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
−8.766889458580715893500065882200, −8.352286345764549994246514734914, −7.30294277597402127371010993392, −6.55977537931332354370886616134, −5.43694041597542558896548099162, −4.89815206241994338836946246987, −4.01734229861403811879319084846, −2.63854890995332164458987902074, −1.60989358505051742037958348957, −0.48683700732759647862798945681,
1.65905580590479644345216927533, 2.25083008138266189984366220974, 3.43584120695676983686940701872, 4.79463370207321211653351825437, 5.15875566930969553695277781355, 6.61572242927532604587133787692, 6.77915271189679497863724320185, 7.86765918572395205470506148031, 8.826991672070667169021254159302, 9.572716355895384901638510742127