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
−16.32996444443100661584801209371, −14.99200650516835733634159953492, −14.25613311472180787644133596704, −12.61414964542809116786633342649, −11.06065026645829599318386215823, −9.891987816169562442578930105613, −9.133466015585412634573809960944, −7.33751890458317553486701039833, −4.79124953153297098143636343104, −2.42736319382287273952130768570,
3.05440350754701718694872391010, 6.13159226112712054656035679676, 7.73083738593641501525207838375, 8.408014615281598237847077483930, 9.906633481779976573688983287679, 12.01127523673784067784751724145, 13.03063515627938353217804082468, 14.52192119774910764806760342760, 15.39577127841221135918675978585, 16.68548436125242120380293371834