| L(s) = 1 | − 5.45i·3-s − 3.94·5-s + 2.61i·7-s − 20.7·9-s + 15.3i·11-s + 1.27·13-s + 21.5i·15-s + 2.27·17-s + 29.0i·19-s + 14.2·21-s − 8.77i·23-s − 9.41·25-s + 64.1i·27-s + 5.38·29-s − 48.1i·31-s + ⋯ |
| L(s) = 1 | − 1.81i·3-s − 0.789·5-s + 0.374i·7-s − 2.30·9-s + 1.39i·11-s + 0.0980·13-s + 1.43i·15-s + 0.133·17-s + 1.52i·19-s + 0.680·21-s − 0.381i·23-s − 0.376·25-s + 2.37i·27-s + 0.185·29-s − 1.55i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5453627121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5453627121\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 5.38T \) |
| good | 3 | \( 1 + 5.45iT - 9T^{2} \) |
| 5 | \( 1 + 3.94T + 25T^{2} \) |
| 7 | \( 1 - 2.61iT - 49T^{2} \) |
| 11 | \( 1 - 15.3iT - 121T^{2} \) |
| 13 | \( 1 - 1.27T + 169T^{2} \) |
| 17 | \( 1 - 2.27T + 289T^{2} \) |
| 19 | \( 1 - 29.0iT - 361T^{2} \) |
| 23 | \( 1 + 8.77iT - 529T^{2} \) |
| 31 | \( 1 + 48.1iT - 961T^{2} \) |
| 37 | \( 1 + 54.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 10.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 0.0320iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 86.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 34.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 74.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 35.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 65.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 41.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 105.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 40.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 97.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 67.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + 34.9T + 9.40e3T^{2} \) |
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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
−11.39797400493778386101345113633, −10.12413211180101816223744123416, −8.880202780249424862329559187898, −7.82377728294874823896933448703, −7.55602458753908769586511217159, −6.50574967356126174304337352095, −5.61335761884388530919105886910, −4.06999259133722725640931404169, −2.52506977152778236759171205549, −1.46837136865316426090019115282,
0.22795371246412992413264891489, 3.13515692296242365643761166674, 3.73045938764179470471492532758, 4.77563634985991118500960224751, 5.60264800694926695006897610341, 6.98939454750049952750551133236, 8.410741007203579114175611197887, 8.834101287331649889963223796603, 9.894374272645253739174042603292, 10.78442773039989610005855968803
