| L(s) = 1 | + (−0.745 − 1.20i)2-s + (−1.61 + 0.613i)3-s + (−0.888 + 1.79i)4-s + (1.55 − 1.03i)5-s + (1.94 + 1.48i)6-s + (−0.159 + 0.385i)7-s + (2.81 − 0.268i)8-s + (2.24 − 1.98i)9-s + (−2.40 − 1.09i)10-s + (0.211 − 1.06i)11-s + (0.339 − 3.44i)12-s + (2.60 − 3.89i)13-s + (0.582 − 0.0955i)14-s + (−1.87 + 2.63i)15-s + (−2.42 − 3.18i)16-s + (−1.09 − 1.09i)17-s + ⋯ |
| L(s) = 1 | + (−0.527 − 0.849i)2-s + (−0.935 + 0.354i)3-s + (−0.444 + 0.895i)4-s + (0.694 − 0.464i)5-s + (0.793 + 0.607i)6-s + (−0.0604 + 0.145i)7-s + (0.995 − 0.0948i)8-s + (0.749 − 0.662i)9-s + (−0.760 − 0.345i)10-s + (0.0636 − 0.319i)11-s + (0.0980 − 0.995i)12-s + (0.721 − 1.07i)13-s + (0.155 − 0.0255i)14-s + (−0.485 + 0.680i)15-s + (−0.605 − 0.795i)16-s + (−0.265 − 0.265i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.593981 - 0.457762i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.593981 - 0.457762i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.745 + 1.20i)T \) |
| 3 | \( 1 + (1.61 - 0.613i)T \) |
| good | 5 | \( 1 + (-1.55 + 1.03i)T + (1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (0.159 - 0.385i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.211 + 1.06i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (-2.60 + 3.89i)T + (-4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (1.09 + 1.09i)T + 17iT^{2} \) |
| 19 | \( 1 + (-4.15 + 6.22i)T + (-7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (-1.51 - 3.66i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-7.27 + 1.44i)T + (26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + 1.30T + 31T^{2} \) |
| 37 | \( 1 + (5.71 - 3.82i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-4.35 + 1.80i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-0.518 + 2.60i)T + (-39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (-0.379 + 0.379i)T - 47iT^{2} \) |
| 53 | \( 1 + (9.70 + 1.93i)T + (48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (-6.01 - 8.99i)T + (-22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (1.92 - 0.383i)T + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (1.89 + 9.54i)T + (-61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (12.7 + 5.26i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (9.04 - 3.74i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (9.12 - 9.12i)T - 79iT^{2} \) |
| 83 | \( 1 + (6.43 + 4.30i)T + (31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (-8.08 - 3.34i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 16.2iT - 97T^{2} \) |
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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
−12.09744798240939966699190631623, −11.25474070174073855222806290933, −10.45754466281923412604335539589, −9.510260551916224828847777135264, −8.756413351022752869715517067229, −7.25358506895689039197310882606, −5.76447197576729223823073061242, −4.77203837250661784522788292465, −3.15870123792674752741413842395, −1.04278569097579713178157838265,
1.61134239960985953939979468450, 4.44105330117388917246202660312, 5.80241199859504667509939542475, 6.47623950400483820570137085709, 7.34719456073205989654260061665, 8.653051952376859199610111965433, 9.933064433743992250800584413230, 10.51126180947091542142677450170, 11.62253843582935948010899104243, 12.82620054990463963869870900595
