| L(s) = 1 | + (0.616 + 1.27i)2-s + (−0.980 − 0.195i)3-s + (−1.23 + 1.56i)4-s + (−0.620 + 0.929i)5-s + (−0.356 − 1.36i)6-s + (−3.49 + 1.44i)7-s + (−2.76 − 0.609i)8-s + (0.923 + 0.382i)9-s + (−1.56 − 0.217i)10-s + (0.138 + 0.696i)11-s + (1.52 − 1.29i)12-s + (3.26 + 4.87i)13-s + (−3.99 − 3.55i)14-s + (0.790 − 0.790i)15-s + (−0.927 − 3.89i)16-s + (−1.76 − 1.76i)17-s + ⋯ |
| L(s) = 1 | + (0.436 + 0.899i)2-s + (−0.566 − 0.112i)3-s + (−0.619 + 0.784i)4-s + (−0.277 + 0.415i)5-s + (−0.145 − 0.558i)6-s + (−1.32 + 0.547i)7-s + (−0.976 − 0.215i)8-s + (0.307 + 0.127i)9-s + (−0.494 − 0.0686i)10-s + (0.0417 + 0.209i)11-s + (0.439 − 0.374i)12-s + (0.904 + 1.35i)13-s + (−1.06 − 0.950i)14-s + (0.204 − 0.204i)15-s + (−0.231 − 0.972i)16-s + (−0.427 − 0.427i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.136745 + 0.805305i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.136745 + 0.805305i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.616 - 1.27i)T \) |
| 3 | \( 1 + (0.980 + 0.195i)T \) |
| good | 5 | \( 1 + (0.620 - 0.929i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (3.49 - 1.44i)T + (4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.138 - 0.696i)T + (-10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-3.26 - 4.87i)T + (-4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (1.76 + 1.76i)T + 17iT^{2} \) |
| 19 | \( 1 + (-3.44 + 2.30i)T + (7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (1.98 - 4.79i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (-1.63 + 8.20i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 - 2.85iT - 31T^{2} \) |
| 37 | \( 1 + (-9.21 - 6.15i)T + (14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (1.40 - 3.39i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-2.84 + 0.565i)T + (39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (4.57 + 4.57i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.62 - 8.18i)T + (-48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (3.01 - 4.51i)T + (-22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (4.52 + 0.899i)T + (56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (2.79 + 0.556i)T + (61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (-0.612 + 0.253i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (7.40 + 3.06i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (0.0773 - 0.0773i)T - 79iT^{2} \) |
| 83 | \( 1 + (-13.2 + 8.84i)T + (31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (5.72 + 13.8i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 11.3iT - 97T^{2} \) |
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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
−13.27489304995096502771823003216, −11.96961955169516128817379656297, −11.45105431730337521513911863474, −9.712828264077520608447843960269, −9.022016516969155801871261211115, −7.50279046538818357579946005481, −6.57895977266589739959373657498, −5.94859783483334026481403320420, −4.45872328492144672005155023365, −3.14696604010223940281780561202,
0.69292946877568684491185818081, 3.17629449522943240478445096323, 4.18698339808676942529133686489, 5.61357358166593966334211994678, 6.49357313126623375345252187035, 8.246852444005042816380014568569, 9.476706057985190952981954192490, 10.42035357776958059305962878588, 10.99426622990034517433016465716, 12.38183970397696158194342944563
