| L(s) = 1 | + (−1.44 − 0.941i)2-s + (−1.08 − 1.34i)3-s + (0.401 + 0.901i)4-s + (−0.477 + 2.18i)5-s + (0.305 + 2.97i)6-s + (0.903 + 0.242i)7-s + (−0.273 + 1.72i)8-s + (−0.638 + 2.93i)9-s + (2.74 − 2.71i)10-s + (0.696 + 3.27i)11-s + (0.780 − 1.52i)12-s + (−2.20 − 3.38i)13-s + (−1.08 − 1.20i)14-s + (3.46 − 1.72i)15-s + (3.34 − 3.71i)16-s + (3.50 + 0.555i)17-s + ⋯ |
| L(s) = 1 | + (−1.02 − 0.665i)2-s + (−0.627 − 0.778i)3-s + (0.200 + 0.450i)4-s + (−0.213 + 0.976i)5-s + (0.124 + 1.21i)6-s + (0.341 + 0.0914i)7-s + (−0.0968 + 0.611i)8-s + (−0.212 + 0.977i)9-s + (0.868 − 0.859i)10-s + (0.210 + 0.988i)11-s + (0.225 − 0.439i)12-s + (−0.610 − 0.940i)13-s + (−0.288 − 0.320i)14-s + (0.894 − 0.446i)15-s + (0.836 − 0.928i)16-s + (0.851 + 0.134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.481769 + 0.0483432i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.481769 + 0.0483432i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.08 + 1.34i)T \) |
| 5 | \( 1 + (0.477 - 2.18i)T \) |
| good | 2 | \( 1 + (1.44 + 0.941i)T + (0.813 + 1.82i)T^{2} \) |
| 7 | \( 1 + (-0.903 - 0.242i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.696 - 3.27i)T + (-10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (2.20 + 3.38i)T + (-5.28 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-3.50 - 0.555i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-3.91 - 5.38i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (2.59 + 0.135i)T + (22.8 + 2.40i)T^{2} \) |
| 29 | \( 1 + (-0.144 + 1.37i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.762 - 7.25i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-10.5 - 5.38i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (0.174 - 0.821i)T + (-37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (1.78 - 6.64i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (7.82 - 6.33i)T + (9.77 - 45.9i)T^{2} \) |
| 53 | \( 1 + (5.44 - 0.862i)T + (50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-2.49 - 0.530i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-7.51 + 1.59i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-3.03 - 2.46i)T + (13.9 + 65.5i)T^{2} \) |
| 71 | \( 1 + (-3.06 + 4.22i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (7.59 - 3.87i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (-0.149 - 0.0157i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-3.11 + 1.19i)T + (61.6 - 55.5i)T^{2} \) |
| 89 | \( 1 + (-2.69 + 8.29i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.64 - 6.96i)T + (-20.1 + 94.8i)T^{2} \) |
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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
−11.90814737314056486175867893309, −11.35529787846059630800222366387, −10.15297095475045452489240185798, −9.919938387758760567200957643124, −7.932490178399954196312390729133, −7.74849442477542967888769837161, −6.30342640633174055624768301296, −5.10464902883751221738348502281, −2.90223115672325873150099599675, −1.50919194704420114963797191283,
0.66209958745329805070834440875, 3.79701988940897300669287455419, 4.97577428585501558174549753730, 6.10317158556103708522400580388, 7.39210077438058050621584024668, 8.419880872381531774643432175255, 9.330491779790251388577697842746, 9.794587986386710502689496296948, 11.28475309125468515761648176877, 11.85659194504162197070693732239
