L(s) = 1 | + (−0.526 − 1.31i)2-s + (2.16 − 1.25i)3-s + (−1.44 + 1.38i)4-s + (−2.19 + 2.19i)5-s + (−2.78 − 2.18i)6-s + (−0.152 − 0.569i)7-s + (2.57 + 1.16i)8-s + (1.63 − 2.83i)9-s + (4.04 + 1.72i)10-s + (−2.85 − 0.764i)11-s + (−1.40 + 4.80i)12-s + (2.37 + 2.71i)13-s + (−0.667 + 0.500i)14-s + (−2.01 + 7.52i)15-s + (0.179 − 3.99i)16-s + (−2.30 − 1.33i)17-s + ⋯ |
L(s) = 1 | + (−0.372 − 0.928i)2-s + (1.25 − 0.723i)3-s + (−0.722 + 0.691i)4-s + (−0.983 + 0.983i)5-s + (−1.13 − 0.893i)6-s + (−0.0576 − 0.215i)7-s + (0.910 + 0.413i)8-s + (0.546 − 0.946i)9-s + (1.27 + 0.546i)10-s + (−0.860 − 0.230i)11-s + (−0.405 + 1.38i)12-s + (0.657 + 0.753i)13-s + (−0.178 + 0.133i)14-s + (−0.520 + 1.94i)15-s + (0.0449 − 0.998i)16-s + (−0.560 − 0.323i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.688109 - 0.452459i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.688109 - 0.452459i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.526 + 1.31i)T \) |
| 13 | \( 1 + (-2.37 - 2.71i)T \) |
good | 3 | \( 1 + (-2.16 + 1.25i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (2.19 - 2.19i)T - 5iT^{2} \) |
| 7 | \( 1 + (0.152 + 0.569i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (2.85 + 0.764i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (2.30 + 1.33i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.11 + 0.835i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (1.03 + 1.78i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.621 + 1.07i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (6.34 + 6.34i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.133 - 0.5i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-5.59 - 1.5i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (3.60 - 6.24i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.16 + 3.16i)T - 47iT^{2} \) |
| 53 | \( 1 - 3.67T + 53T^{2} \) |
| 59 | \( 1 + (-1.82 - 6.80i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (3.97 - 6.88i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.92 + 7.18i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.98 + 0.530i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-0.0440 - 0.0440i)T + 73iT^{2} \) |
| 79 | \( 1 + 2.73iT - 79T^{2} \) |
| 83 | \( 1 + (-10.1 - 10.1i)T + 83iT^{2} \) |
| 89 | \( 1 + (-2.14 + 8.01i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (1.22 + 4.58i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−14.99316516123861635990273209900, −13.85564227480798002665512605459, −13.14964908535024505624154558080, −11.68519828343756302238397691431, −10.76297947868636936510574432952, −9.183438406606072508903695021825, −8.004099666798507673025801503897, −7.21726432500879692583106424966, −3.81537264573178636856392011621, −2.57555361163562246479293078353,
3.83345855683635656007752329412, 5.25468222936889579894861396139, 7.64273346282807461895755750664, 8.446074000671785654891709169571, 9.234488243823770805419356174672, 10.59576910880828179547086758222, 12.64823231216606508618886116510, 13.74902425811294494307865445567, 15.00064008094007994302726448429, 15.76911369042123987095611222204