| L(s) = 1 | + (627. − 362. i)2-s + (5.08e4 − 2.99e4i)3-s + (2.62e5 − 4.54e5i)4-s + (−1.42e7 − 8.24e6i)5-s + (2.10e7 − 3.72e7i)6-s + (−5.08e6 − 8.80e6i)7-s − 3.79e8i·8-s + (1.68e9 − 3.05e9i)9-s − 1.19e10·10-s + (2.97e10 − 1.71e10i)11-s + (−2.75e8 − 3.09e10i)12-s + (6.84e10 − 1.18e11i)13-s + (−6.37e9 − 3.68e9i)14-s + (−9.73e11 + 8.67e9i)15-s + (−1.37e11 − 2.38e11i)16-s + 2.59e12i·17-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (0.861 − 0.507i)3-s + (0.249 − 0.433i)4-s + (−1.46 − 0.844i)5-s + (0.348 − 0.615i)6-s + (−0.0180 − 0.0311i)7-s − 0.353i·8-s + (0.484 − 0.874i)9-s − 1.19·10-s + (1.14 − 0.662i)11-s + (−0.00445 − 0.499i)12-s + (0.496 − 0.860i)13-s + (−0.0220 − 0.0127i)14-s + (−1.68 + 0.0150i)15-s + (−0.125 − 0.216i)16-s + 1.28i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(2.114212159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.114212159\) |
| \(L(11)\) |
|
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 | \( 1 + (-5.08e4 + 2.99e4i)T \) |
| good | 5 | \( 1 + (1.42e7 + 8.24e6i)T + (4.76e13 + 8.25e13i)T^{2} \) |
| 7 | \( 1 + (5.08e6 + 8.80e6i)T + (-3.98e16 + 6.91e16i)T^{2} \) |
| 11 | \( 1 + (-2.97e10 + 1.71e10i)T + (3.36e20 - 5.82e20i)T^{2} \) |
| 13 | \( 1 + (-6.84e10 + 1.18e11i)T + (-9.50e21 - 1.64e22i)T^{2} \) |
| 17 | \( 1 - 2.59e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 + 1.15e13T + 3.75e25T^{2} \) |
| 23 | \( 1 + (2.77e13 + 1.60e13i)T + (8.58e26 + 1.48e27i)T^{2} \) |
| 29 | \( 1 + (5.44e14 - 3.14e14i)T + (8.84e28 - 1.53e29i)T^{2} \) |
| 31 | \( 1 + (8.32e12 - 1.44e13i)T + (-3.35e29 - 5.81e29i)T^{2} \) |
| 37 | \( 1 - 1.18e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + (5.47e15 + 3.16e15i)T + (9.00e31 + 1.56e32i)T^{2} \) |
| 43 | \( 1 + (-1.57e16 - 2.73e16i)T + (-2.33e32 + 4.04e32i)T^{2} \) |
| 47 | \( 1 + (-7.73e15 + 4.46e15i)T + (1.38e33 - 2.39e33i)T^{2} \) |
| 53 | \( 1 - 3.42e16iT - 3.05e34T^{2} \) |
| 59 | \( 1 + (5.59e17 + 3.23e17i)T + (1.30e35 + 2.26e35i)T^{2} \) |
| 61 | \( 1 + (2.60e17 + 4.50e17i)T + (-2.54e35 + 4.40e35i)T^{2} \) |
| 67 | \( 1 + (-8.36e17 + 1.44e18i)T + (-1.66e36 - 2.87e36i)T^{2} \) |
| 71 | \( 1 - 3.29e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 - 6.31e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + (1.46e18 + 2.54e18i)T + (-4.48e37 + 7.76e37i)T^{2} \) |
| 83 | \( 1 + (-1.69e19 + 9.77e18i)T + (1.20e38 - 2.08e38i)T^{2} \) |
| 89 | \( 1 + 5.21e19iT - 9.72e38T^{2} \) |
| 97 | \( 1 + (4.99e18 + 8.64e18i)T + (-2.71e39 + 4.70e39i)T^{2} \) |
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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
−12.98112474517822471193183727101, −12.35829837447830449768953221336, −10.97496842515608017745298779963, −8.820742353213717493863415998450, −8.014624473480877393000177775416, −6.29892522249249277218600263487, −4.14296868969357247715757547038, −3.54088572828241175272009702993, −1.62754560460782378613625585788, −0.40839213772305930000709734014,
2.27738258017692595798822325271, 3.80637001843047013978088126692, 4.26688946081926407665555412664, 6.71931320235417650589286739684, 7.73010728315443180143041620458, 9.134106793158902894441351896417, 10.98860468501502152478533867963, 12.07281347709169945159089253395, 13.85731688062955710214268214600, 14.85652291453846809604479549159
