L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.500 + 0.866i)4-s − 3·8-s + (1.5 − 2.59i)9-s + (1.5 + 2.59i)11-s + 13-s + (0.500 − 0.866i)16-s + (3.5 + 6.06i)17-s + (1.5 + 2.59i)18-s + (−3.5 + 6.06i)19-s − 3·22-s + (3 − 5.19i)23-s + (2.5 + 4.33i)25-s + (−0.5 + 0.866i)26-s − 5·29-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.250 + 0.433i)4-s − 1.06·8-s + (0.5 − 0.866i)9-s + (0.452 + 0.783i)11-s + 0.277·13-s + (0.125 − 0.216i)16-s + (0.848 + 1.47i)17-s + (0.353 + 0.612i)18-s + (−0.802 + 1.39i)19-s − 0.639·22-s + (0.625 − 1.08i)23-s + (0.5 + 0.866i)25-s + (−0.0980 + 0.169i)26-s − 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.711665 + 1.06988i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.711665 + 1.06988i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + (-3.5 + 6.06i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−10.68458612053644865753977119388, −9.882074596398518959537477103817, −8.905146449742066753527586387736, −8.246256936157764178432145919971, −7.25806605948748998293713394908, −6.54522585179525654570302498294, −5.77211743066663245821371098304, −4.14720759242437579204427323377, −3.37986883239747882751581002366, −1.64588763118373417066028485251,
0.831276927847173280375774686574, 2.23190411749249809139188603495, 3.30887562154071612851098618355, 4.81098966747532589402369096897, 5.70699944540611899821201334029, 6.80102784244485052352179526989, 7.65220415425088187959310737100, 8.978064617523603637430001594600, 9.383676610850082375905670500794, 10.52744305472162628133655378335