| L(s) = 1 | + (627. + 362. i)2-s + (−5.87e4 + 5.74e3i)3-s + (2.62e5 + 4.54e5i)4-s + (1.51e7 − 8.77e6i)5-s + (−3.89e7 − 1.76e7i)6-s + (2.43e8 − 4.22e8i)7-s + 3.79e8i·8-s + (3.42e9 − 6.75e8i)9-s + 1.27e10·10-s + (−1.30e10 − 7.52e9i)11-s + (−1.80e10 − 2.51e10i)12-s + (−6.93e10 − 1.20e11i)13-s + (3.05e11 − 1.76e11i)14-s + (−8.42e11 + 6.02e11i)15-s + (−1.37e11 + 2.38e11i)16-s + 1.60e12i·17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.995 + 0.0973i)3-s + (0.249 + 0.433i)4-s + (1.55 − 0.898i)5-s + (−0.643 − 0.292i)6-s + (0.863 − 1.49i)7-s + 0.353i·8-s + (0.981 − 0.193i)9-s + 1.27·10-s + (−0.502 − 0.290i)11-s + (−0.290 − 0.406i)12-s + (−0.502 − 0.870i)13-s + (1.05 − 0.610i)14-s + (−1.46 + 1.04i)15-s + (−0.125 + 0.216i)16-s + 0.794i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0203 + 0.999i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.0203 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(2.548008729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.548008729\) |
| \(L(11)\) |
|
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 | \( 1 + (5.87e4 - 5.74e3i)T \) |
| good | 5 | \( 1 + (-1.51e7 + 8.77e6i)T + (4.76e13 - 8.25e13i)T^{2} \) |
| 7 | \( 1 + (-2.43e8 + 4.22e8i)T + (-3.98e16 - 6.91e16i)T^{2} \) |
| 11 | \( 1 + (1.30e10 + 7.52e9i)T + (3.36e20 + 5.82e20i)T^{2} \) |
| 13 | \( 1 + (6.93e10 + 1.20e11i)T + (-9.50e21 + 1.64e22i)T^{2} \) |
| 17 | \( 1 - 1.60e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 + 7.11e12T + 3.75e25T^{2} \) |
| 23 | \( 1 + (-3.07e12 + 1.77e12i)T + (8.58e26 - 1.48e27i)T^{2} \) |
| 29 | \( 1 + (2.85e14 + 1.64e14i)T + (8.84e28 + 1.53e29i)T^{2} \) |
| 31 | \( 1 + (3.09e14 + 5.36e14i)T + (-3.35e29 + 5.81e29i)T^{2} \) |
| 37 | \( 1 - 7.48e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + (1.49e16 - 8.65e15i)T + (9.00e31 - 1.56e32i)T^{2} \) |
| 43 | \( 1 + (6.52e15 - 1.13e16i)T + (-2.33e32 - 4.04e32i)T^{2} \) |
| 47 | \( 1 + (-1.71e16 - 9.90e15i)T + (1.38e33 + 2.39e33i)T^{2} \) |
| 53 | \( 1 + 6.21e16iT - 3.05e34T^{2} \) |
| 59 | \( 1 + (3.22e17 - 1.86e17i)T + (1.30e35 - 2.26e35i)T^{2} \) |
| 61 | \( 1 + (-3.71e17 + 6.43e17i)T + (-2.54e35 - 4.40e35i)T^{2} \) |
| 67 | \( 1 + (-1.20e18 - 2.09e18i)T + (-1.66e36 + 2.87e36i)T^{2} \) |
| 71 | \( 1 + 4.27e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 - 2.42e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + (-3.15e18 + 5.45e18i)T + (-4.48e37 - 7.76e37i)T^{2} \) |
| 83 | \( 1 + (9.75e16 + 5.63e16i)T + (1.20e38 + 2.08e38i)T^{2} \) |
| 89 | \( 1 - 1.52e19iT - 9.72e38T^{2} \) |
| 97 | \( 1 + (8.99e18 - 1.55e19i)T + (-2.71e39 - 4.70e39i)T^{2} \) |
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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
−13.36879568106939723135222826141, −12.84577146396461913657544434968, −10.95116338814904685707735703576, −10.00231340356993216449817593248, −7.930992206732343168331176359536, −6.29147165608325008684702584851, −5.24985332850680972815341581072, −4.32869756708812372943077447222, −1.81369968195672054523296532246, −0.59710781899040540198248185922,
1.81866345859499318039635020523, 2.39200749155331125254584508497, 4.94269319532665688044475718979, 5.72664155928107436595518955541, 6.84073998020237437476051143530, 9.382511891443082657740197671172, 10.63874770869682964048801107166, 11.68471934622315802808843084138, 12.88955088618269986887287019962, 14.27216930769798262155590711931
