L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−3.19 − 1.84i)5-s + (−1.52 − 2.15i)7-s + 0.999·8-s + (3.19 − 1.84i)10-s + (0.412 + 3.29i)11-s − 5.60i·13-s + (2.63 − 0.245i)14-s + (−0.5 + 0.866i)16-s + (−1.39 − 2.41i)17-s + (−2.14 − 1.23i)19-s + 3.69i·20-s + (−3.05 − 1.28i)22-s + (3.26 + 1.88i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−1.42 − 0.825i)5-s + (−0.578 − 0.815i)7-s + 0.353·8-s + (1.01 − 0.583i)10-s + (0.124 + 0.992i)11-s − 1.55i·13-s + (0.704 − 0.0656i)14-s + (−0.125 + 0.216i)16-s + (−0.338 − 0.585i)17-s + (−0.492 − 0.284i)19-s + 0.825i·20-s + (−0.651 − 0.274i)22-s + (0.680 + 0.392i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.08157599825\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08157599825\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.52 + 2.15i)T \) |
| 11 | \( 1 + (-0.412 - 3.29i)T \) |
good | 5 | \( 1 + (3.19 + 1.84i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 5.60iT - 13T^{2} \) |
| 17 | \( 1 + (1.39 + 2.41i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.14 + 1.23i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.26 - 1.88i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 1.51T + 29T^{2} \) |
| 31 | \( 1 + (1.70 + 2.94i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.85 + 4.95i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 - 7.72iT - 43T^{2} \) |
| 47 | \( 1 + (-8.98 - 5.18i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (9.27 - 5.35i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (11.6 - 6.71i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.13 + 1.23i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.66 - 9.81i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 5.15iT - 71T^{2} \) |
| 73 | \( 1 + (4.53 - 2.62i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.250 + 0.144i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + (-13.7 - 7.93i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 4.58T + 97T^{2} \) |
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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
−9.660432258912941220521732600590, −9.039834159974058828480512170234, −8.046538191958766459081008666538, −7.52721433077990459654470816531, −7.00462174391409477390485605832, −5.77851822523925265770577356282, −4.70976498246521428088114175122, −4.17759917777756416132369127584, −3.04218645515034403392386233281, −1.00238124942084706913503764984,
0.04679359025663189343132581812, 2.01276711321343140177604759677, 3.21129517958780044378559664040, 3.71785085162487272154433625651, 4.74761897911049648295298140919, 6.26739930746099925562799570081, 6.78997254981652311370380163180, 7.79031292669865340305497035032, 8.715592750495988044007651129167, 8.999984907675847781383349339041