| L(s) = 1 | + (−0.942 − 1.04i)2-s + (1.97 + 0.879i)3-s + (0.00145 − 0.0138i)4-s + (−1.79 − 0.380i)5-s + (−0.942 − 2.89i)6-s + (−2.29 + 1.66i)8-s + (1.12 + 1.24i)9-s + (1.29 + 2.23i)10-s + (3.23 + 0.743i)11-s + (0.0151 − 0.0261i)12-s + (1.43 − 4.41i)13-s + (−3.20 − 2.32i)15-s + (3.88 + 0.825i)16-s + (3.66 − 4.06i)17-s + (0.247 − 2.35i)18-s + (−0.606 − 5.76i)19-s + ⋯ |
| L(s) = 1 | + (−0.666 − 0.740i)2-s + (1.14 + 0.508i)3-s + (0.000729 − 0.00694i)4-s + (−0.801 − 0.170i)5-s + (−0.384 − 1.18i)6-s + (−0.811 + 0.589i)8-s + (0.374 + 0.416i)9-s + (0.408 + 0.706i)10-s + (0.974 + 0.224i)11-s + (0.00436 − 0.00755i)12-s + (0.398 − 1.22i)13-s + (−0.827 − 0.601i)15-s + (0.971 + 0.206i)16-s + (0.887 − 0.985i)17-s + (0.0583 − 0.555i)18-s + (−0.139 − 1.32i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.824643 - 0.945476i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.824643 - 0.945476i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (-3.23 - 0.743i)T \) |
| good | 2 | \( 1 + (0.942 + 1.04i)T + (-0.209 + 1.98i)T^{2} \) |
| 3 | \( 1 + (-1.97 - 0.879i)T + (2.00 + 2.22i)T^{2} \) |
| 5 | \( 1 + (1.79 + 0.380i)T + (4.56 + 2.03i)T^{2} \) |
| 13 | \( 1 + (-1.43 + 4.41i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-3.66 + 4.06i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.606 + 5.76i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.359 + 0.623i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.948 - 0.689i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.27 - 0.271i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-1.91 + 0.852i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (0.741 - 0.538i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 8.02T + 43T^{2} \) |
| 47 | \( 1 + (-0.624 - 5.94i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (9.92 - 2.10i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (0.802 - 7.63i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (6.13 + 1.30i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-7.73 - 13.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.29 - 13.2i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.628 + 5.98i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (10.4 + 11.6i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-1.35 - 4.16i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-7.67 + 13.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.745 - 2.29i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.42942816813735365498440305437, −9.533350698815747272400721379391, −9.039559744568058494971604159722, −8.254563396677727810387332665372, −7.40978530639011845815352110790, −5.91440642026420745570246849689, −4.55369047201265801880561208501, −3.39470695115424515636911976777, −2.64709734869042595217210309157, −0.860496502430960681613746551962,
1.68125430774712029634641431898, 3.44861113314494543288664557360, 3.90244564156333126796739899313, 6.03710514271282518142554835086, 6.83498029348512833942985581479, 7.83664522194722261734589540602, 8.120725442715780892067716734369, 9.005538899338887668631623300996, 9.662473036539788212280081976535, 11.08786787746229212988848440581
