| L(s) = 1 | + (−1.26 − 1.02i)2-s + (0.979 − 1.42i)3-s + (0.137 + 0.644i)4-s + (−1.08 + 1.95i)5-s + (−2.70 + 0.804i)6-s + (−3.51 + 0.942i)7-s + (−0.992 + 1.94i)8-s + (−1.07 − 2.79i)9-s + (3.38 − 1.35i)10-s + (−4.73 + 0.497i)11-s + (1.05 + 0.436i)12-s + (−2.32 − 2.87i)13-s + (5.42 + 2.41i)14-s + (1.72 + 3.46i)15-s + (4.46 − 1.98i)16-s + (−2.54 − 1.29i)17-s + ⋯ |
| L(s) = 1 | + (−0.896 − 0.725i)2-s + (0.565 − 0.824i)3-s + (0.0685 + 0.322i)4-s + (−0.486 + 0.873i)5-s + (−1.10 + 0.328i)6-s + (−1.33 + 0.356i)7-s + (−0.350 + 0.688i)8-s + (−0.359 − 0.932i)9-s + (1.07 − 0.429i)10-s + (−1.42 + 0.150i)11-s + (0.304 + 0.125i)12-s + (−0.644 − 0.796i)13-s + (1.45 + 0.645i)14-s + (0.444 + 0.895i)15-s + (1.11 − 0.496i)16-s + (−0.618 − 0.315i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0695330 + 0.163899i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0695330 + 0.163899i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.979 + 1.42i)T \) |
| 5 | \( 1 + (1.08 - 1.95i)T \) |
| good | 2 | \( 1 + (1.26 + 1.02i)T + (0.415 + 1.95i)T^{2} \) |
| 7 | \( 1 + (3.51 - 0.942i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (4.73 - 0.497i)T + (10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (2.32 + 2.87i)T + (-2.70 + 12.7i)T^{2} \) |
| 17 | \( 1 + (2.54 + 1.29i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-6.31 + 2.05i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.919 + 2.39i)T + (-17.0 + 15.3i)T^{2} \) |
| 29 | \( 1 + (-2.82 - 3.14i)T + (-3.03 + 28.8i)T^{2} \) |
| 31 | \( 1 + (0.908 - 1.00i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.242 - 1.53i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-0.830 - 0.0872i)T + (40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (1.23 + 4.59i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-4.08 - 0.213i)T + (46.7 + 4.91i)T^{2} \) |
| 53 | \( 1 + (-2.05 + 1.04i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-1.22 + 11.6i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (-0.806 - 7.67i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (10.4 - 0.546i)T + (66.6 - 7.00i)T^{2} \) |
| 71 | \( 1 + (10.9 + 3.55i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (2.51 - 15.8i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (6.41 - 5.77i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (5.32 + 8.19i)T + (-33.7 + 75.8i)T^{2} \) |
| 89 | \( 1 + (2.03 - 1.47i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.188 + 3.60i)T + (-96.4 - 10.1i)T^{2} \) |
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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
−11.61907861083534983106606200304, −10.43686923770927012604422412007, −9.827340091928967194818908462915, −8.780676800477063701577756722385, −7.73138866034331191985994895304, −6.92640837932257798133622454743, −5.56914291105186325115921148679, −2.97780622156573834529811468382, −2.67194929078626579646103845803, −0.17042892620604310619172128534,
3.12571871785742272782702344567, 4.31456232451709869572564816319, 5.72572712896033692759738578360, 7.30472880853533659277283048123, 7.955185757000061343706705740250, 9.019446477405713356456521994834, 9.627142832966812539682566610499, 10.36193715374856627375564660847, 11.90223599943933187682200413802, 13.01803900924337026147797363357
