| L(s) = 1 | + (−1.41 − 0.0157i)2-s + (1.41 + 1.00i)3-s + (1.99 + 0.0446i)4-s + (−3.04 − 1.62i)5-s + (−1.97 − 1.44i)6-s + (−1.52 − 0.303i)7-s + (−2.82 − 0.0947i)8-s + (0.981 + 2.83i)9-s + (4.27 + 2.34i)10-s + (2.67 − 2.19i)11-s + (2.77 + 2.07i)12-s + (4.59 − 2.45i)13-s + (2.15 + 0.453i)14-s + (−2.66 − 5.35i)15-s + (3.99 + 0.178i)16-s + (6.42 − 2.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0111i)2-s + (0.814 + 0.579i)3-s + (0.999 + 0.0223i)4-s + (−1.36 − 0.727i)5-s + (−0.808 − 0.589i)6-s + (−0.576 − 0.114i)7-s + (−0.999 − 0.0334i)8-s + (0.327 + 0.944i)9-s + (1.35 + 0.742i)10-s + (0.805 − 0.660i)11-s + (0.801 + 0.598i)12-s + (1.27 − 0.680i)13-s + (0.575 + 0.121i)14-s + (−0.687 − 1.38i)15-s + (0.999 + 0.0446i)16-s + (1.55 − 0.645i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.921668 - 0.211639i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.921668 - 0.211639i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0157i)T \) |
| 3 | \( 1 + (-1.41 - 1.00i)T \) |
| good | 5 | \( 1 + (3.04 + 1.62i)T + (2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (1.52 + 0.303i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-2.67 + 2.19i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-4.59 + 2.45i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-6.42 + 2.66i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.298 + 0.983i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-0.163 - 0.109i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-2.73 - 2.24i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (-1.22 + 1.22i)T - 31iT^{2} \) |
| 37 | \( 1 + (1.88 + 6.22i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (4.53 + 3.03i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.466 + 4.73i)T + (-42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (10.2 - 4.23i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-4.04 + 3.31i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (-9.79 - 5.23i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.457 - 4.64i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-0.872 + 0.0859i)T + (65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (-9.11 - 1.81i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (1.24 + 6.26i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-0.950 + 2.29i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (3.96 - 13.0i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-5.57 - 8.34i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (4.54 + 4.54i)T + 97iT^{2} \) |
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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
−11.14236269617451655026130779033, −10.18613019634359391818313694587, −9.248330044469112915833290362715, −8.475606618020993578698504219513, −8.005150532019240578238627793715, −6.95189068718946281314074300828, −5.45877734284348426864193072245, −3.75316481384510671015981076616, −3.25177079753918389896517802147, −0.940504557444821984377841005924,
1.42280966249284522766712416815, 3.17034863517551592157520833567, 3.79055056595024636152580721074, 6.41256002166655948366375637056, 6.81226242743824271366456505329, 7.937759313209020016055424050808, 8.361734166562395743234362537635, 9.500046883360006501103294328036, 10.28031720027289858296857033423, 11.60244043739433702914206268754
