| L(s) = 1 | + (−1.40 − 0.118i)2-s + (1.59 + 0.682i)3-s + (1.97 + 0.333i)4-s + (−2.31 + 2.18i)5-s + (−2.16 − 1.15i)6-s + (−4.62 + 0.540i)7-s + (−2.73 − 0.703i)8-s + (2.06 + 2.17i)9-s + (3.51 − 2.79i)10-s + (−0.848 + 0.253i)11-s + (2.91 + 1.87i)12-s + (4.22 − 0.246i)13-s + (6.57 − 0.213i)14-s + (−5.16 + 1.89i)15-s + (3.77 + 1.31i)16-s + (−0.885 − 0.322i)17-s + ⋯ |
| L(s) = 1 | + (−0.996 − 0.0837i)2-s + (0.918 + 0.394i)3-s + (0.985 + 0.166i)4-s + (−1.03 + 0.975i)5-s + (−0.882 − 0.469i)6-s + (−1.74 + 0.204i)7-s + (−0.968 − 0.248i)8-s + (0.689 + 0.724i)9-s + (1.11 − 0.885i)10-s + (−0.255 + 0.0765i)11-s + (0.840 + 0.542i)12-s + (1.17 − 0.0682i)13-s + (1.75 − 0.0571i)14-s + (−1.33 + 0.488i)15-s + (0.944 + 0.329i)16-s + (−0.214 − 0.0781i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0359158 - 0.142698i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0359158 - 0.142698i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 + 0.118i)T \) |
| 3 | \( 1 + (-1.59 - 0.682i)T \) |
| good | 5 | \( 1 + (2.31 - 2.18i)T + (0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (4.62 - 0.540i)T + (6.81 - 1.61i)T^{2} \) |
| 11 | \( 1 + (0.848 - 0.253i)T + (9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (-4.22 + 0.246i)T + (12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (0.885 + 0.322i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (2.82 + 7.75i)T + (-14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (5.68 + 0.665i)T + (22.3 + 5.30i)T^{2} \) |
| 29 | \( 1 + (1.34 + 2.67i)T + (-17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (0.737 - 0.991i)T + (-8.89 - 29.6i)T^{2} \) |
| 37 | \( 1 + (5.64 + 6.73i)T + (-6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (3.52 - 2.31i)T + (16.2 - 37.6i)T^{2} \) |
| 43 | \( 1 + (0.679 - 2.86i)T + (-38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (-3.65 - 4.90i)T + (-13.4 + 45.0i)T^{2} \) |
| 53 | \( 1 + (5.28 - 3.05i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (9.51 + 2.84i)T + (49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (-4.39 - 1.89i)T + (41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (1.58 - 3.14i)T + (-40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (-1.09 - 6.23i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (2.43 - 13.8i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (11.7 + 7.72i)T + (31.2 + 72.5i)T^{2} \) |
| 83 | \( 1 + (-3.87 + 5.89i)T + (-32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (-1.45 + 8.23i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (3.85 - 4.08i)T + (-5.64 - 96.8i)T^{2} \) |
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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
−10.75797716095929090214446622733, −10.16080110098117722141849918706, −9.220260477916551551495839372469, −8.648432045666944872604420757076, −7.63419158649121735102503121439, −6.90441980509143468301826440140, −6.16791523110201022081676916550, −3.99787316703123427244483791775, −3.22687497382759219645181713007, −2.49899250743752237141352637893,
0.093440763284242058638337574216, 1.64734722099831924757662847319, 3.35863747533429656162887756128, 3.86960608539599998509361533023, 5.96458479814478675044289508519, 6.70410698190096981254644387919, 7.75401301490048696027844931512, 8.392161366991626760376657855119, 8.949651991776819147964055348219, 9.877236903878360942551178560944
