L(s) = 1 | + (−1.02e5 − 2.43e3i)3-s − 1.44e7i·5-s + 1.26e9i·7-s + (1.04e10 + 4.98e8i)9-s − 6.25e10·11-s − 3.73e11·13-s + (−3.52e10 + 1.47e12i)15-s − 7.29e12i·17-s − 1.62e12i·19-s + (3.08e12 − 1.29e14i)21-s − 3.19e14·23-s + 2.67e14·25-s + (−1.06e15 − 7.63e13i)27-s + 6.11e14i·29-s + 5.24e15i·31-s + ⋯ |
L(s) = 1 | + (−0.999 − 0.0238i)3-s − 0.662i·5-s + 1.69i·7-s + (0.998 + 0.0476i)9-s − 0.727·11-s − 0.752·13-s + (−0.0157 + 0.662i)15-s − 0.877i·17-s − 0.0608i·19-s + (0.0403 − 1.69i)21-s − 1.60·23-s + 0.561·25-s + (−0.997 − 0.0713i)27-s + 0.270i·29-s + 1.14i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(0.5058710395\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5058710395\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.02e5 + 2.43e3i)T \) |
good | 5 | \( 1 + 1.44e7iT - 4.76e14T^{2} \) |
| 7 | \( 1 - 1.26e9iT - 5.58e17T^{2} \) |
| 11 | \( 1 + 6.25e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 3.73e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 7.29e12iT - 6.90e25T^{2} \) |
| 19 | \( 1 + 1.62e12iT - 7.14e26T^{2} \) |
| 23 | \( 1 + 3.19e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 6.11e14iT - 5.13e30T^{2} \) |
| 31 | \( 1 - 5.24e15iT - 2.08e31T^{2} \) |
| 37 | \( 1 - 1.14e15T + 8.55e32T^{2} \) |
| 41 | \( 1 - 5.72e16iT - 7.38e33T^{2} \) |
| 43 | \( 1 - 1.59e17iT - 2.00e34T^{2} \) |
| 47 | \( 1 + 5.37e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 4.19e16iT - 1.62e36T^{2} \) |
| 59 | \( 1 - 5.79e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 7.14e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.96e18iT - 2.22e38T^{2} \) |
| 71 | \( 1 + 2.21e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 3.72e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.09e20iT - 7.08e39T^{2} \) |
| 83 | \( 1 - 1.78e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 5.72e20iT - 8.65e40T^{2} \) |
| 97 | \( 1 - 8.41e20T + 5.27e41T^{2} \) |
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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
−11.61404090936338515003314307591, −10.14030583184830315652094322237, −9.104697636215931267839243845429, −7.87624978262549531016265531603, −6.38394281827750572647649476701, −5.31183089013464706121250441633, −4.77698120851883164358920252053, −2.79024978508710089756435021738, −1.63512230697646974205402061233, −0.19406617552657254736952547053,
0.58317625154282486279274865182, 1.96874259842751873157047890400, 3.67703770585353031526738437194, 4.59785797997333708030418800288, 5.98096719569262495704954406722, 7.04908472393984322476660903308, 7.81676638876590225358991692592, 10.09877841074481123156949836725, 10.39003413150434823382747323151, 11.45942399704397354065249452156