L(s) = 1 | + (3.62 + 3.44i)5-s + (5.87 + 3.39i)7-s + (8.57 + 4.94i)11-s + (−17.6 + 10.1i)13-s − 19.1·17-s + 12·19-s + (4.79 + 8.30i)23-s + (1.24 + 24.9i)25-s + (−7.34 − 4.24i)29-s + (19 + 32.9i)31-s + (9.59 + 32.5i)35-s − 6.78i·37-s + (−60.0 + 34.6i)41-s + (−58.7 − 33.9i)43-s + (−38.3 + 66.4i)47-s + ⋯ |
L(s) = 1 | + (0.724 + 0.689i)5-s + (0.839 + 0.484i)7-s + (0.779 + 0.449i)11-s + (−1.35 + 0.782i)13-s − 1.12·17-s + 0.631·19-s + (0.208 + 0.361i)23-s + (0.0498 + 0.998i)25-s + (−0.253 − 0.146i)29-s + (0.612 + 1.06i)31-s + (0.274 + 0.929i)35-s − 0.183i·37-s + (−1.46 + 0.845i)41-s + (−1.36 − 0.788i)43-s + (−0.816 + 1.41i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.932979926\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.932979926\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-3.62 - 3.44i)T \) |
good | 7 | \( 1 + (-5.87 - 3.39i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.57 - 4.94i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (17.6 - 10.1i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 19.1T + 289T^{2} \) |
| 19 | \( 1 - 12T + 361T^{2} \) |
| 23 | \( 1 + (-4.79 - 8.30i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (7.34 + 4.24i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-19 - 32.9i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 6.78iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (60.0 - 34.6i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (58.7 + 33.9i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (38.3 - 66.4i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 + (-72.2 + 41.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-35 + 60.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-93.9 + 54.2i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 118. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 13.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (15 - 25.9i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (67.1 - 116. i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 32.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-82.2 - 47.4i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.572716355895384901638510742127, −8.826991672070667169021254159302, −7.86765918572395205470506148031, −6.77915271189679497863724320185, −6.61572242927532604587133787692, −5.15875566930969553695277781355, −4.79463370207321211653351825437, −3.43584120695676983686940701872, −2.25083008138266189984366220974, −1.65905580590479644345216927533,
0.48683700732759647862798945681, 1.60989358505051742037958348957, 2.63854890995332164458987902074, 4.01734229861403811879319084846, 4.89815206241994338836946246987, 5.43694041597542558896548099162, 6.55977537931332354370886616134, 7.30294277597402127371010993392, 8.352286345764549994246514734914, 8.766889458580715893500065882200