L(s) = 1 | − 3·5-s + (−2.5 − 0.866i)7-s − 3·11-s + (0.5 + 0.866i)13-s + (1.5 + 2.59i)17-s + (−3.5 + 6.06i)19-s − 9·23-s + 4·25-s + (1.5 − 2.59i)29-s + (4 − 6.92i)31-s + (7.5 + 2.59i)35-s + (0.5 − 0.866i)37-s + (1.5 + 2.59i)41-s + (−0.5 + 0.866i)43-s + (5.5 + 4.33i)49-s + ⋯ |
L(s) = 1 | − 1.34·5-s + (−0.944 − 0.327i)7-s − 0.904·11-s + (0.138 + 0.240i)13-s + (0.363 + 0.630i)17-s + (−0.802 + 1.39i)19-s − 1.87·23-s + 0.800·25-s + (0.278 − 0.482i)29-s + (0.718 − 1.24i)31-s + (1.26 + 0.439i)35-s + (0.0821 − 0.142i)37-s + (0.234 + 0.405i)41-s + (−0.0762 + 0.132i)43-s + (0.785 + 0.618i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5946000529\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5946000529\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 5 | \( 1 + 3T + 5T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 9T + 23T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8 + 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.314956312881759123224476564979, −7.974489177529554876701882570338, −7.41477524565974534704403796274, −6.21886383245703373889475541072, −5.92720732385006968741406965400, −4.43031001848698126545053107556, −3.97687565556806584445415874233, −3.24066762113550097094263412158, −2.05884008072415760072867498624, −0.37305863137397643257563003924,
0.52740392271826969054027773658, 2.42744002555340840886467671729, 3.15928926813210391127594788224, 3.99821846522307815142327887860, 4.83751325073459044826877946288, 5.70727236726681934865792631524, 6.69762086587224616322337897264, 7.23354629692893629091474083984, 8.174890139527913313774107040977, 8.517885854922948556694070336518