| L(s) = 1 | + (0.837 + 1.94i)2-s + (−1.69 + 1.79i)4-s + (0.0261 − 0.448i)5-s + (−2.37 + 0.562i)7-s + (−0.937 − 0.341i)8-s + (0.892 − 0.324i)10-s + (−2.86 + 1.88i)11-s + (−3.02 + 0.353i)13-s + (−3.08 − 4.13i)14-s + (0.164 + 2.82i)16-s + (1.96 + 1.65i)17-s + (−5.90 + 4.95i)19-s + (0.761 + 0.807i)20-s + (−6.05 − 3.98i)22-s + (−5.55 − 1.31i)23-s + ⋯ |
| L(s) = 1 | + (0.592 + 1.37i)2-s + (−0.848 + 0.898i)4-s + (0.0116 − 0.200i)5-s + (−0.897 + 0.212i)7-s + (−0.331 − 0.120i)8-s + (0.282 − 0.102i)10-s + (−0.862 + 0.567i)11-s + (−0.838 + 0.0979i)13-s + (−0.823 − 1.10i)14-s + (0.0412 + 0.707i)16-s + (0.477 + 0.400i)17-s + (−1.35 + 1.13i)19-s + (0.170 + 0.180i)20-s + (−1.29 − 0.848i)22-s + (−1.15 − 0.274i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.255054 - 1.15435i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.255054 - 1.15435i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.837 - 1.94i)T + (-1.37 + 1.45i)T^{2} \) |
| 5 | \( 1 + (-0.0261 + 0.448i)T + (-4.96 - 0.580i)T^{2} \) |
| 7 | \( 1 + (2.37 - 0.562i)T + (6.25 - 3.14i)T^{2} \) |
| 11 | \( 1 + (2.86 - 1.88i)T + (4.35 - 10.1i)T^{2} \) |
| 13 | \( 1 + (3.02 - 0.353i)T + (12.6 - 2.99i)T^{2} \) |
| 17 | \( 1 + (-1.96 - 1.65i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (5.90 - 4.95i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (5.55 + 1.31i)T + (20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (-1.87 + 2.51i)T + (-8.31 - 27.7i)T^{2} \) |
| 31 | \( 1 + (-2.39 - 8.01i)T + (-25.9 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.0238 + 0.135i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (-4.79 + 11.1i)T + (-28.1 - 29.8i)T^{2} \) |
| 43 | \( 1 + (-4.05 - 2.03i)T + (25.6 + 34.4i)T^{2} \) |
| 47 | \( 1 + (-0.745 + 2.48i)T + (-39.2 - 25.8i)T^{2} \) |
| 53 | \( 1 + (-0.184 + 0.319i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-8.82 - 5.80i)T + (23.3 + 54.1i)T^{2} \) |
| 61 | \( 1 + (-1.55 - 1.64i)T + (-3.54 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-0.831 - 1.11i)T + (-19.2 + 64.1i)T^{2} \) |
| 71 | \( 1 + (-3.35 + 1.22i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.75 - 1.73i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (4.06 + 9.43i)T + (-54.2 + 57.4i)T^{2} \) |
| 83 | \( 1 + (0.387 + 0.898i)T + (-56.9 + 60.3i)T^{2} \) |
| 89 | \( 1 + (8.14 + 2.96i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.123 - 2.12i)T + (-96.3 + 11.2i)T^{2} \) |
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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
−10.49777556505801660090325554079, −10.11666686563825904031632688775, −8.834289971469260290048940394067, −8.071088138947104974326950639638, −7.25261282078262607607679077003, −6.40882609423304126755592030356, −5.68778673991607113002213789295, −4.77001420282740175509725507017, −3.82990625812455413425573883837, −2.30642102168440111653144128522,
0.47091906163159818240756067552, 2.42552715720312674551853479171, 2.97139070506149242046329149154, 4.13503703564118570632558025792, 5.03257168802291776439724292686, 6.19219918138748642414410825914, 7.23530614932420934960087356756, 8.293506804573466592670444638329, 9.615425182850095240338548698388, 10.01577636926495186279135839857
