| L(s) = 1 | + (−0.265 + 1.71i)3-s + (0.601 − 1.65i)5-s + (3.31 + 0.584i)7-s + (−2.85 − 0.908i)9-s + (1.91 − 0.696i)11-s + (3.10 − 2.60i)13-s + (2.66 + 1.46i)15-s + (−2.93 + 1.69i)17-s + (2.04 + 1.17i)19-s + (−1.88 + 5.52i)21-s + (0.695 + 3.94i)23-s + (1.46 + 1.22i)25-s + (2.31 − 4.65i)27-s + (1.89 − 2.25i)29-s + (−4.69 + 0.827i)31-s + ⋯ |
| L(s) = 1 | + (−0.153 + 0.988i)3-s + (0.268 − 0.738i)5-s + (1.25 + 0.221i)7-s + (−0.953 − 0.302i)9-s + (0.577 − 0.210i)11-s + (0.862 − 0.723i)13-s + (0.688 + 0.378i)15-s + (−0.710 + 0.410i)17-s + (0.468 + 0.270i)19-s + (−0.410 + 1.20i)21-s + (0.145 + 0.822i)23-s + (0.292 + 0.245i)25-s + (0.445 − 0.895i)27-s + (0.351 − 0.418i)29-s + (−0.842 + 0.148i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.52978 + 0.412995i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52978 + 0.412995i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.265 - 1.71i)T \) |
| good | 5 | \( 1 + (-0.601 + 1.65i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-3.31 - 0.584i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.91 + 0.696i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.10 + 2.60i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.93 - 1.69i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.04 - 1.17i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.695 - 3.94i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.89 + 2.25i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (4.69 - 0.827i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.41 - 7.65i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.67 + 7.95i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.29 - 3.55i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (2.19 - 12.4i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 10.0iT - 53T^{2} \) |
| 59 | \( 1 + (-6.19 - 2.25i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.0727 - 0.412i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (7.81 + 9.30i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (7.57 + 13.1i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.87 + 10.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.00 - 5.97i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (12.5 + 10.5i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (2.81 + 1.62i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.7 + 3.90i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.24006725442649052487715105527, −10.41208786666842341002929213039, −9.289335155077285890884555521669, −8.670715141003584325753690867788, −7.899109363599323846376804889944, −6.18910514497789930446279188451, −5.30791213848984195899081790724, −4.55429028885157930499458823918, −3.38000214912977942470800497097, −1.45708267481019516941611806735,
1.40061614841003286956481096335, 2.55668966156962923052560655422, 4.23011358968825246349503725403, 5.48354746408316888228725629305, 6.68937537755979178501225975517, 7.10919376951638040846995289750, 8.311732453858730743562034494139, 9.013624443996772863900575132449, 10.47758165645428111147869353429, 11.32124272972222969355412244845
