| L(s) = 1 | + (−2.34 − 1.25i)3-s + (−2.35 + 2.86i)5-s + (1.77 + 1.18i)7-s + (2.27 + 3.40i)9-s + (−3.00 − 0.911i)11-s + (4.40 − 3.61i)13-s + (9.11 − 3.77i)15-s + (−2.12 − 0.878i)17-s + (0.222 − 2.26i)19-s + (−2.68 − 5.01i)21-s + (−4.14 − 0.825i)23-s + (−1.70 − 8.56i)25-s + (−0.285 − 2.89i)27-s + (−1.05 − 3.49i)29-s + (0.733 − 0.733i)31-s + ⋯ |
| L(s) = 1 | + (−1.35 − 0.724i)3-s + (−1.05 + 1.28i)5-s + (0.670 + 0.448i)7-s + (0.757 + 1.13i)9-s + (−0.905 − 0.274i)11-s + (1.22 − 1.00i)13-s + (2.35 − 0.975i)15-s + (−0.514 − 0.213i)17-s + (0.0511 − 0.519i)19-s + (−0.584 − 1.09i)21-s + (−0.864 − 0.172i)23-s + (−0.340 − 1.71i)25-s + (−0.0549 − 0.557i)27-s + (−0.196 − 0.648i)29-s + (0.131 − 0.131i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.312536 - 0.358890i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.312536 - 0.358890i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (2.34 + 1.25i)T + (1.66 + 2.49i)T^{2} \) |
| 5 | \( 1 + (2.35 - 2.86i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (-1.77 - 1.18i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (3.00 + 0.911i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (-4.40 + 3.61i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (2.12 + 0.878i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.222 + 2.26i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (4.14 + 0.825i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (1.05 + 3.49i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (-0.733 + 0.733i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.40 + 0.138i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (-1.67 + 8.43i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-2.27 + 1.21i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (-4.22 + 10.1i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (0.810 - 2.67i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (-6.48 - 5.32i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (-4.18 + 7.83i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (1.02 - 1.91i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (-4.85 + 7.27i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (9.35 - 6.24i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (-1.50 - 3.63i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-0.375 - 0.0370i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (-5.47 + 1.09i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (4.20 - 4.20i)T - 97iT^{2} \) |
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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
−10.88030044435153593256252894805, −10.39513585837886792630019411452, −8.486511988918627481258854525594, −7.76170302560461545665986599313, −6.97240414254638533861342375187, −6.03909291905767057059177480542, −5.29556865736065908221401422637, −3.84888851656661993969099909287, −2.44756967695468000056183419622, −0.38237351974156344060890263644,
1.24965786084827960736527368960, 4.01307137091810914896808852085, 4.45275059877369382371277045203, 5.29065634173825519614730784736, 6.30309041202574520223511482568, 7.68456161897740351294429164996, 8.410177468258554331934441677194, 9.429095884341360088457466364184, 10.56212165496652572594313819483, 11.21904522930334310182043452466
