1.33i)11-s + (0.860 − 1.25i)13-s + (−0.220 − 0.100i)14-s + (−0.136 − 0.259i)16-s + (−0.405 + 0.203i)17-s + (0.0182 + 0.314i)19-s + (0.512 + 0.283i)20-s + (0.131 − 0.961i)22-s + (0.628 − 1.62i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.239393 - 0.391796i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.239393 - 0.391796i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.984 + 0.273i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (2.73 - 0.106i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (-0.877 - 0.137i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (-0.602 - 4.41i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (-3.10 + 4.52i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (1.67 - 0.838i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (-0.0797 - 1.36i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (-3.01 + 7.80i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (6.07 + 4.34i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (3.66 - 4.19i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (5.28 + 7.09i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (-0.508 + 1.97i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (-3.65 - 3.31i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (3.30 + 3.79i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-1.25 + 7.11i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.734 + 0.300i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (-5.51 + 3.33i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (-4.19 + 2.99i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (-1.74 - 5.82i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (-0.740 + 0.175i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (-4.92 + 4.82i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-0.337 - 1.30i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (-5.30 + 17.7i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (-16.1 - 0.626i)T + (96.7 + 7.51i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.21595795271403830978752076581, −9.169803988844338014072985968828, −8.360357336965720409046052389696, −7.82683124541772046343452494659, −6.90133555226040087805916047263, −5.46610203202605987060302983824, −4.52802477028745756529542320354, −3.66194698237312323836705257628, −1.94486287373828259710457393637, −0.35643956527836630561854291440,
1.23986449450558755296926111757, 3.50167224069462895435478466094, 3.99010645776762225042682565537, 5.18674647080563535000176378988, 6.58408454242506655802684391571, 7.47450926048016218292749723944, 8.141375607441058183077500471197, 8.963329853638364072238799281410, 9.359089730081233626589736091699, 10.95754223142070667041404258652
