| L(s) = 1 | + (−627. − 362. i)2-s + (−2.10e4 + 5.51e4i)3-s + (2.62e5 + 4.54e5i)4-s + (−6.67e5 + 3.85e5i)5-s + (3.31e7 − 2.69e7i)6-s + (2.18e8 − 3.78e8i)7-s − 3.79e8i·8-s + (−2.59e9 − 2.32e9i)9-s + 5.58e8·10-s + (−5.02e9 − 2.90e9i)11-s + (−3.05e10 + 4.88e9i)12-s + (1.09e11 + 1.89e11i)13-s + (−2.74e11 + 1.58e11i)14-s + (−7.18e9 − 4.49e10i)15-s + (−1.37e11 + 2.38e11i)16-s − 4.54e11i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.357 + 0.934i)3-s + (0.249 + 0.433i)4-s + (−0.0683 + 0.0394i)5-s + (0.548 − 0.445i)6-s + (0.773 − 1.34i)7-s − 0.353i·8-s + (−0.745 − 0.667i)9-s + 0.0558·10-s + (−0.193 − 0.111i)11-s + (−0.493 + 0.0789i)12-s + (0.793 + 1.37i)13-s + (−0.947 + 0.547i)14-s + (−0.0124 − 0.0779i)15-s + (−0.125 + 0.216i)16-s − 0.225i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(0.3288169317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3288169317\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (627. + 362. i)T \) |
| 3 | \( 1 + (2.10e4 - 5.51e4i)T \) |
| good | 5 | \( 1 + (6.67e5 - 3.85e5i)T + (4.76e13 - 8.25e13i)T^{2} \) |
| 7 | \( 1 + (-2.18e8 + 3.78e8i)T + (-3.98e16 - 6.91e16i)T^{2} \) |
| 11 | \( 1 + (5.02e9 + 2.90e9i)T + (3.36e20 + 5.82e20i)T^{2} \) |
| 13 | \( 1 + (-1.09e11 - 1.89e11i)T + (-9.50e21 + 1.64e22i)T^{2} \) |
| 17 | \( 1 + 4.54e11iT - 4.06e24T^{2} \) |
| 19 | \( 1 + 5.45e12T + 3.75e25T^{2} \) |
| 23 | \( 1 + (-1.39e13 + 8.08e12i)T + (8.58e26 - 1.48e27i)T^{2} \) |
| 29 | \( 1 + (-4.33e13 - 2.50e13i)T + (8.84e28 + 1.53e29i)T^{2} \) |
| 31 | \( 1 + (1.93e14 + 3.34e14i)T + (-3.35e29 + 5.81e29i)T^{2} \) |
| 37 | \( 1 - 4.87e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + (2.14e16 - 1.23e16i)T + (9.00e31 - 1.56e32i)T^{2} \) |
| 43 | \( 1 + (1.02e15 - 1.76e15i)T + (-2.33e32 - 4.04e32i)T^{2} \) |
| 47 | \( 1 + (7.64e16 + 4.41e16i)T + (1.38e33 + 2.39e33i)T^{2} \) |
| 53 | \( 1 + 1.73e17iT - 3.05e34T^{2} \) |
| 59 | \( 1 + (2.89e17 - 1.66e17i)T + (1.30e35 - 2.26e35i)T^{2} \) |
| 61 | \( 1 + (-2.41e17 + 4.18e17i)T + (-2.54e35 - 4.40e35i)T^{2} \) |
| 67 | \( 1 + (8.55e17 + 1.48e18i)T + (-1.66e36 + 2.87e36i)T^{2} \) |
| 71 | \( 1 + 5.73e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 - 3.36e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + (7.76e18 - 1.34e19i)T + (-4.48e37 - 7.76e37i)T^{2} \) |
| 83 | \( 1 + (-3.36e18 - 1.94e18i)T + (1.20e38 + 2.08e38i)T^{2} \) |
| 89 | \( 1 + 3.05e18iT - 9.72e38T^{2} \) |
| 97 | \( 1 + (-4.63e19 + 8.03e19i)T + (-2.71e39 - 4.70e39i)T^{2} \) |
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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
−13.54902724338301901080323196691, −11.48121606534958644765058449993, −10.88809503767833543073368815859, −9.645905811929198933072534076629, −8.281845975335252216634911567103, −6.66666958417222525192109101473, −4.63910383084352264082314248105, −3.62730135530947663972246741960, −1.55235787457715411993754155517, −0.12115441188711482754965856498,
1.33006586633508837285089810440, 2.53478659404937717480433210297, 5.28647540696668362302688291877, 6.24220265313516695146101095494, 7.935454817863538632511726752585, 8.624645871976337340648771198546, 10.65508692772751898264224582119, 11.85453062342295435292787426830, 13.04377331649485730409930470165, 14.72371178884210014736191754085
