L(s) = 1 | + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (1.38 − 1.75i)5-s − 0.447·7-s + (−0.309 + 0.951i)8-s + (2.15 − 0.609i)10-s + (4.42 + 3.21i)11-s + (−0.573 + 0.416i)13-s + (−0.361 − 0.262i)14-s + (−0.809 + 0.587i)16-s + (0.133 − 0.411i)17-s + (2.49 − 7.67i)19-s + (2.09 + 0.771i)20-s + (1.69 + 5.20i)22-s + (4.65 + 3.37i)23-s + ⋯ |
L(s) = 1 | + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (0.618 − 0.786i)5-s − 0.169·7-s + (−0.109 + 0.336i)8-s + (0.680 − 0.192i)10-s + (1.33 + 0.970i)11-s + (−0.159 + 0.115i)13-s + (−0.0967 − 0.0702i)14-s + (−0.202 + 0.146i)16-s + (0.0324 − 0.0997i)17-s + (0.571 − 1.76i)19-s + (0.469 + 0.172i)20-s + (0.360 + 1.10i)22-s + (0.969 + 0.704i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.11262 + 0.457285i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.11262 + 0.457285i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.38 + 1.75i)T \) |
good | 7 | \( 1 + 0.447T + 7T^{2} \) |
| 11 | \( 1 + (-4.42 - 3.21i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.573 - 0.416i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.133 + 0.411i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.49 + 7.67i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-4.65 - 3.37i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.03 - 3.17i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.407 - 1.25i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (8.14 - 5.91i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (4.80 - 3.48i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 7.09T + 43T^{2} \) |
| 47 | \( 1 + (2.83 + 8.73i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.210 + 0.648i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.75 + 2.00i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.91 - 3.57i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-0.528 + 1.62i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.558 - 1.71i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (10.0 + 7.33i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (2.72 + 8.37i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.545 - 1.67i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (11.2 + 8.18i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.04 - 12.4i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.52989262835958131274706208914, −10.04218709869165987329171726998, −9.246210935040650846623088596035, −8.589020558408213973507910033264, −7.05962933545937125274794110361, −6.64505736192433724983320686861, −5.18800162730189328233888527122, −4.68916627542467161664057121776, −3.26085167102015269089436842894, −1.60561639823574283710200319340,
1.56857866548950018080607151306, 3.07674319760653418226394004779, 3.86968951995979710772997184089, 5.43828532323250386409106436758, 6.22548189217583401823920251842, 7.00469792388741126222686136611, 8.417635392760737964606934738344, 9.504453315886064697556531120907, 10.23513641587685830960029901153, 11.10240234290237586814120222788