| L(s) = 1 | + (2.24 + 0.626i)2-s + (2.95 + 1.78i)4-s + (0.812 − 0.0315i)5-s + (1.20 + 0.188i)7-s + (2.32 + 2.46i)8-s + (1.84 + 0.437i)10-s + (0.262 + 1.91i)11-s + (−1.56 + 2.28i)13-s + (2.59 + 1.18i)14-s + (0.466 + 0.885i)16-s + (3.40 − 1.71i)17-s + (−0.0225 − 0.388i)19-s + (2.45 + 1.35i)20-s + (−0.611 + 4.47i)22-s + (1.72 − 4.46i)23-s + ⋯ |
| L(s) = 1 | + (1.59 + 0.442i)2-s + (1.47 + 0.891i)4-s + (0.363 − 0.0141i)5-s + (0.456 + 0.0714i)7-s + (0.821 + 0.870i)8-s + (0.584 + 0.138i)10-s + (0.0790 + 0.578i)11-s + (−0.435 + 0.634i)13-s + (0.694 + 0.315i)14-s + (0.116 + 0.221i)16-s + (0.826 − 0.415i)17-s + (−0.00518 − 0.0890i)19-s + (0.549 + 0.303i)20-s + (−0.130 + 0.955i)22-s + (0.359 − 0.931i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.64426 + 1.50540i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.64426 + 1.50540i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-2.24 - 0.626i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (-0.812 + 0.0315i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (-1.20 - 0.188i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (-0.262 - 1.91i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (1.56 - 2.28i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-3.40 + 1.71i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (0.0225 + 0.388i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (-1.72 + 4.46i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (-5.20 - 3.72i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (3.58 - 4.11i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (5.86 + 7.87i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (-0.617 + 2.39i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (-0.752 - 0.683i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (1.84 + 2.11i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (1.44 - 8.21i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.578 + 0.236i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (-10.0 + 6.04i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (7.98 - 5.70i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (2.59 + 8.68i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (13.5 - 3.20i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (5.21 - 5.11i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-0.514 - 1.99i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (-1.84 + 6.16i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (-5.74 - 0.222i)T + (96.7 + 7.51i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.70069956322227368797054862629, −9.701206476240271118904371080898, −8.690587770122561188102001491926, −7.40512456104372835174680218716, −6.88489794890507484364709028088, −5.81403927018372807896387615472, −5.03692896748468499286216876505, −4.32599930363726982095017888384, −3.15260610907471210373048377140, −1.96598988102677786197144480970,
1.59526321521918785582044895911, 2.87741732887875167193155934594, 3.73780856032388423736302389950, 4.84732424322220952282070826054, 5.60967975510243228135039492701, 6.27526838696450490971057859692, 7.53443891842693985091045033843, 8.452286260693163961085689163364, 9.813247722005151041836215803652, 10.47752123766690457133156591091
