L(s) = 1 | + (1.16 − 0.806i)2-s + (0.699 − 1.87i)4-s + (−1.86 − 1.23i)5-s + 0.746i·7-s + (−0.699 − 2.74i)8-s + (−3.16 + 0.0603i)10-s − 5.36i·11-s + 2.92·13-s + (0.601 + 0.866i)14-s + (−3.02 − 2.62i)16-s − 2.13i·17-s − 1.73i·19-s + (−3.62 + 2.62i)20-s + (−4.32 − 6.22i)22-s + 7.49i·23-s + ⋯ |
L(s) = 1 | + (0.821 − 0.570i)2-s + (0.349 − 0.936i)4-s + (−0.832 − 0.554i)5-s + 0.282i·7-s + (−0.247 − 0.968i)8-s + (−0.999 + 0.0190i)10-s − 1.61i·11-s + 0.811·13-s + (0.160 + 0.231i)14-s + (−0.755 − 0.655i)16-s − 0.517i·17-s − 0.397i·19-s + (−0.810 + 0.585i)20-s + (−0.921 − 1.32i)22-s + 1.56i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03710 - 1.46386i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03710 - 1.46386i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.16 + 0.806i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.86 + 1.23i)T \) |
good | 7 | \( 1 - 0.746iT - 7T^{2} \) |
| 11 | \( 1 + 5.36iT - 11T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 17 | \( 1 + 2.13iT - 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 - 7.49iT - 23T^{2} \) |
| 29 | \( 1 - 6.74iT - 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 - 1.07T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 7.44T + 43T^{2} \) |
| 47 | \( 1 + 1.73iT - 47T^{2} \) |
| 53 | \( 1 - 7.72T + 53T^{2} \) |
| 59 | \( 1 - 6.85iT - 59T^{2} \) |
| 61 | \( 1 + 6.45iT - 61T^{2} \) |
| 67 | \( 1 - 7.44T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 0.690iT - 73T^{2} \) |
| 79 | \( 1 + 2.64T + 79T^{2} \) |
| 83 | \( 1 + 5.85T + 83T^{2} \) |
| 89 | \( 1 - 7.59T + 89T^{2} \) |
| 97 | \( 1 + 14.1iT - 97T^{2} \) |
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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
−11.41518848774858499655751360745, −10.67211042732258847685743761883, −9.256888749454931679954384454366, −8.548698550449817268483562923857, −7.27915553030978764542257743348, −5.96811472810705703859222832841, −5.18046351619967547590411239353, −3.88992291072403147187779700442, −3.06204124173003800230703701499, −1.03609889396730616685065189693,
2.48315439361145514890695933474, 3.96076908310710617481020453079, 4.50582818968868191961432733130, 6.07489428590315759746169442752, 6.91739668158706076700008634402, 7.73744223028729386820693662336, 8.551113935590183250578355346843, 10.07919996377577255026606062127, 10.97309474461072593005972070460, 11.96288614997974940357715255692