L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.786 − 1.54i)3-s + (−0.499 + 0.866i)4-s + (1.98 + 1.14i)5-s + (−0.942 + 1.45i)6-s + (−2.60 − 0.487i)7-s + 0.999·8-s + (−1.76 + 2.42i)9-s − 2.28i·10-s − 0.297·11-s + (1.72 + 0.0900i)12-s + (−3.20 + 1.66i)13-s + (0.877 + 2.49i)14-s + (0.206 − 3.95i)15-s + (−0.5 − 0.866i)16-s + (−0.446 + 0.773i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.454 − 0.890i)3-s + (−0.249 + 0.433i)4-s + (0.886 + 0.511i)5-s + (−0.384 + 0.593i)6-s + (−0.982 − 0.184i)7-s + 0.353·8-s + (−0.587 + 0.809i)9-s − 0.723i·10-s − 0.0895·11-s + (0.499 + 0.0259i)12-s + (−0.887 + 0.460i)13-s + (0.234 + 0.667i)14-s + (0.0532 − 1.02i)15-s + (−0.125 − 0.216i)16-s + (−0.108 + 0.187i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.112244 + 0.127530i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.112244 + 0.127530i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.786 + 1.54i)T \) |
| 7 | \( 1 + (2.60 + 0.487i)T \) |
| 13 | \( 1 + (3.20 - 1.66i)T \) |
good | 5 | \( 1 + (-1.98 - 1.14i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + 0.297T + 11T^{2} \) |
| 17 | \( 1 + (0.446 - 0.773i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + 7.89T + 19T^{2} \) |
| 23 | \( 1 + (6.76 - 3.90i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.980 - 0.566i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.839 - 1.45i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.32 + 2.49i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.52 - 3.76i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.94 + 3.36i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.21 + 3.00i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.28 - 3.62i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.21 - 3.01i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 8.44iT - 61T^{2} \) |
| 67 | \( 1 - 3.39iT - 67T^{2} \) |
| 71 | \( 1 + (-1.14 - 1.99i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.16 + 10.6i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.46 + 7.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.54iT - 83T^{2} \) |
| 89 | \( 1 + (12.3 - 7.15i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.19 - 2.06i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.92594182269723600416862699899, −10.20080015547825243343862664571, −9.561112254559191633862909369587, −8.420344590328598259439182394795, −7.34710963550657976869782230716, −6.47673397447611102120967659262, −5.84643622812149658853331516659, −4.27810567862643146559207913153, −2.69773801427267943954999904274, −1.90319810293240666125428654937,
0.10631522992482555112535302164, 2.46279470837477500971608548660, 4.13045780710380729473600028859, 5.10007976621611376841315830561, 6.06639343557437125198543942294, 6.52070060465671678923666171426, 8.086774735856287950295019583507, 8.982889561256306701156163922916, 9.856243464253920677175146628188, 10.04920990731372075393401767112