| L(s) = 1 | + (627. − 362. i)2-s + (−2.00e4 + 5.55e4i)3-s + (2.62e5 − 4.54e5i)4-s + (2.60e6 + 1.50e6i)5-s + (7.50e6 + 4.20e7i)6-s + (3.31e7 + 5.74e7i)7-s − 3.79e8i·8-s + (−2.67e9 − 2.23e9i)9-s + 2.17e9·10-s + (−3.97e10 + 2.29e10i)11-s + (1.99e10 + 2.36e10i)12-s + (1.19e11 − 2.06e11i)13-s + (4.16e10 + 2.40e10i)14-s + (−1.35e11 + 1.14e11i)15-s + (−1.37e11 − 2.38e11i)16-s + 2.24e12i·17-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.340 + 0.940i)3-s + (0.249 − 0.433i)4-s + (0.266 + 0.154i)5-s + (0.124 + 0.696i)6-s + (0.117 + 0.203i)7-s − 0.353i·8-s + (−0.768 − 0.639i)9-s + 0.217·10-s + (−1.53 + 0.884i)11-s + (0.322 + 0.382i)12-s + (0.864 − 1.49i)13-s + (0.143 + 0.0830i)14-s + (−0.235 + 0.198i)15-s + (−0.125 − 0.216i)16-s + 1.11i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 + 0.867i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(0.9371986277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9371986277\) |
| \(L(11)\) |
|
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 | \( 1 + (2.00e4 - 5.55e4i)T \) |
| good | 5 | \( 1 + (-2.60e6 - 1.50e6i)T + (4.76e13 + 8.25e13i)T^{2} \) |
| 7 | \( 1 + (-3.31e7 - 5.74e7i)T + (-3.98e16 + 6.91e16i)T^{2} \) |
| 11 | \( 1 + (3.97e10 - 2.29e10i)T + (3.36e20 - 5.82e20i)T^{2} \) |
| 13 | \( 1 + (-1.19e11 + 2.06e11i)T + (-9.50e21 - 1.64e22i)T^{2} \) |
| 17 | \( 1 - 2.24e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 + 1.35e12T + 3.75e25T^{2} \) |
| 23 | \( 1 + (6.66e12 + 3.84e12i)T + (8.58e26 + 1.48e27i)T^{2} \) |
| 29 | \( 1 + (6.67e13 - 3.85e13i)T + (8.84e28 - 1.53e29i)T^{2} \) |
| 31 | \( 1 + (-6.87e14 + 1.19e15i)T + (-3.35e29 - 5.81e29i)T^{2} \) |
| 37 | \( 1 + 4.13e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + (2.22e16 + 1.28e16i)T + (9.00e31 + 1.56e32i)T^{2} \) |
| 43 | \( 1 + (2.36e15 + 4.08e15i)T + (-2.33e32 + 4.04e32i)T^{2} \) |
| 47 | \( 1 + (-4.48e16 + 2.59e16i)T + (1.38e33 - 2.39e33i)T^{2} \) |
| 53 | \( 1 + 3.59e16iT - 3.05e34T^{2} \) |
| 59 | \( 1 + (6.50e17 + 3.75e17i)T + (1.30e35 + 2.26e35i)T^{2} \) |
| 61 | \( 1 + (3.71e17 + 6.43e17i)T + (-2.54e35 + 4.40e35i)T^{2} \) |
| 67 | \( 1 + (6.49e17 - 1.12e18i)T + (-1.66e36 - 2.87e36i)T^{2} \) |
| 71 | \( 1 + 2.16e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 + 1.41e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + (-5.33e18 - 9.23e18i)T + (-4.48e37 + 7.76e37i)T^{2} \) |
| 83 | \( 1 + (-8.69e18 + 5.02e18i)T + (1.20e38 - 2.08e38i)T^{2} \) |
| 89 | \( 1 - 1.83e19iT - 9.72e38T^{2} \) |
| 97 | \( 1 + (1.44e19 + 2.49e19i)T + (-2.71e39 + 4.70e39i)T^{2} \) |
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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
−13.46317503160244002658996993941, −12.28078493357684759513136636569, −10.63718282951406109035418933425, −10.16808065361365725495712355719, −8.209428373975391959904265990075, −6.00936117081416555993912101754, −5.06825676956918938615046593850, −3.64101711474065950662816778454, −2.27915400498495361291627836764, −0.20459580383475919351592754352,
1.46422435358160475530640896475, 2.94356209206192330186677566587, 4.93238696265743433819867515807, 6.10226547031908784059725873851, 7.33114956467356493737735631770, 8.639400821036882628734676853314, 10.89086312645601964230554743223, 11.99559432772727588800478340679, 13.55975665880236905294642613607, 13.77198396617547222532367247715
