| L(s) = 1 | + (−2.46 − 0.968i)2-s + (−0.844 + 1.51i)3-s + (3.68 + 3.41i)4-s + (0.558 + 0.380i)5-s + (3.54 − 2.91i)6-s + (−2.61 − 0.401i)7-s + (−3.47 − 7.21i)8-s + (−1.57 − 2.55i)9-s + (−1.00 − 1.47i)10-s + (0.397 + 2.63i)11-s + (−8.27 + 2.68i)12-s + (−1.84 + 1.47i)13-s + (6.06 + 3.52i)14-s + (−1.04 + 0.522i)15-s + (0.835 + 11.1i)16-s + (−6.22 − 1.92i)17-s + ⋯ |
| L(s) = 1 | + (−1.74 − 0.684i)2-s + (−0.487 + 0.873i)3-s + (1.84 + 1.70i)4-s + (0.249 + 0.170i)5-s + (1.44 − 1.18i)6-s + (−0.988 − 0.151i)7-s + (−1.22 − 2.55i)8-s + (−0.524 − 0.851i)9-s + (−0.318 − 0.467i)10-s + (0.119 + 0.795i)11-s + (−2.38 + 0.774i)12-s + (−0.512 + 0.408i)13-s + (1.62 + 0.941i)14-s + (−0.270 + 0.135i)15-s + (0.208 + 2.78i)16-s + (−1.50 − 0.465i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0365709 + 0.133193i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0365709 + 0.133193i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.844 - 1.51i)T \) |
| 7 | \( 1 + (2.61 + 0.401i)T \) |
| good | 2 | \( 1 + (2.46 + 0.968i)T + (1.46 + 1.36i)T^{2} \) |
| 5 | \( 1 + (-0.558 - 0.380i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (-0.397 - 2.63i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (1.84 - 1.47i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (6.22 + 1.92i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (3.81 - 2.20i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.902 - 2.92i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (4.05 - 0.924i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (2.82 + 1.63i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.670 - 0.622i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (0.0490 - 0.0236i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-9.68 - 4.66i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (2.86 - 7.30i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-7.93 + 8.55i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (4.53 - 3.09i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-5.32 - 5.73i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (5.29 - 9.16i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.60 - 1.05i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (2.97 - 1.16i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (5.34 + 9.25i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.527 - 0.661i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-9.48 - 1.42i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 - 5.34iT - 97T^{2} \) |
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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
−12.97832732826600702260009093670, −11.99252482319246159359609941242, −11.07452281723444575836651211255, −10.21775890256295162977137924158, −9.532901730673826076533220213845, −8.902843272840238836536625516133, −7.26609582264693223024886927461, −6.32260963968504191757681202218, −4.08578038587665811022329001437, −2.44486086791595189953018093942,
0.22871726208908582745171817837, 2.23264117556588483875891023010, 5.66665251867380403603909879990, 6.49169809192123541722950127322, 7.28333662395526765055551745322, 8.556775440197456258434760740038, 9.203336301377249381212417662353, 10.54633440793450679022936013874, 11.17623368256912309806671814487, 12.58048944434989558900677686361
