| L(s) = 1 | + (1.00 + 1.41i)3-s + (−3.61 + 1.31i)5-s + (−0.621 − 3.52i)7-s + (−0.999 + 2.82i)9-s + (0.498 + 0.181i)11-s + (−2.55 − 2.14i)13-s + (−5.47 − 3.79i)15-s + (4.03 − 6.98i)17-s + (−0.371 − 0.644i)19-s + (4.36 − 4.40i)21-s + (−0.753 + 4.27i)23-s + (7.49 − 6.29i)25-s + (−4.99 + 1.41i)27-s + (−0.0122 + 0.0102i)29-s + (1.55 − 8.79i)31-s + ⋯ |
| L(s) = 1 | + (0.577 + 0.816i)3-s + (−1.61 + 0.588i)5-s + (−0.234 − 1.33i)7-s + (−0.333 + 0.942i)9-s + (0.150 + 0.0546i)11-s + (−0.707 − 0.593i)13-s + (−1.41 − 0.979i)15-s + (0.977 − 1.69i)17-s + (−0.0853 − 0.147i)19-s + (0.952 − 0.961i)21-s + (−0.157 + 0.891i)23-s + (1.49 − 1.25i)25-s + (−0.962 + 0.272i)27-s + (−0.00227 + 0.00191i)29-s + (0.278 − 1.57i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.558261 - 0.475491i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.558261 - 0.475491i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.00 - 1.41i)T \) |
| good | 5 | \( 1 + (3.61 - 1.31i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.621 + 3.52i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.498 - 0.181i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (2.55 + 2.14i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.03 + 6.98i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.371 + 0.644i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.753 - 4.27i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.0122 - 0.0102i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.55 + 8.79i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.44 + 2.51i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (7.56 + 6.35i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (6.12 + 2.22i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.19 - 6.75i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 4.81T + 53T^{2} \) |
| 59 | \( 1 + (8.77 - 3.19i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.66 + 9.44i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.29 - 1.92i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.465 + 0.806i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.910 + 1.57i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.62 - 8.08i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-11.5 + 9.68i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-2.65 - 4.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.23 + 0.811i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.02134401931343593918142394354, −9.315225520385255307415510109253, −7.925749255649454103909340432490, −7.63066880028858833445938179671, −6.96611788184008898252999970666, −5.24220535141954136252971559882, −4.28624018396702484117412105964, −3.59041473124710322427972966239, −2.86506322209101640357946194561, −0.33779354536186984906066708140,
1.53359729048699449329065861378, 2.93979603496542882943723606203, 3.83278641974147165043454636832, 4.98207439334583539848704104515, 6.19572586161772883032184636777, 7.03690439139690616595669099649, 8.204623720651239811037307340035, 8.335202338209897660610391579500, 9.134251939142605912633501680128, 10.29402010403776309210793128927
