| L(s) = 1 | + (−1.39 − 0.240i)2-s + (−0.116 − 1.18i)3-s + (1.88 + 0.669i)4-s + (0.0176 − 0.0329i)5-s + (−0.122 + 1.68i)6-s + (−0.0229 − 2.64i)7-s + (−2.46 − 1.38i)8-s + (1.54 − 0.307i)9-s + (−0.0324 + 0.0417i)10-s + (4.05 − 3.33i)11-s + (0.574 − 2.31i)12-s + (2.08 − 1.11i)13-s + (−0.603 + 3.69i)14-s + (−0.0411 − 0.0170i)15-s + (3.10 + 2.52i)16-s + (−3.49 + 1.44i)17-s + ⋯ |
| L(s) = 1 | + (−0.985 − 0.169i)2-s + (−0.0674 − 0.685i)3-s + (0.942 + 0.334i)4-s + (0.00788 − 0.0147i)5-s + (−0.0498 + 0.686i)6-s + (−0.00868 − 0.999i)7-s + (−0.871 − 0.489i)8-s + (0.515 − 0.102i)9-s + (−0.0102 + 0.0131i)10-s + (1.22 − 1.00i)11-s + (0.165 − 0.668i)12-s + (0.578 − 0.309i)13-s + (−0.161 + 0.986i)14-s + (−0.0106 − 0.00440i)15-s + (0.775 + 0.630i)16-s + (−0.848 + 0.351i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.625789 - 0.895797i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.625789 - 0.895797i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.240i)T \) |
| 7 | \( 1 + (0.0229 + 2.64i)T \) |
| good | 3 | \( 1 + (0.116 + 1.18i)T + (-2.94 + 0.585i)T^{2} \) |
| 5 | \( 1 + (-0.0176 + 0.0329i)T + (-2.77 - 4.15i)T^{2} \) |
| 11 | \( 1 + (-4.05 + 3.33i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.08 + 1.11i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (3.49 - 1.44i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.867 - 2.85i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-1.85 + 2.77i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-2.45 - 2.01i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (-2.17 - 2.17i)T + 31iT^{2} \) |
| 37 | \( 1 + (-5.03 + 1.52i)T + (30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (8.84 + 5.90i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (5.19 + 0.512i)T + (42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (-9.12 + 3.77i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (3.20 - 2.62i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (-2.58 + 4.83i)T + (-32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.424 - 4.30i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-0.114 - 1.15i)T + (-65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (2.15 - 10.8i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (1.46 - 0.291i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-4.31 + 10.4i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (8.87 + 2.69i)T + (69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (8.46 + 12.6i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-6.93 + 6.93i)T - 97iT^{2} \) |
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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
−9.941442256769976274128721291490, −8.790833407518690457407889903407, −8.382811948282832365246891810775, −7.19305981305631988294364090586, −6.75837295581999486930290237462, −5.98458716790991159962528977528, −4.17047251974492143791025742332, −3.29379080360309521671555999073, −1.62764923328668487517587442849, −0.821176497659721207408556113314,
1.51680209488320470628631196235, 2.68456567671659154141209372586, 4.19698511373825290207316112274, 5.08361141414717812165658439571, 6.45706916243413583573409003328, 6.79761616099942426794270066252, 8.066956016843355396247197208639, 9.006594168301707626718482173563, 9.402265400150231133800372475366, 10.09630873315810099250462381919
