| L(s) = 1 | + (0.909 − 1.08i)2-s + (−2.95 − 0.512i)3-s + (−0.347 − 1.96i)4-s + (2.71 − 7.45i)5-s + (−3.24 + 2.73i)6-s + (0.0787 − 0.446i)7-s + (−2.44 − 1.41i)8-s + (8.47 + 3.02i)9-s + (−5.60 − 9.71i)10-s + (4.82 + 13.2i)11-s + (0.0171 + 5.99i)12-s + (−9.80 + 8.22i)13-s + (−0.412 − 0.491i)14-s + (−11.8 + 20.6i)15-s + (−3.75 + 1.36i)16-s + (28.5 − 16.4i)17-s + ⋯ |
| L(s) = 1 | + (0.454 − 0.541i)2-s + (−0.985 − 0.170i)3-s + (−0.0868 − 0.492i)4-s + (0.542 − 1.49i)5-s + (−0.540 + 0.456i)6-s + (0.0112 − 0.0638i)7-s + (−0.306 − 0.176i)8-s + (0.941 + 0.336i)9-s + (−0.560 − 0.971i)10-s + (0.438 + 1.20i)11-s + (0.00143 + 0.499i)12-s + (−0.754 + 0.632i)13-s + (−0.0294 − 0.0351i)14-s + (−0.789 + 1.37i)15-s + (−0.234 + 0.0855i)16-s + (1.67 − 0.969i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0495 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0495 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.801091 - 0.841840i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.801091 - 0.841840i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.909 + 1.08i)T \) |
| 3 | \( 1 + (2.95 + 0.512i)T \) |
| good | 5 | \( 1 + (-2.71 + 7.45i)T + (-19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-0.0787 + 0.446i)T + (-46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-4.82 - 13.2i)T + (-92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (9.80 - 8.22i)T + (29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (-28.5 + 16.4i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-0.202 + 0.351i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-14.2 + 2.51i)T + (497. - 180. i)T^{2} \) |
| 29 | \( 1 + (16.8 - 20.0i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-4.33 - 24.6i)T + (-903. + 328. i)T^{2} \) |
| 37 | \( 1 + (-3.84 - 6.65i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (15.9 + 18.9i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (16.8 - 6.13i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (46.7 + 8.24i)T + (2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 - 0.261iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-18.8 + 51.7i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (18.1 - 103. i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (49.4 - 41.5i)T + (779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (94.6 - 54.6i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-31.4 + 54.5i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (14.7 + 12.3i)T + (1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-36.7 + 43.7i)T + (-1.19e3 - 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-89.7 - 51.8i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (52.8 - 19.2i)T + (7.20e3 - 6.04e3i)T^{2} \) |
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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
−14.63659495007173232695284856805, −13.30975533988698384722493862788, −12.30444002368090790684641842240, −11.91472224811346262774028532794, −10.12242134535871983312392153510, −9.292237550333178996523448569262, −7.15083503525383830451260416508, −5.38485266564028263576716573466, −4.64240913726474890267412423156, −1.41090898549712650034754932274,
3.40575499655363090947144870374, 5.58580739548615630688803097536, 6.35309309687222240865104765000, 7.66851459705219236856100251635, 9.848784475471328167653082518452, 10.83710250122808227529506411666, 11.92125065442921931289912108490, 13.28492442479197956528118945505, 14.53930511992143102586237651661, 15.17380170452530639676301767123
