L(s) = 1 | + (−0.596 + 1.28i)2-s + (−1.72 + 0.141i)3-s + (−1.28 − 1.52i)4-s + (−0.236 − 1.18i)5-s + (0.848 − 2.29i)6-s + (1.67 − 0.692i)7-s + (2.73 − 0.738i)8-s + (2.95 − 0.488i)9-s + (1.66 + 0.405i)10-s + (−0.195 + 0.292i)11-s + (2.44 + 2.45i)12-s + (1.51 + 0.302i)13-s + (−0.109 + 2.55i)14-s + (0.575 + 2.01i)15-s + (−0.681 + 3.94i)16-s + (5.35 − 5.35i)17-s + ⋯ |
L(s) = 1 | + (−0.421 + 0.906i)2-s + (−0.996 + 0.0817i)3-s + (−0.644 − 0.764i)4-s + (−0.105 − 0.530i)5-s + (0.346 − 0.938i)6-s + (0.632 − 0.261i)7-s + (0.965 − 0.261i)8-s + (0.986 − 0.162i)9-s + (0.525 + 0.128i)10-s + (−0.0590 + 0.0883i)11-s + (0.704 + 0.709i)12-s + (0.421 + 0.0838i)13-s + (−0.0292 + 0.683i)14-s + (0.148 + 0.520i)15-s + (−0.170 + 0.985i)16-s + (1.29 − 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.709538 + 0.0424883i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.709538 + 0.0424883i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.596 - 1.28i)T \) |
| 3 | \( 1 + (1.72 - 0.141i)T \) |
good | 5 | \( 1 + (0.236 + 1.18i)T + (-4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (-1.67 + 0.692i)T + (4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (0.195 - 0.292i)T + (-4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-1.51 - 0.302i)T + (12.0 + 4.97i)T^{2} \) |
| 17 | \( 1 + (-5.35 + 5.35i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.449 - 0.0895i)T + (17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (5.61 + 2.32i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-2.38 + 1.59i)T + (11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 + (1.66 + 8.34i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (1.93 - 4.66i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (1.78 - 2.67i)T + (-16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.866i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.141 + 0.0944i)T + (20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (-8.97 + 1.78i)T + (54.5 - 22.5i)T^{2} \) |
| 61 | \( 1 + (11.0 - 7.39i)T + (23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + (2.80 + 4.19i)T + (-25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (2.24 + 5.41i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (6.08 - 14.6i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-5.00 - 5.00i)T + 79iT^{2} \) |
| 83 | \( 1 + (1.64 - 8.26i)T + (-76.6 - 31.7i)T^{2} \) |
| 89 | \( 1 + (-4.01 - 9.68i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 0.511iT - 97T^{2} \) |
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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
−12.45187260809630299653658592297, −11.55522392821828260113024850638, −10.41097256771733025170492844903, −9.634455769734687078178412335686, −8.323993365814189291211668025003, −7.42965687085025395911923356322, −6.26728381140708148568589656251, −5.20158307148620324564938382409, −4.38169532067596726017502944158, −0.987975810983212597406034078412,
1.52722101667084434799634785241, 3.48086987585125704097769352122, 4.87940675491322407228529239294, 6.16689227758494047666957953710, 7.61052189444444472856161324520, 8.501446078409477810469339377978, 10.12308560994619043882581818370, 10.49483764127039861800354177737, 11.63505435556059084009166142255, 12.06884711944059315526848731820