L(s) = 1 | + (0.5 + 0.866i)2-s + (0.500 − 0.866i)4-s + 5-s + (0.5 − 2.59i)7-s + 3·8-s + (0.5 + 0.866i)10-s − 5·11-s + (2.5 + 4.33i)13-s + (2.5 − 0.866i)14-s + (0.500 + 0.866i)16-s + (1.5 + 2.59i)17-s + (−0.5 + 0.866i)19-s + (0.500 − 0.866i)20-s + (−2.5 − 4.33i)22-s − 3·23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.250 − 0.433i)4-s + 0.447·5-s + (0.188 − 0.981i)7-s + 1.06·8-s + (0.158 + 0.273i)10-s − 1.50·11-s + (0.693 + 1.20i)13-s + (0.668 − 0.231i)14-s + (0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s + (−0.114 + 0.198i)19-s + (0.111 − 0.193i)20-s + (−0.533 − 0.923i)22-s − 0.625·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.57795 + 0.176127i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.57795 + 0.176127i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 2.59i)T \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (1.5 + 2.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.5 - 11.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.5 + 7.79i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.01002145692832672929896698278, −11.45864290406942219613481034488, −10.50724460328676014828734711336, −9.934778265525707550009118246164, −8.278093257197619904958850562194, −7.30876594798100087675587758522, −6.27681967613607538491015499824, −5.27745395674239555413144214031, −4.03357147087678414271010530640, −1.82747189575830799731420572209,
2.23682937701810946979391624099, 3.23360641549339548483224109574, 5.00571261370781038522520340926, 5.94431177626814035491798849008, 7.66094480501915480945253857803, 8.335743490566554719418023111861, 9.820741524510833736270812329267, 10.73552797197154587147109636488, 11.61049754995393916376844565011, 12.64440623041172584319540162341