L(s) = 1 | + (0.551 + 0.955i)2-s + (0.454 + 1.67i)3-s + (0.391 − 0.678i)4-s − 0.105·5-s + (−1.34 + 1.35i)6-s + 3.07·8-s + (−2.58 + 1.52i)9-s + (−0.0581 − 0.100i)10-s + 3.33·11-s + (1.31 + 0.346i)12-s + (1.23 + 2.14i)13-s + (−0.0479 − 0.176i)15-s + (0.909 + 1.57i)16-s + (0.806 + 1.39i)17-s + (−2.87 − 1.63i)18-s + (−3.84 + 6.65i)19-s + ⋯ |
L(s) = 1 | + (0.389 + 0.675i)2-s + (0.262 + 0.964i)3-s + (0.195 − 0.339i)4-s − 0.0471·5-s + (−0.549 + 0.553i)6-s + 1.08·8-s + (−0.862 + 0.506i)9-s + (−0.0183 − 0.0318i)10-s + 1.00·11-s + (0.378 + 0.0999i)12-s + (0.343 + 0.595i)13-s + (−0.0123 − 0.0455i)15-s + (0.227 + 0.393i)16-s + (0.195 + 0.338i)17-s + (−0.678 − 0.384i)18-s + (−0.881 + 1.52i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0325 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0325 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43439 + 1.48188i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43439 + 1.48188i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.454 - 1.67i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.551 - 0.955i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + 0.105T + 5T^{2} \) |
| 11 | \( 1 - 3.33T + 11T^{2} \) |
| 13 | \( 1 + (-1.23 - 2.14i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.806 - 1.39i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.84 - 6.65i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 1.89T + 23T^{2} \) |
| 29 | \( 1 + (-4.64 + 8.04i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.63 + 8.02i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.991 + 1.71i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.74 + 6.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.77 - 6.53i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.59 + 2.76i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.98 - 8.64i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.22 + 3.86i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.83 + 4.91i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.98 - 8.63i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 + (2.36 + 4.09i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.84 + 6.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.584 + 1.01i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.01 + 5.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.90 + 3.29i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.33018962807507029378772943654, −10.20738206499657641897633555327, −9.775514781736352577563097724993, −8.531014922032146649629389794168, −7.74511391503461439724437759708, −6.24960740176496462747553837295, −5.91426152007024201786279460188, −4.40019129210135170238558074242, −3.91543598529055002234176551908, −1.97437562527834566999589728948,
1.34118799244652637934992588969, 2.66693953124041462758002435508, 3.58929104641524734217677124779, 4.92204200857210029562907939122, 6.48732501739678464433656291531, 7.05341780745715430789923810993, 8.209858634475618131336337701233, 8.858278874482561887528569176842, 10.24013077568616745870123735855, 11.23806576730002292674023602935