| L(s) = 1 | + (1.97 − 0.950i)2-s + (−0.880 − 1.49i)3-s + (1.74 − 2.18i)4-s + (−1.72 − 1.59i)5-s + (−3.15 − 2.10i)6-s + (0.702 − 2.55i)7-s + (0.386 − 1.69i)8-s + (−1.45 + 2.62i)9-s + (−4.91 − 1.51i)10-s + (1.75 + 1.19i)11-s + (−4.79 − 0.675i)12-s + (0.407 + 0.278i)13-s + (−1.03 − 5.69i)14-s + (−0.868 + 3.97i)15-s + (0.396 + 1.73i)16-s + (−1.06 + 0.160i)17-s + ⋯ |
| L(s) = 1 | + (1.39 − 0.671i)2-s + (−0.508 − 0.861i)3-s + (0.871 − 1.09i)4-s + (−0.770 − 0.715i)5-s + (−1.28 − 0.859i)6-s + (0.265 − 0.964i)7-s + (0.136 − 0.599i)8-s + (−0.483 + 0.875i)9-s + (−1.55 − 0.479i)10-s + (0.528 + 0.360i)11-s + (−1.38 − 0.195i)12-s + (0.113 + 0.0771i)13-s + (−0.277 − 1.52i)14-s + (−0.224 + 1.02i)15-s + (0.0990 + 0.434i)16-s + (−0.257 + 0.0388i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.540698 - 2.05628i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.540698 - 2.05628i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.880 + 1.49i)T \) |
| 7 | \( 1 + (-0.702 + 2.55i)T \) |
| good | 2 | \( 1 + (-1.97 + 0.950i)T + (1.24 - 1.56i)T^{2} \) |
| 5 | \( 1 + (1.72 + 1.59i)T + (0.373 + 4.98i)T^{2} \) |
| 11 | \( 1 + (-1.75 - 1.19i)T + (4.01 + 10.2i)T^{2} \) |
| 13 | \( 1 + (-0.407 - 0.278i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (1.06 - 0.160i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (1.55 + 2.69i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.66 + 4.23i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (-0.161 + 0.0242i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 - 6.34T + 31T^{2} \) |
| 37 | \( 1 + (-2.77 + 7.05i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-1.49 + 0.460i)T + (33.8 - 23.0i)T^{2} \) |
| 43 | \( 1 + (-9.57 - 2.95i)T + (35.5 + 24.2i)T^{2} \) |
| 47 | \( 1 + (-4.11 + 1.98i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (-3.75 - 9.57i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-1.01 - 4.44i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (0.814 + 1.02i)T + (-13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 + 9.89T + 67T^{2} \) |
| 71 | \( 1 + (5.48 - 6.88i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (3.61 - 2.46i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 - 6.83T + 79T^{2} \) |
| 83 | \( 1 + (10.0 - 6.88i)T + (30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (-1.07 - 14.3i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (-7.39 + 12.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.12962195368221513924229995330, −10.47293629944541689574374602707, −8.771231175679652087542805263314, −7.78141564348214379010721533504, −6.80080846200469769754290228775, −5.80704113191990972727002166516, −4.45120819484843818021873908841, −4.26271559799901346332801159311, −2.48640901093338324904017932426, −0.983239562013856939713789719078,
2.99894283354203569281888575388, 3.85931965505328408057929912900, 4.74366043244825895987859237868, 5.83140184792538626927344955028, 6.34037056834918376533450734550, 7.53283600330650493236128427031, 8.664088489137337160094216585128, 9.798095491055550514465501452122, 10.96898527370416054197048989035, 11.79426058274328166566665820747
