L(s) = 1 | + (1.68 − 0.971i)5-s + (2.61 + 1.50i)7-s + (2.20 − 3.81i)11-s + (−2.65 − 4.60i)13-s − 4.16i·17-s + 4.66i·19-s + (1.30 + 2.26i)23-s + (−0.613 + 1.06i)25-s + (−1.13 − 0.655i)29-s + (0.0648 − 0.0374i)31-s + 5.86·35-s + 8.43·37-s + (−8.16 + 4.71i)41-s + (5.06 + 2.92i)43-s + (2.40 − 4.16i)47-s + ⋯ |
L(s) = 1 | + (0.752 − 0.434i)5-s + (0.988 + 0.570i)7-s + (0.664 − 1.15i)11-s + (−0.737 − 1.27i)13-s − 1.00i·17-s + 1.06i·19-s + (0.272 + 0.471i)23-s + (−0.122 + 0.212i)25-s + (−0.210 − 0.121i)29-s + (0.0116 − 0.00672i)31-s + 0.991·35-s + 1.38·37-s + (−1.27 + 0.735i)41-s + (0.772 + 0.446i)43-s + (0.350 − 0.607i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.87939 - 0.606822i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.87939 - 0.606822i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.68 + 0.971i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.61 - 1.50i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.20 + 3.81i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.65 + 4.60i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.16iT - 17T^{2} \) |
| 19 | \( 1 - 4.66iT - 19T^{2} \) |
| 23 | \( 1 + (-1.30 - 2.26i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.13 + 0.655i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.0648 + 0.0374i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8.43T + 37T^{2} \) |
| 41 | \( 1 + (8.16 - 4.71i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.06 - 2.92i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.40 + 4.16i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 8.96iT - 53T^{2} \) |
| 59 | \( 1 + (1.74 + 3.02i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.32 + 10.9i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.38 - 1.37i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.910T + 71T^{2} \) |
| 73 | \( 1 - 6.86T + 73T^{2} \) |
| 79 | \( 1 + (-10.8 - 6.28i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.61 - 14.9i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 8.96iT - 89T^{2} \) |
| 97 | \( 1 + (9.03 - 15.6i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.866198865053123429820136570369, −9.349362111114605921503067236077, −8.295207984131670539640515464565, −7.82958861428003758122859515173, −6.46128407800004607750679253344, −5.41555594821738231213938914831, −5.16290369898685910731782031936, −3.59618034568178791556112435362, −2.38394952242419834417358769055, −1.09161177236160635990807837007,
1.58791220282600512945658551011, 2.42423673896309010824122390634, 4.23180340865107049787717246138, 4.65364004537000205767857182199, 6.00902512917119041436023875705, 6.93709449622409934590469779327, 7.44940284725482051041616629806, 8.712125147118362797149289604050, 9.462177637057513965903045455308, 10.24339379491342800182182624259