| L(s) = 1 | + (1.50 + 1.26i)2-s + (0.326 + 1.85i)4-s + (3.47 + 1.26i)5-s + (−0.407 + 2.31i)7-s + (0.118 − 0.205i)8-s + (3.64 + 6.31i)10-s + (−2.04 + 0.745i)11-s + (−3.61 + 3.03i)13-s + (−3.54 + 2.97i)14-s + (3.97 − 1.44i)16-s + (−1.46 − 2.54i)17-s + (3.11 − 5.39i)19-s + (−1.20 + 6.85i)20-s + (−4.03 − 1.46i)22-s + (−0.0901 − 0.511i)23-s + ⋯ |
| L(s) = 1 | + (1.06 + 0.895i)2-s + (0.163 + 0.925i)4-s + (1.55 + 0.566i)5-s + (−0.154 + 0.873i)7-s + (0.0419 − 0.0727i)8-s + (1.15 + 1.99i)10-s + (−0.617 + 0.224i)11-s + (−1.00 + 0.840i)13-s + (−0.946 + 0.794i)14-s + (0.992 − 0.361i)16-s + (−0.355 − 0.616i)17-s + (0.714 − 1.23i)19-s + (−0.270 + 1.53i)20-s + (−0.859 − 0.312i)22-s + (−0.0187 − 0.106i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.12940 + 2.39279i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.12940 + 2.39279i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.50 - 1.26i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (-3.47 - 1.26i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.407 - 2.31i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (2.04 - 0.745i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (3.61 - 3.03i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.46 + 2.54i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.11 + 5.39i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0901 + 0.511i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.67 - 2.24i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.747 + 4.23i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (1.20 + 2.08i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.91 - 1.60i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1 - 0.363i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.0412 + 0.233i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 4.66T + 53T^{2} \) |
| 59 | \( 1 + (12.5 + 4.55i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.638 - 3.61i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-10.9 + 9.19i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.601 - 1.04i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.34 + 4.05i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (9.80 + 8.22i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.65 - 7.26i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-0.349 + 0.605i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.65 + 2.42i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.50984938606945278451407932485, −9.578696419762578646119724367397, −9.165387689340921434782959961874, −7.56878110992236554107520164187, −6.78180216802383861575655621221, −6.17463000750522432559330607215, −5.21855512161456302866753910816, −4.78715124933758586004548095178, −2.98226978638077791889148029052, −2.18353243566683180971786423464,
1.36235173811013166272104627780, 2.45761220700379642066382227535, 3.50294710750217477675815095163, 4.74751334684969392858139884762, 5.39896156623780067430726539298, 6.16032367260358192432750980339, 7.54565607532432448167081943881, 8.515927738857142961638368199229, 9.937154224060857342945644024766, 10.12739169550036213695942541745
