| L(s) = 1 | + (0.201 + 1.39i)2-s + (1.40 − 1.01i)3-s + (−1.91 + 0.563i)4-s + (1.55 − 1.85i)5-s + (1.70 + 1.75i)6-s + (0.763 + 0.278i)7-s + (−1.17 − 2.57i)8-s + (0.935 − 2.85i)9-s + (2.91 + 1.80i)10-s + (2.23 + 2.65i)11-s + (−2.11 + 2.74i)12-s + (−2.83 − 0.500i)13-s + (−0.235 + 1.12i)14-s + (0.299 − 4.18i)15-s + (3.36 − 2.16i)16-s + (−3.55 + 6.15i)17-s + ⋯ |
| L(s) = 1 | + (0.142 + 0.989i)2-s + (0.809 − 0.586i)3-s + (−0.959 + 0.281i)4-s + (0.696 − 0.830i)5-s + (0.695 + 0.718i)6-s + (0.288 + 0.105i)7-s + (−0.415 − 0.909i)8-s + (0.311 − 0.950i)9-s + (0.921 + 0.571i)10-s + (0.672 + 0.801i)11-s + (−0.611 + 0.791i)12-s + (−0.787 − 0.138i)13-s + (−0.0628 + 0.300i)14-s + (0.0772 − 1.08i)15-s + (0.840 − 0.541i)16-s + (−0.861 + 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61393 + 0.367190i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61393 + 0.367190i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.201 - 1.39i)T \) |
| 3 | \( 1 + (-1.40 + 1.01i)T \) |
| good | 5 | \( 1 + (-1.55 + 1.85i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.763 - 0.278i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-2.23 - 2.65i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.83 + 0.500i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (3.55 - 6.15i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.16 + 2.98i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.58 - 2.39i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (6.59 - 1.16i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.55 + 0.930i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (3.59 + 2.07i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.742 - 4.20i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.77 - 2.10i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-2.80 - 1.01i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 7.38iT - 53T^{2} \) |
| 59 | \( 1 + (-4.72 + 5.62i)T + (-10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.0900 - 0.247i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.20 + 0.388i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.69 + 4.67i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.97 + 3.41i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.76 - 10.0i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (7.86 - 1.38i)T + (77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-2.14 - 3.72i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.28 + 3.59i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.80749335436215677853053487725, −11.89678828114289799565345138370, −9.783725688882330904711069260650, −9.277484701158689446934046182023, −8.311079774717569766494181497861, −7.39994258031705950092107043089, −6.35044242690666096882873241524, −5.15320340089194949476399887284, −3.90866700029250151354688124940, −1.79288266933691519040834420989,
2.15242467540552866038409834861, 3.18825822358471566800701452861, 4.42742705605341108374801184783, 5.72555294135997622893594313036, 7.37126301992527313740347037652, 8.704837967419393978651466373582, 9.598914570552209234593523758268, 10.19339905294258922819224185937, 11.19809245106009353662591156208, 12.02730382978487531967023484520
