| L(s) = 1 | + (0.241 + 1.68i)3-s + (−0.959 − 0.281i)5-s + (2.58 − 2.97i)7-s + (0.105 − 0.0309i)9-s + (1.84 − 1.18i)11-s + (2.08 + 2.40i)13-s + (0.241 − 1.68i)15-s + (0.820 − 1.79i)17-s + (0.904 + 1.97i)19-s + (5.63 + 3.62i)21-s + (−3.74 + 3.00i)23-s + (0.841 + 0.540i)25-s + (2.19 + 4.80i)27-s + (0.693 − 1.51i)29-s + (0.122 − 0.852i)31-s + ⋯ |
| L(s) = 1 | + (0.139 + 0.971i)3-s + (−0.429 − 0.125i)5-s + (0.975 − 1.12i)7-s + (0.0351 − 0.0103i)9-s + (0.555 − 0.357i)11-s + (0.577 + 0.666i)13-s + (0.0624 − 0.434i)15-s + (0.198 − 0.435i)17-s + (0.207 + 0.454i)19-s + (1.23 + 0.790i)21-s + (−0.779 + 0.625i)23-s + (0.168 + 0.108i)25-s + (0.422 + 0.925i)27-s + (0.128 − 0.281i)29-s + (0.0220 − 0.153i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.58534 + 0.347140i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58534 + 0.347140i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (3.74 - 3.00i)T \) |
| good | 3 | \( 1 + (-0.241 - 1.68i)T + (-2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (-2.58 + 2.97i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.84 + 1.18i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.08 - 2.40i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.820 + 1.79i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.904 - 1.97i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.693 + 1.51i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.122 + 0.852i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.64 + 0.483i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (1.93 + 0.568i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.39 - 9.72i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 + (-6.84 + 7.89i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (8.37 + 9.66i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.26 + 8.81i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-11.5 - 7.44i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-2.28 - 1.46i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-0.192 - 0.420i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (6.61 + 7.63i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (7.69 - 2.25i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (0.0222 + 0.154i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (11.4 + 3.37i)T + (81.6 + 52.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.27758597284201668995162986948, −10.15355313736086803972234431891, −9.498517428161243555351262837229, −8.396221081652963772310926875250, −7.62116474100822701090682389334, −6.52782648149568912687814334837, −5.06496109029840721791154693549, −4.16609132119277764226653604597, −3.60303011847313057117869357812, −1.37041394103026499374859778541,
1.42104772654170486659645160738, 2.57954713488230069143871390108, 4.16351538946117405878388632470, 5.40568004836901711124784893873, 6.42885824508704770850766081458, 7.43524609662377103357952464588, 8.245972928142294466178049109078, 8.831749879459785660148995847265, 10.20051375404337656163879030443, 11.19587199036063751634928925035
