L(s) = 1 | + (−18.9 − 18.9i)3-s + (−36.4 − 42.3i)5-s + (−112. + 112. i)7-s + 473. i·9-s − 269. i·11-s + (−403. + 403. i)13-s + (−112. + 1.49e3i)15-s + (1.09e3 + 1.09e3i)17-s − 1.80e3·19-s + 4.25e3·21-s + (−2.83e3 − 2.83e3i)23-s + (−470. + 3.08e3i)25-s + (4.35e3 − 4.35e3i)27-s − 7.85e3i·29-s − 4.68e3i·31-s + ⋯ |
L(s) = 1 | + (−1.21 − 1.21i)3-s + (−0.651 − 0.758i)5-s + (−0.867 + 0.867i)7-s + 1.94i·9-s − 0.671i·11-s + (−0.662 + 0.662i)13-s + (−0.129 + 1.71i)15-s + (0.922 + 0.922i)17-s − 1.14·19-s + 2.10·21-s + (−1.11 − 1.11i)23-s + (−0.150 + 0.988i)25-s + (1.15 − 1.15i)27-s − 1.73i·29-s − 0.875i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.4468997248\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4468997248\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (36.4 + 42.3i)T \) |
good | 3 | \( 1 + (18.9 + 18.9i)T + 243iT^{2} \) |
| 7 | \( 1 + (112. - 112. i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 + 269. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (403. - 403. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (-1.09e3 - 1.09e3i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 + 1.80e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (2.83e3 + 2.83e3i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 + 7.85e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 4.68e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (-5.15e3 - 5.15e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.65e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-3.23e3 - 3.23e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (-6.00e3 + 6.00e3i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (6.07e3 - 6.07e3i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 - 3.93e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (4.36e4 - 4.36e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 7.04e3iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.44e4 + 2.44e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 + 3.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-2.56e4 - 2.56e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + 1.37e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-4.79e4 - 4.79e4i)T + 8.58e9iT^{2} \) |
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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
−12.10007515328918625545192941387, −11.40937332066314812881694302194, −10.02441068483406502371567585278, −8.578557505064571907056698560108, −7.70193997265988431213818232151, −6.27802285406029938407131028821, −5.85005841326093720840290428311, −4.30147064806867136692602898037, −2.20318703460563196023505327293, −0.59661923270852835988912320909,
0.32647490309767431685035942877, 3.27109094712267594056600536920, 4.20166214200518099234937034036, 5.38430366632860734590683768095, 6.65796112967817124617031893599, 7.54255817531910954593852213668, 9.507550417271977384617483305265, 10.26902102461594606485132485218, 10.75473754845182322455311915199, 11.89286900061115663734899771310