| L(s) = 1 | + (−0.323 − 1.37i)2-s + (−0.881 + 0.471i)3-s + (−1.79 + 0.890i)4-s + (−1.61 − 1.97i)5-s + (0.934 + 1.06i)6-s + (2.12 − 1.41i)7-s + (1.80 + 2.17i)8-s + (0.555 − 0.831i)9-s + (−2.19 + 2.86i)10-s + (−4.64 + 1.40i)11-s + (1.15 − 1.62i)12-s + (−1.38 − 1.13i)13-s + (−2.64 − 2.46i)14-s + (2.35 + 0.976i)15-s + (2.41 − 3.18i)16-s + (−6.63 + 2.74i)17-s + ⋯ |
| L(s) = 1 | + (−0.228 − 0.973i)2-s + (−0.509 + 0.272i)3-s + (−0.895 + 0.445i)4-s + (−0.724 − 0.882i)5-s + (0.381 + 0.433i)6-s + (0.803 − 0.536i)7-s + (0.638 + 0.769i)8-s + (0.185 − 0.277i)9-s + (−0.693 + 0.906i)10-s + (−1.40 + 0.424i)11-s + (0.334 − 0.470i)12-s + (−0.384 − 0.315i)13-s + (−0.706 − 0.659i)14-s + (0.608 + 0.252i)15-s + (0.603 − 0.797i)16-s + (−1.60 + 0.666i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0243260 + 0.0362229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0243260 + 0.0362229i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.323 + 1.37i)T \) |
| 3 | \( 1 + (0.881 - 0.471i)T \) |
| good | 5 | \( 1 + (1.61 + 1.97i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (-2.12 + 1.41i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (4.64 - 1.40i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (1.38 + 1.13i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (6.63 - 2.74i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.496 - 5.04i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (-1.56 + 0.310i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (2.71 - 8.95i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-2.30 - 2.30i)T + 31iT^{2} \) |
| 37 | \( 1 + (6.80 + 0.669i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (1.43 + 7.23i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (4.98 + 2.66i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (1.74 + 4.20i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (0.385 + 1.26i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (-8.00 + 6.57i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (3.44 + 6.45i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (0.149 + 0.280i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (-2.04 - 3.06i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (10.5 + 7.05i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.746i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-10.1 + 0.996i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (16.2 + 3.23i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-9.82 - 9.82i)T + 97iT^{2} \) |
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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
−10.65062695456070742353659224231, −10.24634195659005761035356786583, −8.803513732425418619699323467233, −8.208540387750859439805475530064, −7.24684098010521736453073734429, −5.18480959460746318604259388165, −4.73677989469692202084218753566, −3.65600292635196369542452738904, −1.80341583057459167751777649511, −0.03260397105420053012877431054,
2.57848313738356239811309274518, 4.50655443175312533734334962271, 5.25936641523342466377625263107, 6.47304316915526812694654960542, 7.28987250508043697655476184308, 7.999870036652197360980523354480, 8.938233452170719552801256905426, 10.18409041652293514121303986003, 11.24562323017607640645162874558, 11.54719325083472836631287008835
