| L(s) = 1 | + (−1.46 + 0.924i)3-s + (−0.217 − 1.23i)5-s + (−2.00 − 1.68i)7-s + (1.29 − 2.70i)9-s + (0.0755 − 0.428i)11-s + (0.140 + 0.0512i)13-s + (1.45 + 1.60i)15-s + (2.14 + 3.72i)17-s + (−3.46 + 5.99i)19-s + (4.49 + 0.612i)21-s + (−1.58 + 1.32i)23-s + (3.22 − 1.17i)25-s + (0.610 + 5.16i)27-s + (−8.62 + 3.13i)29-s + (−0.439 + 0.368i)31-s + ⋯ |
| L(s) = 1 | + (−0.845 + 0.533i)3-s + (−0.0972 − 0.551i)5-s + (−0.758 − 0.636i)7-s + (0.430 − 0.902i)9-s + (0.0227 − 0.129i)11-s + (0.0390 + 0.0142i)13-s + (0.376 + 0.414i)15-s + (0.521 + 0.902i)17-s + (−0.793 + 1.37i)19-s + (0.981 + 0.133i)21-s + (−0.329 + 0.276i)23-s + (0.645 − 0.234i)25-s + (0.117 + 0.993i)27-s + (−1.60 + 0.582i)29-s + (−0.0789 + 0.0662i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.151334 + 0.351532i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.151334 + 0.351532i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.46 - 0.924i)T \) |
| good | 5 | \( 1 + (0.217 + 1.23i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (2.00 + 1.68i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.0755 + 0.428i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.140 - 0.0512i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.14 - 3.72i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.46 - 5.99i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.58 - 1.32i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (8.62 - 3.13i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (0.439 - 0.368i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-1.47 - 2.55i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.70 + 1.71i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.773 - 4.38i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (5.17 + 4.34i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 7.31T + 53T^{2} \) |
| 59 | \( 1 + (-1.82 - 10.3i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-6.60 - 5.54i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (3.27 + 1.19i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.25 - 2.18i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.40 - 9.36i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.9 + 5.09i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (3.90 - 1.42i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (2.90 - 5.03i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.44 - 13.8i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.33347113598754120608078047857, −9.915528673209750105115105472155, −8.910841260943077076569102580792, −7.979741054744779239177446909592, −6.84624475284337012861077065909, −6.06115985175389839858372876659, −5.26200762859948197338912794648, −4.08574300948762965813810597194, −3.52434691761988371492119570059, −1.39842273645938366693564300919,
0.21234371484566165405966620618, 2.12807231995720635874198730218, 3.19256098099668116149572844923, 4.66240042219846884638001066506, 5.59033436197821782479409450243, 6.49075902160804839368985804376, 7.03535771956925151299304814060, 7.969464070579398488762471234724, 9.169709236221367020453479048768, 9.859272458999046306928079795463
