| L(s) = 1 | + (0.183 + 1.03i)2-s + (−1.72 − 0.0916i)3-s + (0.834 − 0.303i)4-s + (−1.33 − 1.12i)5-s + (−0.221 − 1.81i)6-s + (−2.31 − 0.841i)7-s + (1.52 + 2.63i)8-s + (2.98 + 0.317i)9-s + (0.920 − 1.59i)10-s + (−0.960 + 0.806i)11-s + (−1.47 + 0.449i)12-s + (−0.789 + 4.47i)13-s + (0.450 − 2.55i)14-s + (2.21 + 2.06i)15-s + (−1.09 + 0.921i)16-s + (3.32 − 5.75i)17-s + ⋯ |
| L(s) = 1 | + (0.129 + 0.734i)2-s + (−0.998 − 0.0529i)3-s + (0.417 − 0.151i)4-s + (−0.598 − 0.501i)5-s + (−0.0904 − 0.740i)6-s + (−0.873 − 0.317i)7-s + (0.538 + 0.932i)8-s + (0.994 + 0.105i)9-s + (0.291 − 0.504i)10-s + (−0.289 + 0.243i)11-s + (−0.424 + 0.129i)12-s + (−0.219 + 1.24i)13-s + (0.120 − 0.682i)14-s + (0.570 + 0.532i)15-s + (−0.274 + 0.230i)16-s + (0.806 − 1.39i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.558592 + 0.172015i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.558592 + 0.172015i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.72 + 0.0916i)T \) |
| good | 2 | \( 1 + (-0.183 - 1.03i)T + (-1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (1.33 + 1.12i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (2.31 + 0.841i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (0.960 - 0.806i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.789 - 4.47i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.32 + 5.75i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.124 + 0.215i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.791 - 0.287i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.0889 + 0.504i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.770 + 0.280i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (1.30 - 2.25i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.41 - 8.02i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.31 + 2.78i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-4.98 - 1.81i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + (2.30 + 1.93i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (2.70 + 0.986i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.75 + 9.93i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (0.0447 - 0.0774i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.66 - 4.60i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.829 - 4.70i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.39 - 7.91i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (3.35 + 5.80i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.20 + 3.52i)T + (16.8 - 95.5i)T^{2} \) |
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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
−16.98055354576521716765959420225, −16.29261293252171031524415064835, −15.66354308199993085027933152609, −13.94936806137024838968969533632, −12.33994372064912666209240580689, −11.37358457724085833965091123629, −9.798430816246424384668931499777, −7.53102393364548771339000669493, −6.44636889622299531201869147288, −4.80572426485302466151161265436,
3.46941127797150886877192563780, 5.99333211160525166453632063784, 7.52478072853952657584684600133, 10.15434837582262845810476258272, 10.91786897002129528975749832501, 12.24640299228232124653570192383, 12.87597695430516544210829255854, 15.25824199555523598898068478136, 16.04171777458762816651730397564, 17.27041545161542903522268396776
