| L(s) = 1 | + (0.795 + 0.948i)2-s + (0.595 + 1.62i)3-s + (0.0810 − 0.459i)4-s + (2.00 − 0.996i)5-s + (−1.06 + 1.85i)6-s + (−2.24 + 0.395i)7-s + (2.64 − 1.52i)8-s + (−2.28 + 1.93i)9-s + (2.53 + 1.10i)10-s + (−4.87 − 1.77i)11-s + (0.795 − 0.142i)12-s + (−0.993 + 1.18i)13-s + (−2.15 − 1.81i)14-s + (2.81 + 2.66i)15-s + (2.67 + 0.974i)16-s + (−4.61 − 2.66i)17-s + ⋯ |
| L(s) = 1 | + (0.562 + 0.670i)2-s + (0.344 + 0.938i)3-s + (0.0405 − 0.229i)4-s + (0.895 − 0.445i)5-s + (−0.436 + 0.759i)6-s + (−0.847 + 0.149i)7-s + (0.935 − 0.539i)8-s + (−0.763 + 0.646i)9-s + (0.802 + 0.349i)10-s + (−1.46 − 0.534i)11-s + (0.229 − 0.0410i)12-s + (−0.275 + 0.328i)13-s + (−0.577 − 0.484i)14-s + (0.726 + 0.687i)15-s + (0.669 + 0.243i)16-s + (−1.11 − 0.646i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37909 + 0.798763i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37909 + 0.798763i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.595 - 1.62i)T \) |
| 5 | \( 1 + (-2.00 + 0.996i)T \) |
| good | 2 | \( 1 + (-0.795 - 0.948i)T + (-0.347 + 1.96i)T^{2} \) |
| 7 | \( 1 + (2.24 - 0.395i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (4.87 + 1.77i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (0.993 - 1.18i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (4.61 + 2.66i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.28 - 3.96i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.06 - 0.892i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.94 + 4.15i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.228 - 1.29i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.18 - 1.84i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.26 - 2.74i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.88 - 7.92i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (6.68 - 1.17i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 6.64iT - 53T^{2} \) |
| 59 | \( 1 + (-2.83 + 1.03i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.999 - 5.67i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.22 + 2.65i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.0130 - 0.0226i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.00 + 2.89i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (12.6 - 10.6i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.96 - 8.29i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-2.32 - 4.02i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.48 + 6.81i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−13.50847449101488578568307808534, −13.00463009077936856993049037713, −11.12115884384428019297841323524, −10.03270032863773804403476062778, −9.505142998791089454000920175846, −8.147582473700589399805095290682, −6.51835660360564820883197091041, −5.46826999718259806309382468813, −4.68012553254084501913305100337, −2.78802322866191422568888715626,
2.31386215855653061871985118218, 3.07830865331834297661175858101, 5.11534543540800732636876067804, 6.64488775925737903759728795520, 7.49529097487257926648762252486, 8.875837312081998223807436239907, 10.24131353581929359566648598674, 11.10557945447168774910639147553, 12.50428168163127322497859270349, 13.16968785689840889944514830045
