| L(s) = 1 | + (0.663 − 1.24i)2-s + (−1.73 + 0.0789i)3-s + (−1.11 − 1.65i)4-s + (1.18 − 2.53i)5-s + (−1.04 + 2.21i)6-s + (−1.66 + 0.294i)7-s + (−2.81 + 0.297i)8-s + (2.98 − 0.273i)9-s + (−2.38 − 3.16i)10-s + (−3.10 + 1.44i)11-s + (2.06 + 2.77i)12-s + (−0.388 − 4.44i)13-s + (−0.739 + 2.27i)14-s + (−1.84 + 4.48i)15-s + (−1.49 + 3.71i)16-s + (−1.31 + 2.27i)17-s + ⋯ |
| L(s) = 1 | + (0.469 − 0.883i)2-s + (−0.998 + 0.0455i)3-s + (−0.559 − 0.828i)4-s + (0.529 − 1.13i)5-s + (−0.428 + 0.903i)6-s + (−0.630 + 0.111i)7-s + (−0.994 + 0.105i)8-s + (0.995 − 0.0911i)9-s + (−0.754 − 1.00i)10-s + (−0.937 + 0.437i)11-s + (0.596 + 0.802i)12-s + (−0.107 − 1.23i)13-s + (−0.197 + 0.609i)14-s + (−0.477 + 1.15i)15-s + (−0.373 + 0.927i)16-s + (−0.318 + 0.552i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.170562 + 0.677226i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.170562 + 0.677226i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.663 + 1.24i)T \) |
| 3 | \( 1 + (1.73 - 0.0789i)T \) |
| good | 5 | \( 1 + (-1.18 + 2.53i)T + (-3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (1.66 - 0.294i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (3.10 - 1.44i)T + (7.07 - 8.42i)T^{2} \) |
| 13 | \( 1 + (0.388 + 4.44i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (1.31 - 2.27i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0505 - 0.188i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.44 - 0.254i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (9.05 + 0.792i)T + (28.5 + 5.03i)T^{2} \) |
| 31 | \( 1 + (-0.660 + 3.74i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.404 - 1.50i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (3.67 - 4.38i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-4.29 + 2.00i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (1.85 + 10.4i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-7.37 + 7.37i)T - 53iT^{2} \) |
| 59 | \( 1 + (-4.07 + 8.74i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (-4.68 + 6.68i)T + (-20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (0.802 + 9.17i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (3.63 + 2.09i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-13.2 + 7.64i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.93 - 1.62i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-4.10 - 0.359i)T + (81.7 + 14.4i)T^{2} \) |
| 89 | \( 1 + (3.94 - 2.28i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-14.7 - 5.36i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.63660273094231789003206014154, −9.953615657275832284017293337575, −9.279835208204742622292884343046, −7.985321968633469668644635192989, −6.41857350322236096659146752723, −5.37623744784112546072580389550, −5.07733281191383706830003166553, −3.68924658645793025556064417155, −2.00370174601794764071671659237, −0.41262342286783670037297639270,
2.67253213827608563045383224466, 4.04241463722688300635090059490, 5.29530049433274366246556920911, 6.09918753760468395665293890447, 6.90679332060971289377208559282, 7.39461431925513210174664529470, 9.006222449510859664687818671692, 9.921340717675635474013487693023, 10.87627094415157360709132383216, 11.62914443450144271288761148506
