L(s) = 1 | + (−1.97 − 1.13i)2-s + (1.59 + 2.75i)4-s + 1.43·5-s − 2.69i·8-s + (−2.82 − 1.63i)10-s − 3.23i·11-s + (−4.43 − 2.55i)13-s + (0.119 − 0.207i)16-s + (−0.545 + 0.945i)17-s + (3.88 − 2.24i)19-s + (2.28 + 3.95i)20-s + (−3.68 + 6.37i)22-s + 4.00i·23-s − 2.94·25-s + (5.82 + 10.0i)26-s + ⋯ |
L(s) = 1 | + (−1.39 − 0.804i)2-s + (0.795 + 1.37i)4-s + 0.641·5-s − 0.951i·8-s + (−0.894 − 0.516i)10-s − 0.975i·11-s + (−1.22 − 0.709i)13-s + (0.0298 − 0.0517i)16-s + (−0.132 + 0.229i)17-s + (0.891 − 0.514i)19-s + (0.510 + 0.883i)20-s + (−0.784 + 1.35i)22-s + 0.835i·23-s − 0.588·25-s + (1.14 + 1.97i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4074800953\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4074800953\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.97 + 1.13i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 1.43T + 5T^{2} \) |
| 11 | \( 1 + 3.23iT - 11T^{2} \) |
| 13 | \( 1 + (4.43 + 2.55i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.545 - 0.945i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.88 + 2.24i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 4.00iT - 23T^{2} \) |
| 29 | \( 1 + (1.02 - 0.593i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.24 - 1.87i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.119 - 0.207i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.71 + 6.43i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.82 + 6.62i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.11 - 3.65i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.07 - 3.50i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.73 + 8.20i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.82 + 1.63i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.330 + 0.571i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.82iT - 71T^{2} \) |
| 73 | \( 1 + (6.33 + 3.65i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.83 - 3.16i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.45 + 9.44i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.84 + 11.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.69 - 1.55i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.330637051398199148112714773824, −8.713017811516132591827633905544, −7.73857577315718776381498302024, −7.21556065378604234554780141022, −5.84768441618448981930022369296, −5.14621691020365684649714001782, −3.47857525762594818758090445672, −2.64153832684846225547251553185, −1.60262062401629105918000610695, −0.26684101023497472966883172841,
1.49227346147455446249620186889, 2.48607716739405860820816413414, 4.25303825616517625479334756779, 5.30581827189682691458645615132, 6.22005906902759258172677596089, 7.06434901594932666822886957262, 7.53837577298757939569259839216, 8.427531683909406519467716812486, 9.473771863727379356432573708070, 9.672694987428570724292429301463