| L(s) = 1 | + (−2.17 − 2.59i)2-s + (−2.99 − 0.0504i)3-s + (−1.30 + 7.38i)4-s + (−1.19 − 3.28i)5-s + (6.40 + 7.90i)6-s + (−1.88 − 10.7i)7-s + (10.2 − 5.92i)8-s + (8.99 + 0.302i)9-s + (−5.92 + 10.2i)10-s + (0.00845 − 0.0232i)11-s + (4.27 − 22.0i)12-s + (6.61 + 5.55i)13-s + (−23.6 + 28.2i)14-s + (3.42 + 9.91i)15-s + (−9.59 − 3.49i)16-s + (−3.55 − 2.05i)17-s + ⋯ |
| L(s) = 1 | + (−1.08 − 1.29i)2-s + (−0.999 − 0.0168i)3-s + (−0.325 + 1.84i)4-s + (−0.239 − 0.656i)5-s + (1.06 + 1.31i)6-s + (−0.269 − 1.52i)7-s + (1.28 − 0.740i)8-s + (0.999 + 0.0336i)9-s + (−0.592 + 1.02i)10-s + (0.000768 − 0.00211i)11-s + (0.356 − 1.83i)12-s + (0.508 + 0.427i)13-s + (−1.69 + 2.01i)14-s + (0.228 + 0.660i)15-s + (−0.599 − 0.218i)16-s + (−0.209 − 0.120i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0325082 - 0.355662i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0325082 - 0.355662i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.99 + 0.0504i)T \) |
| good | 2 | \( 1 + (2.17 + 2.59i)T + (-0.694 + 3.93i)T^{2} \) |
| 5 | \( 1 + (1.19 + 3.28i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (1.88 + 10.7i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (-0.00845 + 0.0232i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-6.61 - 5.55i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (3.55 + 2.05i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (12.2 + 21.2i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-7.41 - 1.30i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-12.4 - 14.8i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (1.89 - 10.7i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (5.30 - 9.19i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-49.7 + 59.2i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (18.6 + 6.77i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-45.8 + 8.08i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 39.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-17.6 - 48.4i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-1.84 - 10.4i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-81.7 - 68.5i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (42.1 + 24.3i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (8.15 + 14.1i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-55.8 + 46.8i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (59.6 + 71.0i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-38.0 + 21.9i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (88.2 + 32.1i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−17.00611080804080191961084896164, −16.10634009170172795772004522745, −13.39278063648640943386378879749, −12.35358470944991135596415183120, −11.07697282226890428462893125871, −10.38859749163581936493351361018, −8.914657444566343362921122010163, −7.10536545284400575231454497058, −4.24291892994181507673228199030, −0.795455365553394133549567097715,
5.67172418859031670886391691590, 6.57036893846826254802442919311, 8.189162651482107326984126588893, 9.647678334913479480617839556938, 11.00819710410905577837251025959, 12.53887845487082187526571101435, 14.84650448541235147511595830809, 15.56510875853222601652688926253, 16.49386394918226439368257514829, 17.70434204043162146036904526361
