| L(s) = 1 | + (2.05 − 3.43i)2-s + (−3.67 + 3.67i)3-s + (−7.56 − 14.0i)4-s + (−34.5 + 34.5i)5-s + (5.06 + 20.1i)6-s − 15.7·7-s + (−63.9 − 2.95i)8-s − 27i·9-s + (47.6 + 189. i)10-s + (43.6 + 43.6i)11-s + (79.6 + 23.9i)12-s + (−152. − 152. i)13-s + (−32.2 + 53.9i)14-s − 253. i·15-s + (−141. + 213. i)16-s + 343.·17-s + ⋯ |
| L(s) = 1 | + (0.513 − 0.858i)2-s + (−0.408 + 0.408i)3-s + (−0.473 − 0.881i)4-s + (−1.38 + 1.38i)5-s + (0.140 + 0.559i)6-s − 0.320·7-s + (−0.998 − 0.0461i)8-s − 0.333i·9-s + (0.476 + 1.89i)10-s + (0.360 + 0.360i)11-s + (0.552 + 0.166i)12-s + (−0.899 − 0.899i)13-s + (−0.164 + 0.275i)14-s − 1.12i·15-s + (−0.552 + 0.833i)16-s + 1.19·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.151344 + 0.284474i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.151344 + 0.284474i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.05 + 3.43i)T \) |
| 3 | \( 1 + (3.67 - 3.67i)T \) |
| good | 5 | \( 1 + (34.5 - 34.5i)T - 625iT^{2} \) |
| 7 | \( 1 + 15.7T + 2.40e3T^{2} \) |
| 11 | \( 1 + (-43.6 - 43.6i)T + 1.46e4iT^{2} \) |
| 13 | \( 1 + (152. + 152. i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 - 343.T + 8.35e4T^{2} \) |
| 19 | \( 1 + (412. - 412. i)T - 1.30e5iT^{2} \) |
| 23 | \( 1 + 317.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (185. + 185. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 - 84.5iT - 9.23e5T^{2} \) |
| 37 | \( 1 + (-0.769 + 0.769i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 - 622. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (-822. - 822. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 - 3.36e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (725. - 725. i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + (-2.83e3 - 2.83e3i)T + 1.21e7iT^{2} \) |
| 61 | \( 1 + (479. + 479. i)T + 1.38e7iT^{2} \) |
| 67 | \( 1 + (1.16e3 - 1.16e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 + 8.08e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 2.91e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 5.37e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-4.36e3 + 4.36e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 1.59e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 9.73e3T + 8.85e7T^{2} \) |
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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
−14.82140662831924652191368459173, −14.61044046907945248227794261735, −12.50528291558279983485318778645, −11.83794682950728151853792310388, −10.64940170353655276253894223633, −9.974857522530181183774960744837, −7.77143185675780711235048769751, −6.14812879743071256385887363223, −4.21770562897231132423785088153, −3.07383744937685454806747453752,
0.17414166587173731452179106303, 4.01609651216188923728070574631, 5.18543600968518539822506269772, 6.91093686064191484714737024252, 8.058490414377954154592574394102, 9.126574647289192338450711845075, 11.65467181608887313101515135904, 12.29484167140563972826723200043, 13.19369876514248123261077610530, 14.65678484430818503755919002727
