| L(s) = 1 | + (0.781 + 0.623i)2-s + (−0.861 − 1.50i)3-s + (0.222 + 0.974i)4-s + (0.100 + 1.33i)5-s + (0.263 − 1.71i)6-s + (0.473 − 2.60i)7-s + (−0.433 + 0.900i)8-s + (−1.51 + 2.58i)9-s + (−0.754 + 1.10i)10-s + (−4.31 + 1.69i)11-s + (1.27 − 1.17i)12-s + (−5.95 + 2.33i)13-s + (1.99 − 1.73i)14-s + (1.92 − 1.30i)15-s + (−0.900 + 0.433i)16-s + (−3.36 + 1.03i)17-s + ⋯ |
| L(s) = 1 | + (0.552 + 0.440i)2-s + (−0.497 − 0.867i)3-s + (0.111 + 0.487i)4-s + (0.0447 + 0.597i)5-s + (0.107 − 0.698i)6-s + (0.178 − 0.983i)7-s + (−0.153 + 0.318i)8-s + (−0.505 + 0.862i)9-s + (−0.238 + 0.349i)10-s + (−1.30 + 0.510i)11-s + (0.367 − 0.338i)12-s + (−1.65 + 0.647i)13-s + (0.532 − 0.465i)14-s + (0.495 − 0.335i)15-s + (−0.225 + 0.108i)16-s + (−0.815 + 0.251i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.122619 + 0.512794i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.122619 + 0.512794i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.781 - 0.623i)T \) |
| 3 | \( 1 + (0.861 + 1.50i)T \) |
| 7 | \( 1 + (-0.473 + 2.60i)T \) |
| good | 5 | \( 1 + (-0.100 - 1.33i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (4.31 - 1.69i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (5.95 - 2.33i)T + (9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (3.36 - 1.03i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-5.72 - 3.30i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.86 + 4.16i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (1.46 + 4.73i)T + (-23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 - 10.8iT - 31T^{2} \) |
| 37 | \( 1 + (0.694 + 0.644i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (9.25 - 6.30i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (2.10 + 1.43i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (3.31 - 4.15i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (-0.277 - 0.299i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (-6.78 + 3.26i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-4.38 - 1.00i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 - 2.61T + 67T^{2} \) |
| 71 | \( 1 + (11.7 - 2.68i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (12.0 + 4.72i)T + (53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 - 6.42T + 79T^{2} \) |
| 83 | \( 1 + (-2.66 + 6.78i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-0.268 + 0.0404i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (-9.14 + 5.27i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.42516905021407630899660762712, −10.04412283157424363436098939650, −8.369984225170066258129603341154, −7.53675892084827039042461405748, −7.10497863383759879082865377584, −6.40421035319564571039001871343, −5.13473565776581417902482004101, −4.61621077184976336938593556576, −3.02621846664411342433268798848, −1.97185485433243658685754153830,
0.20328901029959076925108224034, 2.38777815382943716679618833750, 3.24544017064111196956233419548, 4.70347342443073919946546366565, 5.30861158868591138870866980436, 5.60962267017796832438883855812, 7.13359457106139993291418768631, 8.276504269459916962252109960296, 9.256941658278411375657510331517, 9.826074335761598091470604971585
