| L(s) = 1 | + (−0.138 − 1.40i)2-s + (2.62 − 0.658i)3-s + (−1.96 + 0.389i)4-s + (−2.26 − 1.07i)5-s + (−1.28 − 3.60i)6-s + (−0.826 − 2.72i)7-s + (0.819 + 2.70i)8-s + (3.82 − 2.04i)9-s + (−1.19 + 3.33i)10-s + (−2.11 + 0.314i)11-s + (−4.89 + 2.31i)12-s + (0.782 − 2.18i)13-s + (−3.72 + 1.54i)14-s + (−6.65 − 1.32i)15-s + (3.69 − 1.52i)16-s + (4.41 − 0.877i)17-s + ⋯ |
| L(s) = 1 | + (−0.0978 − 0.995i)2-s + (1.51 − 0.379i)3-s + (−0.980 + 0.194i)4-s + (−1.01 − 0.479i)5-s + (−0.526 − 1.47i)6-s + (−0.312 − 1.03i)7-s + (0.289 + 0.957i)8-s + (1.27 − 0.681i)9-s + (−0.377 + 1.05i)10-s + (−0.638 + 0.0947i)11-s + (−1.41 + 0.668i)12-s + (0.217 − 0.606i)13-s + (−0.994 + 0.411i)14-s + (−1.71 − 0.341i)15-s + (0.924 − 0.381i)16-s + (1.07 − 0.212i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.560854 - 1.31125i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.560854 - 1.31125i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.138 + 1.40i)T \) |
| good | 3 | \( 1 + (-2.62 + 0.658i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (2.26 + 1.07i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (0.826 + 2.72i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (2.11 - 0.314i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (-0.782 + 2.18i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (-4.41 + 0.877i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (-3.49 - 3.16i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-2.93 + 0.288i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (-4.77 - 6.43i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-2.44 - 1.01i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (0.110 - 2.25i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (7.40 - 9.02i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (-2.59 + 10.3i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (7.88 + 5.26i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (6.89 - 9.29i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (-3.93 - 10.9i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (-1.81 + 3.03i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (-4.53 - 2.71i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (-4.71 + 8.82i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (-1.42 + 4.69i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (-4.85 - 7.26i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (0.750 + 15.2i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (3.99 + 0.393i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (-5.45 + 13.1i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−11.89458739829421708377770837601, −10.50207398839187869244680475927, −9.842616936881268808597580736543, −8.621735485063456541259915018359, −7.997413467839322975196223906321, −7.33622990579446013459202575731, −4.94020126309702638284071985611, −3.61297414889779509774466391205, −3.09012905072238770217393916183, −1.13283204435462624733595625631,
2.86890125134199002355127880236, 3.80465644419239421096544244798, 5.15640403664740063630488640093, 6.65137136573401347666902693954, 7.83421115445288425240811024982, 8.238632282583778195526741274780, 9.274298189778591724399355371703, 9.908620069763820700317443401777, 11.43222032180755238076103897926, 12.64438538479564353648375873756
