| L(s) = 1 | + (1.67 − 0.450i)3-s + (−0.436 − 1.19i)5-s + (3.53 − 0.622i)7-s + (2.59 − 1.50i)9-s + (−4.59 − 1.67i)11-s + (1.75 + 1.47i)13-s + (−1.27 − 1.80i)15-s + (0.393 + 0.227i)17-s + (−5.43 + 3.13i)19-s + (5.62 − 2.63i)21-s + (0.629 − 3.57i)23-s + (2.58 − 2.16i)25-s + (3.66 − 3.68i)27-s + (6.09 + 7.26i)29-s + (−0.352 − 0.0621i)31-s + ⋯ |
| L(s) = 1 | + (0.965 − 0.260i)3-s + (−0.195 − 0.536i)5-s + (1.33 − 0.235i)7-s + (0.864 − 0.502i)9-s + (−1.38 − 0.504i)11-s + (0.485 + 0.407i)13-s + (−0.327 − 0.467i)15-s + (0.0954 + 0.0551i)17-s + (−1.24 + 0.719i)19-s + (1.22 − 0.574i)21-s + (0.131 − 0.744i)23-s + (0.516 − 0.433i)25-s + (0.704 − 0.709i)27-s + (1.13 + 1.34i)29-s + (−0.0633 − 0.0111i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.86163 - 0.704459i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86163 - 0.704459i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.67 + 0.450i)T \) |
| good | 5 | \( 1 + (0.436 + 1.19i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-3.53 + 0.622i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (4.59 + 1.67i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.75 - 1.47i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.393 - 0.227i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.43 - 3.13i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.629 + 3.57i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-6.09 - 7.26i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.352 + 0.0621i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.46 + 4.26i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.66 - 5.56i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.37 - 6.52i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.475 + 2.69i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 4.30iT - 53T^{2} \) |
| 59 | \( 1 + (9.78 - 3.56i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.68 - 15.2i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.26 - 1.50i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.92 - 5.06i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.44 + 5.96i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.89 - 7.02i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-9.97 + 8.37i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-0.480 + 0.277i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.40 + 1.96i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.86001191461487955502219968662, −10.30860721136147846363523929611, −8.767693157885660872510166772345, −8.369023371755169396554913401732, −7.74725521199443344117796725581, −6.48920249700659495687311891931, −5.01412657375530296945751290145, −4.20971563285830507464556368492, −2.73598262697316758567041504428, −1.39786556105037928799990579583,
2.00696654131437934645745693873, 3.01322170649520032133485017742, 4.42980689163213690327614284347, 5.23837266575876239500068307708, 6.84317001676177985649114043604, 7.981264015123543203180145569745, 8.199549536851419458080022874201, 9.395747755060894883924509011758, 10.57837605260982536841522394578, 10.88398056697983943286338703373
