| L(s) = 1 | + (1.90 − 0.619i)2-s + (−2.89 − 0.776i)3-s + (0.0135 − 0.00987i)4-s + (−5.21 − 1.69i)5-s + (−6.00 + 0.315i)6-s + (4.52 − 3.28i)7-s + (−4.69 + 6.45i)8-s + (7.79 + 4.49i)9-s − 10.9·10-s + (−0.0470 + 0.0180i)12-s + (3.00 + 9.24i)13-s + (6.58 − 9.06i)14-s + (13.7 + 8.95i)15-s + (−4.96 + 15.2i)16-s + (16.9 + 5.52i)17-s + (17.6 + 3.74i)18-s + ⋯ |
| L(s) = 1 | + (0.953 − 0.309i)2-s + (−0.965 − 0.258i)3-s + (0.00339 − 0.00246i)4-s + (−1.04 − 0.338i)5-s + (−1.00 + 0.0525i)6-s + (0.646 − 0.469i)7-s + (−0.586 + 0.807i)8-s + (0.866 + 0.499i)9-s − 1.09·10-s + (−0.00392 + 0.00150i)12-s + (0.230 + 0.710i)13-s + (0.470 − 0.647i)14-s + (0.919 + 0.597i)15-s + (−0.310 + 0.955i)16-s + (0.999 + 0.324i)17-s + (0.980 + 0.208i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.27106 + 0.465980i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27106 + 0.465980i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.89 + 0.776i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-1.90 + 0.619i)T + (3.23 - 2.35i)T^{2} \) |
| 5 | \( 1 + (5.21 + 1.69i)T + (20.2 + 14.6i)T^{2} \) |
| 7 | \( 1 + (-4.52 + 3.28i)T + (15.1 - 46.6i)T^{2} \) |
| 13 | \( 1 + (-3.00 - 9.24i)T + (-136. + 99.3i)T^{2} \) |
| 17 | \( 1 + (-16.9 - 5.52i)T + (233. + 169. i)T^{2} \) |
| 19 | \( 1 + (-15.0 - 10.9i)T + (111. + 343. i)T^{2} \) |
| 23 | \( 1 - 12.3iT - 529T^{2} \) |
| 29 | \( 1 + (1.45 + 2.00i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-15.2 - 46.8i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-31.8 + 23.1i)T + (423. - 1.30e3i)T^{2} \) |
| 41 | \( 1 + (33.2 - 45.7i)T + (-519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 - 43.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (33.9 - 46.7i)T + (-682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (41.0 - 13.3i)T + (2.27e3 - 1.65e3i)T^{2} \) |
| 59 | \( 1 + (52.9 + 72.8i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-9.53 + 29.3i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 - 34.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-35.7 - 11.6i)T + (4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (9.81 - 7.12i)T + (1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-19.5 - 60.0i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (9.22 + 2.99i)T + (5.57e3 + 4.04e3i)T^{2} \) |
| 89 | \( 1 - 34.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (11.6 + 35.9i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
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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
−11.49654886044961699302124610377, −11.00071250700162483221262550580, −9.647765855800597267970910067701, −8.161095557986442473875745956972, −7.61787933475193804995201786008, −6.23549881459813431645039394840, −5.11528738161494785254510276978, −4.41530330213035595595438169596, −3.48577235563719942946880892688, −1.32448698846827370000132225603,
0.59638576433340924846668735860, 3.25831693176116742175872273777, 4.30320434867097584961856544951, 5.19836604151707364532629948430, 5.92691372388609125748276062742, 7.09756228760439052377489501833, 8.021266980934706120781528252457, 9.426431819258986659677705066023, 10.39339385520327280252318172631, 11.52337246040348114299543966660
