L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s − 0.999·6-s + (−2.59 + 0.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.866 + 0.499i)12-s + 5i·13-s + (−2 + 1.73i)14-s + (−0.5 − 0.866i)16-s + (−5.19 − 3i)17-s + (0.866 + 0.499i)18-s + (−3.5 − 6.06i)19-s + (2.5 + 0.866i)21-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s − 0.408·6-s + (−0.981 + 0.188i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.249 + 0.144i)12-s + 1.38i·13-s + (−0.534 + 0.462i)14-s + (−0.125 − 0.216i)16-s + (−1.26 − 0.727i)17-s + (0.204 + 0.117i)18-s + (−0.802 − 1.39i)19-s + (0.545 + 0.188i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1167428751\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1167428751\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.59 - 0.5i)T \) |
good | 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 + (5.19 + 3i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + (5.19 - 3i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.19 - 3i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (11.2 + 6.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (4.33 + 2.5i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.5 + 6.06i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 18iT - 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−10.38963666283643177149840789905, −9.341971002257943217135281529011, −8.906749707367554599323910224848, −7.32122843624675234187254006531, −6.63606439730410024265162017165, −6.11229106859604829865623657888, −4.87496310485378857507731095309, −4.18614271572534912084264657829, −2.89362032017199742150163091233, −1.86335582811478623288106145197,
0.04152500858884396692286968463, 2.27533344562555721772520295581, 3.65504791490066104670725455104, 4.17858275485684785170424899570, 5.51813395073120131674058364817, 6.08544171431597726056541653554, 6.78513075267249828473255134447, 7.906804340161256200066437413825, 8.631960660828422160330465007529, 9.881877512546522592373298904386