L(s) = 1 | + (−0.105 + 0.0611i)2-s + (1.73 − 0.0838i)3-s + (−0.992 + 1.71i)4-s − 0.529·5-s + (−0.178 + 0.114i)6-s − 0.487i·8-s + (2.98 − 0.290i)9-s + (0.0560 − 0.0323i)10-s + 4.20i·11-s + (−1.57 + 3.05i)12-s + (1.74 − 1.00i)13-s + (−0.915 + 0.0444i)15-s + (−1.95 − 3.38i)16-s + (2.19 + 3.79i)17-s + (−0.298 + 0.213i)18-s + (4.54 + 2.62i)19-s + ⋯ |
L(s) = 1 | + (−0.0749 + 0.0432i)2-s + (0.998 − 0.0484i)3-s + (−0.496 + 0.859i)4-s − 0.236·5-s + (−0.0727 + 0.0468i)6-s − 0.172i·8-s + (0.995 − 0.0967i)9-s + (0.0177 − 0.0102i)10-s + 1.26i·11-s + (−0.454 + 0.882i)12-s + (0.484 − 0.279i)13-s + (−0.236 + 0.0114i)15-s + (−0.488 − 0.846i)16-s + (0.532 + 0.921i)17-s + (−0.0703 + 0.0503i)18-s + (1.04 + 0.601i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.38905 + 0.855597i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.38905 + 0.855597i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.73 + 0.0838i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.105 - 0.0611i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 0.529T + 5T^{2} \) |
| 11 | \( 1 - 4.20iT - 11T^{2} \) |
| 13 | \( 1 + (-1.74 + 1.00i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.19 - 3.79i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.54 - 2.62i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 6.27iT - 23T^{2} \) |
| 29 | \( 1 + (7.27 + 4.20i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.03 - 0.595i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.61 + 2.79i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0994 - 0.172i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.96 + 6.86i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.98 + 8.63i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.65 - 2.10i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.71 + 11.6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-11.3 + 6.55i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.29 + 5.69i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.50iT - 71T^{2} \) |
| 73 | \( 1 + (4.86 - 2.80i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.286 + 0.495i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.42 - 9.39i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.43 + 11.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.493 + 0.285i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.45205176002768435387137380856, −9.854632887305571340450540318274, −9.597682628478672240087153746979, −8.361062649043522858641711601959, −7.76204585800264796301598740996, −7.12860245404230854187802235079, −5.44144080209036532131115261049, −3.97439823923337406152638347297, −3.53413755294744091034200806444, −1.91773903380148788422866370649,
1.08090340569352673696338980292, 2.81276582880084920461016828813, 3.96457892952831906980325860328, 5.12068336499668059685279764614, 6.22421749007465921504915089718, 7.46334040651890504430822866606, 8.441028420385909415888126422705, 9.169491309902542080271484126316, 9.831770188334186500252896516810, 10.88591387659476334946695371019