| L(s) = 1 | + (−0.203 + 1.98i)2-s + (2.70 − 2.21i)3-s + (−3.91 − 0.808i)4-s + (7.86 + 2.38i)5-s + (3.86 + 5.82i)6-s + (−2.00 − 10.0i)7-s + (2.40 − 7.63i)8-s + (0.624 − 3.14i)9-s + (−6.34 + 15.1i)10-s + (−0.496 − 5.04i)11-s + (−12.3 + 6.49i)12-s + (4.42 + 14.5i)13-s + (20.4 − 1.93i)14-s + (26.5 − 10.9i)15-s + (14.6 + 6.33i)16-s + (3.79 − 9.16i)17-s + ⋯ |
| L(s) = 1 | + (−0.101 + 0.994i)2-s + (0.900 − 0.738i)3-s + (−0.979 − 0.202i)4-s + (1.57 + 0.477i)5-s + (0.643 + 0.970i)6-s + (−0.286 − 1.43i)7-s + (0.300 − 0.953i)8-s + (0.0694 − 0.349i)9-s + (−0.634 + 1.51i)10-s + (−0.0451 − 0.458i)11-s + (−1.03 + 0.541i)12-s + (0.340 + 1.12i)13-s + (1.45 − 0.138i)14-s + (1.76 − 0.732i)15-s + (0.918 + 0.395i)16-s + (0.223 − 0.539i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.85751 + 0.392614i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85751 + 0.392614i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.203 - 1.98i)T \) |
| good | 3 | \( 1 + (-2.70 + 2.21i)T + (1.75 - 8.82i)T^{2} \) |
| 5 | \( 1 + (-7.86 - 2.38i)T + (20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (2.00 + 10.0i)T + (-45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (0.496 + 5.04i)T + (-118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (-4.42 - 14.5i)T + (-140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (-3.79 + 9.16i)T + (-204. - 204. i)T^{2} \) |
| 19 | \( 1 + (14.2 - 26.6i)T + (-200. - 300. i)T^{2} \) |
| 23 | \( 1 + (9.19 - 6.14i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (1.72 - 17.5i)T + (-824. - 164. i)T^{2} \) |
| 31 | \( 1 + (36.5 + 36.5i)T + 961iT^{2} \) |
| 37 | \( 1 + (29.8 + 55.8i)T + (-760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (31.5 - 21.0i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-29.0 - 23.8i)T + (360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (-15.5 + 37.5i)T + (-1.56e3 - 1.56e3i)T^{2} \) |
| 53 | \( 1 + (2.26 + 22.9i)T + (-2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (-20.4 - 6.19i)T + (2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (25.8 - 21.1i)T + (725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (-21.1 - 25.7i)T + (-875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (81.7 - 16.2i)T + (4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (4.57 + 0.909i)T + (4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-4.68 - 11.3i)T + (-4.41e3 + 4.41e3i)T^{2} \) |
| 83 | \( 1 + (-13.8 - 7.40i)T + (3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (61.7 - 92.3i)T + (-3.03e3 - 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-26.4 + 26.4i)T - 9.40e3iT^{2} \) |
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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
−13.56195849626834082639696606156, −12.92786214281829487143048385377, −10.68273722479625793758567298261, −9.782010307021967212529957868039, −8.823889015468145044490923487367, −7.53446821264433750831479041450, −6.78056156607325966756746582528, −5.75230141454877652816832626067, −3.78521615222541287013894184335, −1.72205732839223778948013160957,
2.08436475164194297231722858834, 3.08128957096581237568891843986, 4.89060496302480155181909877158, 5.93574565985187001411242135080, 8.593463858256421931376848808993, 8.950119615336377737782913389357, 9.834553694669463189703976080846, 10.56787919995365583512524051699, 12.27794052006944081911929902091, 12.94635896212844126575428118819
