| L(s) = 1 | + (1.41 − 0.0480i)2-s + (−0.478 − 3.32i)3-s + (1.99 − 0.135i)4-s + (−1.40 − 2.18i)5-s + (−0.836 − 4.68i)6-s + (0.0455 + 0.0709i)7-s + (2.81 − 0.287i)8-s + (−7.97 + 2.34i)9-s + (−2.09 − 3.02i)10-s + (−0.836 + 1.30i)11-s + (−1.40 − 6.57i)12-s + (−0.0479 − 0.333i)13-s + (0.0678 + 0.0980i)14-s + (−6.60 + 5.72i)15-s + (3.96 − 0.542i)16-s + (0.818 − 0.944i)17-s + ⋯ |
| L(s) = 1 | + (0.999 − 0.0339i)2-s + (−0.276 − 1.92i)3-s + (0.997 − 0.0679i)4-s + (−0.628 − 0.977i)5-s + (−0.341 − 1.91i)6-s + (0.0172 + 0.0268i)7-s + (0.994 − 0.101i)8-s + (−2.65 + 0.780i)9-s + (−0.661 − 0.955i)10-s + (−0.252 + 0.392i)11-s + (−0.406 − 1.89i)12-s + (−0.0133 − 0.0925i)13-s + (0.0181 + 0.0262i)14-s + (−1.70 + 1.47i)15-s + (0.990 − 0.135i)16-s + (0.198 − 0.229i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.264764 - 2.08561i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.264764 - 2.08561i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 + 0.0480i)T \) |
| 89 | \( 1 + (-7.83 + 5.25i)T \) |
| good | 3 | \( 1 + (0.478 + 3.32i)T + (-2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (1.40 + 2.18i)T + (-2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (-0.0455 - 0.0709i)T + (-2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.836 - 1.30i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (0.0479 + 0.333i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-0.818 + 0.944i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-4.14 + 1.21i)T + (15.9 - 10.2i)T^{2} \) |
| 23 | \( 1 + (0.955 + 3.25i)T + (-19.3 + 12.4i)T^{2} \) |
| 29 | \( 1 + (-5.04 + 3.24i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (2.50 - 8.54i)T + (-26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + 9.53T + 37T^{2} \) |
| 41 | \( 1 + (-6.99 - 1.00i)T + (39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (6.45 + 4.14i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (0.651 - 4.53i)T + (-45.0 - 13.2i)T^{2} \) |
| 53 | \( 1 + (-3.72 + 0.535i)T + (50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.77 + 12.3i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (1.20 + 2.63i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.12 + 0.161i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-4.67 - 3.00i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-3.31 - 0.974i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (13.7 + 4.05i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (5.02 - 5.80i)T + (-11.8 - 82.1i)T^{2} \) |
| 97 | \( 1 + (-14.0 + 9.00i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.37815621235618956910323244688, −8.699421117617656402685686434750, −8.010714201438748848138915625781, −7.21717541079977566354129984058, −6.58691101442806649652981336164, −5.45298009367717058533555885314, −4.83733904385193159471368294646, −3.22669766176757432763821608058, −2.02394959695152237232475842630, −0.828101365218289121386297123816,
2.88568380592569902568309145829, 3.52601743891830222924165996606, 4.25036428467323814143841590383, 5.32351965328324717793815382466, 5.94461045508328163674414266835, 7.16082099856172830134862601408, 8.190264004421667818861854926040, 9.407068390214239096797259179402, 10.37382380642749641122793550701, 10.75962655154775289318297805754
