L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (−1.30 + 0.951i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s − 1.61·10-s + (−0.809 + 3.21i)11-s − 12-s + (−1.61 − 1.17i)13-s + (−0.309 + 0.951i)14-s + (−0.499 − 1.53i)15-s + (−0.809 + 0.587i)16-s + (−2.11 + 1.53i)17-s + ⋯ |
L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (−0.585 + 0.425i)5-s + (−0.330 + 0.239i)6-s + (0.116 + 0.359i)7-s + (−0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s − 0.511·10-s + (−0.243 + 0.969i)11-s − 0.288·12-s + (−0.448 − 0.326i)13-s + (−0.0825 + 0.254i)14-s + (−0.129 − 0.397i)15-s + (−0.202 + 0.146i)16-s + (−0.513 + 0.373i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.356586 + 1.29650i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.356586 + 1.29650i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.309 - 0.951i)T \) |
| 7 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (0.809 - 3.21i)T \) |
good | 5 | \( 1 + (1.30 - 0.951i)T + (1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (1.61 + 1.17i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (2.11 - 1.53i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.572 + 1.76i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 1.85T + 23T^{2} \) |
| 29 | \( 1 + (-2 - 6.15i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.92 + 1.40i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.881 + 2.71i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.118 + 0.363i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + (0.763 - 2.35i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-6.85 - 4.97i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.236 + 0.726i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-8.23 + 5.98i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + (-2 + 1.45i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.76 - 11.5i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (4.47 + 3.24i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (5.85 - 4.25i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 8.79T + 89T^{2} \) |
| 97 | \( 1 + (9.85 + 7.15i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.37111533174086263351819245437, −10.73711981229992986741113008412, −9.633021014223147566824932463567, −8.667338906967442852236704298479, −7.53608366864868532736802266152, −6.87278809737402159254171160802, −5.58476438239570726864493347485, −4.75112110472995897211315561201, −3.74973675352364082519574199288, −2.51500954380065041470914203025,
0.70270736443764762160574086859, 2.41950158009148225673486235856, 3.79050837996209070029185658774, 4.80530822766002509728142264104, 5.86073361327136568668475428090, 6.91148099580451337456724469555, 7.902604287268975596664115099304, 8.775498069870787359004688037747, 9.986805533200307085198503255144, 11.00436422149492465629207169757