L(s) = 1 | + (−0.849 − 1.47i)2-s + (−0.444 + 0.769i)4-s − 0.949·5-s − 1.88·8-s + (0.806 + 1.39i)10-s + 0.588·11-s + (−2.50 − 4.34i)13-s + (2.49 + 4.31i)16-s + (3.79 + 6.56i)17-s + (−2.23 + 3.86i)19-s + (0.421 − 0.730i)20-s + (−0.5 − 0.866i)22-s − 2.47·23-s − 4.09·25-s + (−4.26 + 7.38i)26-s + ⋯ |
L(s) = 1 | + (−0.600 − 1.04i)2-s + (−0.222 + 0.384i)4-s − 0.424·5-s − 0.667·8-s + (0.255 + 0.441i)10-s + 0.177·11-s + (−0.696 − 1.20i)13-s + (0.623 + 1.07i)16-s + (0.919 + 1.59i)17-s + (−0.511 + 0.886i)19-s + (0.0943 − 0.163i)20-s + (−0.106 − 0.184i)22-s − 0.516·23-s − 0.819·25-s + (−0.836 + 1.44i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5852492164\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5852492164\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.849 + 1.47i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + 0.949T + 5T^{2} \) |
| 11 | \( 1 - 0.588T + 11T^{2} \) |
| 13 | \( 1 + (2.50 + 4.34i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.79 - 6.56i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.23 - 3.86i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 + (-2.73 + 4.74i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.03 - 5.26i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.49 + 6.05i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.527 - 0.913i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.49 - 6.05i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.73 - 6.47i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.46 - 5.99i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.21 - 9.03i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.82 - 10.0i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.93 + 10.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4.30T + 71T^{2} \) |
| 73 | \( 1 + (-2.23 - 3.86i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.666 - 1.15i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.84 + 4.92i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.421 - 0.730i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.70 - 2.94i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.05464094396187451262716024396, −9.006291720849097637331034556054, −8.082886093642741159463082666874, −7.69399047887802155520889706039, −6.16085395317524717262878424907, −5.67388148992940014583711296643, −4.16116522374061159909204771151, −3.39139154140835941680404301797, −2.31801222719081076095649801449, −1.16588928774620046349954382398,
0.32566413189893949728223362840, 2.30482839566329170636242778910, 3.51344545125187962424754222031, 4.68979944488585700893446590713, 5.54826549457739740629359462481, 6.72385333701727512778809335350, 7.08046351754981669883672478986, 7.87830667622589733434736180031, 8.657440133527404543597563398000, 9.472551638081976506183437489024