| L(s) = 1 | + (0.115 − 0.0204i)2-s + (2.32 + 1.89i)3-s + (−3.74 + 1.36i)4-s + (3.98 − 4.74i)5-s + (0.308 + 0.172i)6-s + (−7.49 − 2.72i)7-s + (−0.814 + 0.469i)8-s + (1.80 + 8.81i)9-s + (0.364 − 0.631i)10-s + (−7.35 − 8.76i)11-s + (−11.2 − 3.93i)12-s + (−1.43 + 8.13i)13-s + (−0.924 − 0.162i)14-s + (18.2 − 3.47i)15-s + (12.1 − 10.1i)16-s + (20.7 + 11.9i)17-s + ⋯ |
| L(s) = 1 | + (0.0579 − 0.0102i)2-s + (0.774 + 0.632i)3-s + (−0.936 + 0.340i)4-s + (0.796 − 0.949i)5-s + (0.0513 + 0.0287i)6-s + (−1.07 − 0.389i)7-s + (−0.101 + 0.0587i)8-s + (0.200 + 0.979i)9-s + (0.0364 − 0.0631i)10-s + (−0.668 − 0.796i)11-s + (−0.940 − 0.328i)12-s + (−0.110 + 0.625i)13-s + (−0.0660 − 0.0116i)14-s + (1.21 − 0.231i)15-s + (0.758 − 0.636i)16-s + (1.21 + 0.703i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.02422 + 0.140661i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02422 + 0.140661i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.32 - 1.89i)T \) |
| good | 2 | \( 1 + (-0.115 + 0.0204i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-3.98 + 4.74i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (7.49 + 2.72i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (7.35 + 8.76i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (1.43 - 8.13i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (-20.7 - 11.9i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-6.12 - 10.6i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (5.04 + 13.8i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (17.5 - 3.08i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (12.2 - 4.45i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (-8.53 + 14.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-30.2 - 5.33i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (30.5 - 25.6i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-19.9 + 54.8i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + 91.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (13.2 - 15.8i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-32.8 - 11.9i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-8.95 + 50.7i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-2.28 - 1.31i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (34.5 + 59.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-26.2 - 149. i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-53.2 + 9.38i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (141. - 81.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-43.6 + 36.6i)T + (1.63e3 - 9.26e3i)T^{2} \) |
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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
−16.66606750653154830315570718752, −16.45469504597157656800941159064, −14.42815753526893586104744796091, −13.44283923593709608199195960224, −12.72666842577417724078143463704, −10.12771047356672348475837738363, −9.314969096779293389482621719630, −8.162864927770901227885894675844, −5.40963570547488441420115295059, −3.64900048729990847664860925001,
2.96464390508232112487338350892, 5.82358998776106063487829702330, 7.45939787227419116393143265389, 9.389006647536693178505479281101, 10.05604798958408524797711846286, 12.55696730883190689123489553057, 13.45537972311170807161575541133, 14.41215034085298365096373545156, 15.45918737119804070473201391853, 17.61536884244553827017291105118
