| L(s) = 1 | + (1.47 − 2.61i)3-s + (−4.32 − 0.763i)5-s + (2.73 − 2.29i)7-s + (−4.66 − 7.69i)9-s + (14.4 − 2.54i)11-s + (−3.67 + 1.33i)13-s + (−8.36 + 10.1i)15-s + (−5.96 − 3.44i)17-s + (−14.1 − 24.4i)19-s + (−1.97 − 10.5i)21-s + (−0.832 + 0.992i)23-s + (−5.33 − 1.94i)25-s + (−26.9 + 0.868i)27-s + (−12.9 + 35.7i)29-s + (−41.7 − 35.0i)31-s + ⋯ |
| L(s) = 1 | + (0.490 − 0.871i)3-s + (−0.865 − 0.152i)5-s + (0.391 − 0.328i)7-s + (−0.518 − 0.855i)9-s + (1.31 − 0.231i)11-s + (−0.282 + 0.102i)13-s + (−0.557 + 0.679i)15-s + (−0.350 − 0.202i)17-s + (−0.744 − 1.28i)19-s + (−0.0940 − 0.501i)21-s + (−0.0361 + 0.0431i)23-s + (−0.213 − 0.0777i)25-s + (−0.999 + 0.0321i)27-s + (−0.448 + 1.23i)29-s + (−1.34 − 1.12i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.576i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.817 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.408659 - 1.28915i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.408659 - 1.28915i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.47 + 2.61i)T \) |
| good | 5 | \( 1 + (4.32 + 0.763i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-2.73 + 2.29i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (-14.4 + 2.54i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (3.67 - 1.33i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (5.96 + 3.44i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (14.1 + 24.4i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (0.832 - 0.992i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (12.9 - 35.7i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (41.7 + 35.0i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-18.5 + 32.1i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (14.0 + 38.6i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-0.615 - 3.49i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-27.5 - 32.7i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 - 47.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-61.1 - 10.7i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (8.50 - 7.13i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-105. + 38.2i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (90.9 + 52.5i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (48.5 + 84.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-104. - 37.9i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-44.8 + 123. i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-87.8 + 50.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (28.7 + 163. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−10.94267137387798793846236919352, −9.190076284089872915321784114635, −8.858755120053839179588631154206, −7.61960112181644736076762159220, −7.13247203431106632786365697020, −6.05521262448255663666558501029, −4.45920651576095376172448342332, −3.55840420623316216655019672200, −2.02309883607938529388858061545, −0.53247861034914073699559289127,
2.00484123747845236575724646820, 3.63881453883487860382994775913, 4.16469901465851506368882692800, 5.39995754887296619409296448454, 6.69831887238121546868647276781, 7.931878321065824063119665742980, 8.528646861258989439751360484036, 9.500931980010675558220951399000, 10.32677334012718948286788560489, 11.40507360820182482799560378824
