L(s) = 1 | + (1.34 + 0.776i)5-s − 2.95i·7-s − 3.57·11-s + (2.39 + 4.14i)13-s + (−3.37 − 1.95i)17-s + (−2.61 + 3.48i)19-s + (−2.64 − 4.57i)23-s + (−1.29 − 2.24i)25-s + (7.08 − 4.09i)29-s − 7.52i·31-s + (2.29 − 3.97i)35-s + 2.32·37-s + (−3.70 − 2.14i)41-s + (3.48 + 2.01i)43-s + (−4.08 − 7.06i)47-s + ⋯ |
L(s) = 1 | + (0.601 + 0.347i)5-s − 1.11i·7-s − 1.07·11-s + (0.663 + 1.14i)13-s + (−0.819 − 0.473i)17-s + (−0.599 + 0.800i)19-s + (−0.550 − 0.954i)23-s + (−0.259 − 0.448i)25-s + (1.31 − 0.759i)29-s − 1.35i·31-s + (0.387 − 0.671i)35-s + 0.381·37-s + (−0.579 − 0.334i)41-s + (0.531 + 0.306i)43-s + (−0.595 − 1.03i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.184038713\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.184038713\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (2.61 - 3.48i)T \) |
good | 5 | \( 1 + (-1.34 - 0.776i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 2.95iT - 7T^{2} \) |
| 11 | \( 1 + 3.57T + 11T^{2} \) |
| 13 | \( 1 + (-2.39 - 4.14i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.37 + 1.95i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (2.64 + 4.57i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.08 + 4.09i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7.52iT - 31T^{2} \) |
| 37 | \( 1 - 2.32T + 37T^{2} \) |
| 41 | \( 1 + (3.70 + 2.14i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.48 - 2.01i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.08 + 7.06i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.03 + 1.17i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.14 + 5.44i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.24 + 10.8i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.02 + 5.21i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.72 - 4.71i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.18 - 8.98i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.60 - 4.96i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 17.9T + 83T^{2} \) |
| 89 | \( 1 + (7.70 - 4.44i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.08 + 10.5i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.363288546017747771671287542399, −8.014612590535962963932606887212, −6.80600950770646157858602775322, −6.52607792734871298543742649104, −5.60348022993924861744113558434, −4.39323648322228325209820918329, −4.05420145116938670147412145770, −2.64281854378359230169453104338, −1.91278997141321175155091978023, −0.36539031229742724082296518166,
1.38827435694262315912341967623, 2.49754954327410948516062494445, 3.15456852189224522332532986738, 4.52050652433076104526594343369, 5.38538026539869749174192614143, 5.77310062984244384247435806464, 6.63148138298472245240012226685, 7.68212831922480098387524994343, 8.561424574500219077774020561615, 8.801048426514025911614509557076