| L(s) = 1 | + (0.135 − 0.765i)2-s + (1.31 + 0.477i)4-s + (1.82 − 1.52i)5-s + (2.35 − 0.855i)7-s + (1.32 − 2.28i)8-s + (−0.924 − 1.60i)10-s + (2.40 + 2.01i)11-s + (0.232 + 1.31i)13-s + (−0.337 − 1.91i)14-s + (0.564 + 0.473i)16-s + (−3.13 − 5.43i)17-s + (−4.03 + 6.98i)19-s + (3.11 − 1.13i)20-s + (1.87 − 1.57i)22-s + (−3.81 − 1.38i)23-s + ⋯ |
| L(s) = 1 | + (0.0954 − 0.541i)2-s + (0.655 + 0.238i)4-s + (0.814 − 0.683i)5-s + (0.888 − 0.323i)7-s + (0.466 − 0.808i)8-s + (−0.292 − 0.506i)10-s + (0.725 + 0.608i)11-s + (0.0643 + 0.365i)13-s + (−0.0902 − 0.512i)14-s + (0.141 + 0.118i)16-s + (−0.760 − 1.31i)17-s + (−0.925 + 1.60i)19-s + (0.696 − 0.253i)20-s + (0.399 − 0.334i)22-s + (−0.794 − 0.289i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.16353 - 1.08656i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.16353 - 1.08656i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.135 + 0.765i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (-1.82 + 1.52i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-2.35 + 0.855i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-2.40 - 2.01i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.232 - 1.31i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (3.13 + 5.43i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.03 - 6.98i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.81 + 1.38i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.61 - 9.14i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.66 + 0.968i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (2.76 + 4.79i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.23 + 6.99i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.79 - 1.50i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-4.33 + 1.57i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 0.135T + 53T^{2} \) |
| 59 | \( 1 + (3.06 - 2.57i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.321 - 0.116i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.75 - 9.96i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (4.09 + 7.09i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.15 + 10.6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.708 + 4.01i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.158 + 0.899i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-1.86 + 3.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.59 + 3.85i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.44912049256686848853577017625, −9.457816770771850386467903571425, −8.738052355965738044553288658379, −7.57187460998424217072608162499, −6.83267667009456231878459923656, −5.75533209286886649517177532630, −4.63492904127834711333898510504, −3.77810445682981362632806245385, −2.08871632371829729387039493029, −1.54347570342700470900959227319,
1.77062003982160361070525066953, 2.58037171104200349394521106795, 4.22274264918978537461648589215, 5.43641202638811118178192307693, 6.29672273126929262804675508181, 6.64648543164297434837216948186, 7.953793510879037845578772385203, 8.577788614390858415067992058804, 9.725385492862916172156841051316, 10.76758539485224255330184080504
