L(s) = 1 | + 1.73i·3-s + (−1.5 + 2.59i)5-s + (2 − 1.73i)7-s − 2.99·9-s + (4.5 − 2.59i)11-s + (−4.5 − 2.59i)15-s + (−1.5 − 2.59i)17-s + (1.5 + 0.866i)19-s + (2.99 + 3.46i)21-s + (−4.5 − 2.59i)23-s + (−2 − 3.46i)25-s − 5.19i·27-s + (−1.5 + 0.866i)31-s + (4.5 + 7.79i)33-s + (1.5 + 7.79i)35-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + (−0.670 + 1.16i)5-s + (0.755 − 0.654i)7-s − 0.999·9-s + (1.35 − 0.783i)11-s + (−1.16 − 0.670i)15-s + (−0.363 − 0.630i)17-s + (0.344 + 0.198i)19-s + (0.654 + 0.755i)21-s + (−0.938 − 0.541i)23-s + (−0.400 − 0.692i)25-s − 0.999i·27-s + (−0.269 + 0.155i)31-s + (0.783 + 1.35i)33-s + (0.253 + 1.31i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.795524 + 0.493545i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.795524 + 0.493545i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 5 | \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.5 + 2.59i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.5 + 2.59i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 - 2.59i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (10.5 + 6.06i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (10.5 - 6.06i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−14.30007466802251090881176146207, −14.10908677261187547868823734370, −11.77099264527247962141301237020, −11.19547474441001663919566442074, −10.31977830992065478285507198937, −8.959569177307192509690606165204, −7.65784074094312859218918253664, −6.29060002996217269992566457595, −4.42674995746821934470783251129, −3.34339564043339402970541273188,
1.66083755492376087956416999134, 4.29558771480786242961523833261, 5.78976907741528653497454627437, 7.34333967932112187354646184389, 8.414372359329517044140273470889, 9.215859163200377806247973577386, 11.35856485317969817843575583503, 12.11139840696310783391186017406, 12.69589717085870242385032194163, 14.06615680336555087338634410833