| 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 + (−2.23 + 10.7i)11-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 + ⋯ |
| 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 + (−0.203 + 0.979i)11-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.115i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00172759 - 0.0297289i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00172759 - 0.0297289i\) |
| \(L(2)\) |
|
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 | \( 1 + (2.23 - 10.7i)T \) |
| 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
−17.35253234626591376034672892548, −15.95672904433066207733019637571, −15.57931948728915869302873427789, −12.98780329340782475851959035071, −12.30999079306226766693462315800, −10.79654708420554249525626404499, −9.413212354252283188995801304006, −7.930423600689101870231692824023, −6.78171977873196538521412207979, −4.60058385696875865140689475670,
0.05399574569636960424370906896, 4.26144291725557982852523893967, 6.46749482148093291399462106998, 8.131475059773975078207790468224, 9.722653400740790148133653384052, 10.87498310323776639517266291707, 11.58427161713621373106148820947, 13.27395477625904035191492908585, 15.02716669319368556105104999686, 16.37752320523106621410213809995
