| L(s) = 1 | + (−1.81 − 0.846i)2-s + (1.29 + 1.54i)4-s + (−0.828 + 2.07i)5-s + (−2.21 + 0.193i)7-s + (−0.00834 − 0.0311i)8-s + (3.26 − 3.07i)10-s + (−1.09 − 0.193i)11-s + (−1.28 − 2.74i)13-s + (4.18 + 1.52i)14-s + (0.688 − 3.90i)16-s + (1.56 − 5.82i)17-s + (3.84 + 2.22i)19-s + (−4.28 + 1.41i)20-s + (1.82 + 1.27i)22-s + (0.293 − 3.34i)23-s + ⋯ |
| L(s) = 1 | + (−1.28 − 0.598i)2-s + (0.647 + 0.772i)4-s + (−0.370 + 0.928i)5-s + (−0.836 + 0.0731i)7-s + (−0.00295 − 0.0110i)8-s + (1.03 − 0.970i)10-s + (−0.330 − 0.0582i)11-s + (−0.355 − 0.762i)13-s + (1.11 + 0.406i)14-s + (0.172 − 0.976i)16-s + (0.378 − 1.41i)17-s + (0.882 + 0.509i)19-s + (−0.957 + 0.315i)20-s + (0.389 + 0.272i)22-s + (0.0611 − 0.698i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.194059 - 0.317253i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.194059 - 0.317253i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.828 - 2.07i)T \) |
| good | 2 | \( 1 + (1.81 + 0.846i)T + (1.28 + 1.53i)T^{2} \) |
| 7 | \( 1 + (2.21 - 0.193i)T + (6.89 - 1.21i)T^{2} \) |
| 11 | \( 1 + (1.09 + 0.193i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (1.28 + 2.74i)T + (-8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (-1.56 + 5.82i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.84 - 2.22i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.293 + 3.34i)T + (-22.6 - 3.99i)T^{2} \) |
| 29 | \( 1 + (-1.74 + 0.635i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-7.13 + 5.98i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (4.53 + 1.21i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.07 + 2.95i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (5.86 - 4.10i)T + (14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (1.10 + 12.5i)T + (-46.2 + 8.16i)T^{2} \) |
| 53 | \( 1 + (6.34 - 6.34i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.242 + 1.37i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.30 + 1.93i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-8.07 + 3.76i)T + (43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (-6.78 + 3.91i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.167 - 0.0447i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (0.697 + 1.91i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.27 + 4.86i)T + (-53.3 - 63.5i)T^{2} \) |
| 89 | \( 1 + (6.70 - 11.6i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.92 - 14.1i)T + (-33.1 + 91.1i)T^{2} \) |
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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
−10.68751274391087494016491533961, −9.980557658215746819405851990353, −9.509932838728921551586281643401, −8.186506688041365794225153994511, −7.55186014950364070518492382807, −6.55181429103701985858757440501, −5.17190755729166266335293546875, −3.28456689484082263029861544490, −2.55467588890636392636564420361, −0.40172112780369220179626686640,
1.32530854238570347964182688263, 3.50474034774669692640438689237, 4.85438190904118879911919019466, 6.19935079097084525279404389789, 7.10482951975202979041533185351, 8.048404393958680023978436356708, 8.734327077600818777214967035212, 9.618900048272906244443470573464, 10.14655957980689892588588910036, 11.40910765636473911576651699639
