| L(s) = 1 | + (−0.686 + 0.727i)2-s + (0.851 + 1.50i)3-s + (−0.0581 − 0.998i)4-s + (−3.77 − 0.441i)5-s + (−1.68 − 0.416i)6-s + (−3.13 + 1.57i)7-s + (0.766 + 0.642i)8-s + (−1.55 + 2.56i)9-s + (2.91 − 2.44i)10-s + (−1.41 + 3.27i)11-s + (1.45 − 0.937i)12-s + (6.00 − 1.42i)13-s + (1.00 − 3.36i)14-s + (−2.54 − 6.07i)15-s + (−0.993 + 0.116i)16-s + (−0.0237 − 0.134i)17-s + ⋯ |
| L(s) = 1 | + (−0.485 + 0.514i)2-s + (0.491 + 0.870i)3-s + (−0.0290 − 0.499i)4-s + (−1.68 − 0.197i)5-s + (−0.686 − 0.169i)6-s + (−1.18 + 0.594i)7-s + (0.270 + 0.227i)8-s + (−0.517 + 0.855i)9-s + (0.921 − 0.772i)10-s + (−0.426 + 0.988i)11-s + (0.420 − 0.270i)12-s + (1.66 − 0.394i)13-s + (0.268 − 0.898i)14-s + (−0.657 − 1.56i)15-s + (−0.248 + 0.0290i)16-s + (−0.00576 − 0.0326i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0621200 + 0.529137i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0621200 + 0.529137i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.686 - 0.727i)T \) |
| 3 | \( 1 + (-0.851 - 1.50i)T \) |
| good | 5 | \( 1 + (3.77 + 0.441i)T + (4.86 + 1.15i)T^{2} \) |
| 7 | \( 1 + (3.13 - 1.57i)T + (4.18 - 5.61i)T^{2} \) |
| 11 | \( 1 + (1.41 - 3.27i)T + (-7.54 - 8.00i)T^{2} \) |
| 13 | \( 1 + (-6.00 + 1.42i)T + (11.6 - 5.83i)T^{2} \) |
| 17 | \( 1 + (0.0237 + 0.134i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (0.591 - 3.35i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (-1.62 - 0.814i)T + (13.7 + 18.4i)T^{2} \) |
| 29 | \( 1 + (0.375 + 1.25i)T + (-24.2 + 15.9i)T^{2} \) |
| 31 | \( 1 + (-0.265 + 0.174i)T + (12.2 - 28.4i)T^{2} \) |
| 37 | \( 1 + (7.16 + 2.60i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (-4.36 - 4.62i)T + (-2.38 + 40.9i)T^{2} \) |
| 43 | \( 1 + (-1.92 - 2.58i)T + (-12.3 + 41.1i)T^{2} \) |
| 47 | \( 1 + (-3.99 - 2.62i)T + (18.6 + 43.1i)T^{2} \) |
| 53 | \( 1 + (2.78 + 4.82i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.85 - 4.30i)T + (-40.4 + 42.9i)T^{2} \) |
| 61 | \( 1 + (0.784 - 13.4i)T + (-60.5 - 7.08i)T^{2} \) |
| 67 | \( 1 + (2.79 - 9.34i)T + (-55.9 - 36.8i)T^{2} \) |
| 71 | \( 1 + (-3.05 + 2.56i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (10.7 + 8.98i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-2.36 + 2.51i)T + (-4.59 - 78.8i)T^{2} \) |
| 83 | \( 1 + (-0.597 + 0.632i)T + (-4.82 - 82.8i)T^{2} \) |
| 89 | \( 1 + (-6.42 - 5.39i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (6.91 - 0.807i)T + (94.3 - 22.3i)T^{2} \) |
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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
−13.31448910480592244181396372791, −12.30756543580145115023563002514, −11.11042234855182317991678623622, −10.16906402676387669028906827400, −9.043608644861718199113822919851, −8.316158398794571827838103039232, −7.36373656315250177875535739935, −5.83830300554885301257338903252, −4.30582219090560638162527019910, −3.24971551875167577500708680394,
0.56449495562195402653180643624, 3.18420969925162340316956103557, 3.77708929002976908582526354464, 6.44990115669816690365588327800, 7.30716955912208883066984841609, 8.352269624059604624343438013020, 8.994270053101859754810928290762, 10.72768513833571217921292909433, 11.33405322499696044408186688791, 12.39291923329711209043088412619
