| L(s) = 1 | + (−3.46 + 0.610i)2-s + (1.47 − 2.61i)3-s + (7.86 − 2.86i)4-s + (3.16 − 3.77i)5-s + (−3.51 + 9.94i)6-s + (−3.18 − 1.16i)7-s + (−13.2 + 7.67i)8-s + (−4.64 − 7.71i)9-s + (−8.66 + 15.0i)10-s + (10.2 + 12.1i)11-s + (4.13 − 24.7i)12-s + (−0.621 + 3.52i)13-s + (11.7 + 2.07i)14-s + (−5.18 − 13.8i)15-s + (15.7 − 13.1i)16-s + (−9.99 − 5.77i)17-s + ⋯ |
| L(s) = 1 | + (−1.73 + 0.305i)2-s + (0.492 − 0.870i)3-s + (1.96 − 0.715i)4-s + (0.633 − 0.754i)5-s + (−0.586 + 1.65i)6-s + (−0.455 − 0.165i)7-s + (−1.66 + 0.959i)8-s + (−0.515 − 0.856i)9-s + (−0.866 + 1.50i)10-s + (0.928 + 1.10i)11-s + (0.344 − 2.06i)12-s + (−0.0477 + 0.271i)13-s + (0.839 + 0.148i)14-s + (−0.345 − 0.922i)15-s + (0.981 − 0.823i)16-s + (−0.588 − 0.339i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.521620 - 0.216620i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.521620 - 0.216620i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.47 + 2.61i)T \) |
| good | 2 | \( 1 + (3.46 - 0.610i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-3.16 + 3.77i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (3.18 + 1.16i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (-10.2 - 12.1i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (0.621 - 3.52i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (9.99 + 5.77i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-9.59 - 16.6i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-5.28 - 14.5i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (11.0 - 1.94i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (-23.0 + 8.38i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (-21.2 + 36.8i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-6.57 - 1.15i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (36.6 - 30.7i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (17.5 - 48.2i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + 61.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (28.8 - 34.4i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-4.52 - 1.64i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-2.06 + 11.7i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (76.5 + 44.1i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-21.5 - 37.3i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (25.8 + 146. i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-78.1 + 13.7i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (-69.3 + 40.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (141. - 118. i)T + (1.63e3 - 9.26e3i)T^{2} \) |
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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
−17.33490089950261382446941904110, −16.30947074509048920836434177658, −14.75545708639462647184996182262, −13.16132710940801043541997614649, −11.75519412796959712587717277440, −9.686802574840437190256082400552, −9.116606847275694874873859110521, −7.62251896223824859749978314143, −6.43899239237412988677898611414, −1.60555146991293249521318549836,
2.87624351522045982711212073039, 6.53733651151336654691464271873, 8.457887835148245974239732887899, 9.422336087316621194145851509076, 10.43608970068773046807620935789, 11.39913735749875804504402323509, 13.79256535987735000806683938887, 15.25480789250001165581228192492, 16.41738972629419425899058013235, 17.32757971693782118515689144918
