L(s) = 1 | + (2.16 + 1.57i)2-s + (1.60 + 4.93i)4-s + (−0.946 + 0.687i)5-s + (0.309 + 0.951i)7-s + (−2.64 + 8.12i)8-s − 3.13·10-s + (−3.31 + 0.0332i)11-s + (1.36 + 0.989i)13-s + (−0.828 + 2.54i)14-s + (−10.1 + 7.36i)16-s + (4.52 − 3.28i)17-s + (1.34 − 4.14i)19-s + (−4.91 − 3.56i)20-s + (−7.24 − 5.15i)22-s − 0.119·23-s + ⋯ |
L(s) = 1 | + (1.53 + 1.11i)2-s + (0.801 + 2.46i)4-s + (−0.423 + 0.307i)5-s + (0.116 + 0.359i)7-s + (−0.933 + 2.87i)8-s − 0.992·10-s + (−0.999 + 0.0100i)11-s + (0.377 + 0.274i)13-s + (−0.221 + 0.681i)14-s + (−2.53 + 1.84i)16-s + (1.09 − 0.797i)17-s + (0.308 − 0.950i)19-s + (−1.09 − 0.797i)20-s + (−1.54 − 1.09i)22-s − 0.0249·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 - 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.918480 + 2.99828i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.918480 + 2.99828i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (3.31 - 0.0332i)T \) |
good | 2 | \( 1 + (-2.16 - 1.57i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (0.946 - 0.687i)T + (1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (-1.36 - 0.989i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.52 + 3.28i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.34 + 4.14i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 0.119T + 23T^{2} \) |
| 29 | \( 1 + (1.35 + 4.18i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.10 - 3.71i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.80 - 5.56i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (3.40 - 10.4i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 5.97T + 43T^{2} \) |
| 47 | \( 1 + (-3.88 + 11.9i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.337 + 0.245i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.223 + 0.688i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-7.16 + 5.20i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 8.78T + 67T^{2} \) |
| 71 | \( 1 + (2.59 - 1.88i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (4.50 + 13.8i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (11.7 + 8.50i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-2.79 + 2.03i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + (-4.85 - 3.52i)T + (29.9 + 92.2i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.40714988926095565013605343901, −9.971328944750166260209933338908, −8.624857981883990835169712328150, −7.79652840762197353142686221722, −7.20356123080026971075232754109, −6.24964997455371435532522560200, −5.31450409395628465282702303998, −4.69202980812486479924649405200, −3.43988526657839576073359426644, −2.69054672421491056727607819540,
1.09377973920724194085332296775, 2.50203757762851506923560653470, 3.64001162326336721704421699662, 4.27817821690862066355311471453, 5.45845694671549416157900952816, 5.92834819099885328646929105730, 7.38217804768617994533733249770, 8.333612452204028055755268515375, 9.828067183980883860507831049657, 10.43456629100307035545229956317