| L(s) = 1 | + (−0.652 − 0.429i)2-s + (−0.550 − 1.27i)4-s + (−1.01 − 1.07i)5-s + (3.77 − 0.441i)7-s + (−0.459 + 2.60i)8-s + (0.200 + 1.13i)10-s + (1.44 + 4.84i)11-s + (0.261 + 4.48i)13-s + (−2.65 − 1.33i)14-s + (−0.485 + 0.515i)16-s + (4.30 + 1.56i)17-s + (4.19 − 1.52i)19-s + (−0.814 + 1.88i)20-s + (1.13 − 3.78i)22-s + (−3.43 − 0.401i)23-s + ⋯ |
| L(s) = 1 | + (−0.461 − 0.303i)2-s + (−0.275 − 0.637i)4-s + (−0.454 − 0.481i)5-s + (1.42 − 0.166i)7-s + (−0.162 + 0.922i)8-s + (0.0635 + 0.360i)10-s + (0.437 + 1.45i)11-s + (0.0724 + 1.24i)13-s + (−0.709 − 0.356i)14-s + (−0.121 + 0.128i)16-s + (1.04 + 0.380i)17-s + (0.962 − 0.350i)19-s + (−0.182 + 0.422i)20-s + (0.241 − 0.806i)22-s + (−0.715 − 0.0836i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20034 - 0.263926i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20034 - 0.263926i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.652 + 0.429i)T + (0.792 + 1.83i)T^{2} \) |
| 5 | \( 1 + (1.01 + 1.07i)T + (-0.290 + 4.99i)T^{2} \) |
| 7 | \( 1 + (-3.77 + 0.441i)T + (6.81 - 1.61i)T^{2} \) |
| 11 | \( 1 + (-1.44 - 4.84i)T + (-9.19 + 6.04i)T^{2} \) |
| 13 | \( 1 + (-0.261 - 4.48i)T + (-12.9 + 1.50i)T^{2} \) |
| 17 | \( 1 + (-4.30 - 1.56i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-4.19 + 1.52i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (3.43 + 0.401i)T + (22.3 + 5.30i)T^{2} \) |
| 29 | \( 1 + (0.583 - 0.293i)T + (17.3 - 23.2i)T^{2} \) |
| 31 | \( 1 + (-0.393 + 0.527i)T + (-8.89 - 29.6i)T^{2} \) |
| 37 | \( 1 + (-0.766 + 0.642i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (0.570 - 0.375i)T + (16.2 - 37.6i)T^{2} \) |
| 43 | \( 1 + (-8.16 - 1.93i)T + (38.4 + 19.2i)T^{2} \) |
| 47 | \( 1 + (4.73 + 6.36i)T + (-13.4 + 45.0i)T^{2} \) |
| 53 | \( 1 + (-2.07 - 3.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.51 + 5.04i)T + (-49.2 - 32.4i)T^{2} \) |
| 61 | \( 1 + (2.68 - 6.23i)T + (-41.8 - 44.3i)T^{2} \) |
| 67 | \( 1 + (-4.73 - 2.37i)T + (40.0 + 53.7i)T^{2} \) |
| 71 | \( 1 + (-1.06 - 6.03i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (0.764 - 4.33i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (9.39 + 6.17i)T + (31.2 + 72.5i)T^{2} \) |
| 83 | \( 1 + (-1.35 - 0.891i)T + (32.8 + 76.2i)T^{2} \) |
| 89 | \( 1 + (-0.181 + 1.02i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (3.42 - 3.62i)T + (-5.64 - 96.8i)T^{2} \) |
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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
−10.16845722524255793535002421540, −9.569943201725254874279293480736, −8.683442040499821607908605478964, −7.923244029918822556919934418624, −7.06316495254274509140686462000, −5.68098354053661539832290933810, −4.69043834269134934910647644455, −4.21402829247815230209878881343, −2.04561547723065660224131106630, −1.22628033155649267384805236127,
0.974672334507125158497478640242, 3.06663216195434977388881801622, 3.73972358904872291090422091235, 5.13619792979055756906567733685, 6.00928257286495375775346945993, 7.46436700617226333459417147486, 7.86644576340617017355862234944, 8.451220265930111974989945309991, 9.419618605637260429359507348572, 10.48391691597502968573179074243
