| L(s) = 1 | + (0.669 + 0.743i)2-s + (−1.22 − 1.22i)3-s + (−0.104 + 0.994i)4-s + (−3.11 − 2.80i)5-s + (0.0906 − 1.72i)6-s + (−0.209 − 0.470i)7-s + (−0.809 + 0.587i)8-s + (−3.18e−5 + 2.99i)9-s − 4.19i·10-s + (−2.04 − 2.61i)11-s + (1.34 − 1.09i)12-s + (−1.03 − 4.87i)13-s + (0.209 − 0.470i)14-s + (0.380 + 7.25i)15-s + (−0.978 − 0.207i)16-s + (−0.180 − 0.554i)17-s + ⋯ |
| L(s) = 1 | + (0.473 + 0.525i)2-s + (−0.707 − 0.707i)3-s + (−0.0522 + 0.497i)4-s + (−1.39 − 1.25i)5-s + (0.0370 − 0.706i)6-s + (−0.0791 − 0.177i)7-s + (−0.286 + 0.207i)8-s + (−1.06e−5 + 0.999i)9-s − 1.32i·10-s + (−0.616 − 0.787i)11-s + (0.388 − 0.314i)12-s + (−0.287 − 1.35i)13-s + (0.0559 − 0.125i)14-s + (0.0981 + 1.87i)15-s + (−0.244 − 0.0519i)16-s + (−0.0436 − 0.134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.375595 - 0.548307i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.375595 - 0.548307i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.669 - 0.743i)T \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 11 | \( 1 + (2.04 + 2.61i)T \) |
| good | 5 | \( 1 + (3.11 + 2.80i)T + (0.522 + 4.97i)T^{2} \) |
| 7 | \( 1 + (0.209 + 0.470i)T + (-4.68 + 5.20i)T^{2} \) |
| 13 | \( 1 + (1.03 + 4.87i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (0.180 + 0.554i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-3.00 - 4.13i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-4.73 + 2.73i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.05 - 0.915i)T + (19.4 - 21.5i)T^{2} \) |
| 31 | \( 1 + (-0.878 + 0.186i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (2.90 + 2.11i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (9.65 + 4.29i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.868 + 0.501i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-10.2 + 1.07i)T + (45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (3.65 + 1.18i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (2.41 + 0.253i)T + (57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-2.21 + 10.4i)T + (-55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-2.79 - 4.84i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.42 - 0.461i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.11 + 1.53i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.57 + 2.31i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-4.64 - 0.987i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + 3.64iT - 89T^{2} \) |
| 97 | \( 1 + (4.79 + 5.32i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−12.41483405354993991262968390025, −11.57693400649551925560271466229, −10.54005941252890446898480837327, −8.628029733987863623207192265424, −7.935994310509316905073645176191, −7.20502038215002066706227428758, −5.57103998843298551138964278493, −4.98397461430679274489693362290, −3.46672091869524573541812218990, −0.54507672397957127502088693564,
2.90641780877432408565082481803, 4.06136501213633489266469733534, 4.99403871983312436524730834010, 6.63162975885608934048641755879, 7.35777373849889761011377122053, 9.143904647486785198795537241245, 10.21234921673142399734619231549, 11.07766824559038793152394936143, 11.65305044468169832586565051194, 12.30559704399487174317496474881
