| 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
−17.27041545161542903522268396776, −16.04171777458762816651730397564, −15.25824199555523598898068478136, −12.87597695430516544210829255854, −12.24640299228232124653570192383, −10.91786897002129528975749832501, −10.15434837582262845810476258272, −7.52478072853952657584684600133, −5.99333211160525166453632063784, −3.46941127797150886877192563780,
4.80572426485302466151161265436, 6.44636889622299531201869147288, 7.53102393364548771339000669493, 9.798430816246424384668931499777, 11.37358457724085833965091123629, 12.33994372064912666209240580689, 13.94936806137024838968969533632, 15.66354308199993085027933152609, 16.29261293252171031524415064835, 16.98055354576521716765959420225
