| L(s) = 1 | + (−0.974 + 1.34i)2-s + (2.52 − 1.61i)3-s + (0.386 + 1.18i)4-s + (0.410 + 0.565i)5-s + (−0.300 + 4.96i)6-s + (0.806 + 2.48i)7-s + (−8.28 − 2.69i)8-s + (3.79 − 8.16i)9-s − 1.15·10-s + (−4.26 − 10.1i)11-s + (2.89 + 2.38i)12-s + (−13.8 − 10.0i)13-s + (−4.11 − 1.33i)14-s + (1.95 + 0.766i)15-s + (7.63 − 5.54i)16-s + (9.47 + 13.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.487 + 0.670i)2-s + (0.843 − 0.537i)3-s + (0.0966 + 0.297i)4-s + (0.0821 + 0.113i)5-s + (−0.0500 + 0.827i)6-s + (0.115 + 0.354i)7-s + (−1.03 − 0.336i)8-s + (0.421 − 0.906i)9-s − 0.115·10-s + (−0.387 − 0.921i)11-s + (0.241 + 0.198i)12-s + (−1.06 − 0.773i)13-s + (−0.293 − 0.0954i)14-s + (0.130 + 0.0511i)15-s + (0.477 − 0.346i)16-s + (0.557 + 0.766i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.625i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.946892 + 0.332479i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.946892 + 0.332479i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.52 + 1.61i)T \) |
| 11 | \( 1 + (4.26 + 10.1i)T \) |
| good | 2 | \( 1 + (0.974 - 1.34i)T + (-1.23 - 3.80i)T^{2} \) |
| 5 | \( 1 + (-0.410 - 0.565i)T + (-7.72 + 23.7i)T^{2} \) |
| 7 | \( 1 + (-0.806 - 2.48i)T + (-39.6 + 28.8i)T^{2} \) |
| 13 | \( 1 + (13.8 + 10.0i)T + (52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (-9.47 - 13.0i)T + (-89.3 + 274. i)T^{2} \) |
| 19 | \( 1 + (4.92 - 15.1i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 - 23.1iT - 529T^{2} \) |
| 29 | \( 1 + (-5.10 + 1.65i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-3.28 - 2.38i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-19.6 - 60.4i)T + (-1.10e3 + 804. i)T^{2} \) |
| 41 | \( 1 + (-64.1 - 20.8i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + 22.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + (70.2 + 22.8i)T + (1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (25.1 - 34.6i)T + (-868. - 2.67e3i)T^{2} \) |
| 59 | \( 1 + (27.3 - 8.88i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-37.4 + 27.2i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + 77.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-24.2 - 33.3i)T + (-1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (17.5 + 53.9i)T + (-4.31e3 + 3.13e3i)T^{2} \) |
| 79 | \( 1 + (-41.1 - 29.9i)T + (1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (34.8 + 47.9i)T + (-2.12e3 + 6.55e3i)T^{2} \) |
| 89 | \( 1 + 38.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (13.1 + 9.57i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−16.68548436125242120380293371834, −15.39577127841221135918675978585, −14.52192119774910764806760342760, −13.03063515627938353217804082468, −12.01127523673784067784751724145, −9.906633481779976573688983287679, −8.408014615281598237847077483930, −7.73083738593641501525207838375, −6.13159226112712054656035679676, −3.05440350754701718694872391010,
2.42736319382287273952130768570, 4.79124953153297098143636343104, 7.33751890458317553486701039833, 9.133466015585412634573809960944, 9.891987816169562442578930105613, 11.06065026645829599318386215823, 12.61414964542809116786633342649, 14.25613311472180787644133596704, 14.99200650516835733634159953492, 16.32996444443100661584801209371
