| L(s) = 1 | + (9.88 − 30.4i)2-s + (−394. + 286. i)3-s + (−828. − 601. i)4-s + (250. + 771. i)5-s + (4.82e3 + 1.48e4i)6-s + (3.75e4 + 2.72e4i)7-s + (−2.65e4 + 1.92e4i)8-s + (1.88e4 − 5.79e4i)9-s + 2.59e4·10-s + (−5.24e5 + 1.00e5i)11-s + 4.99e5·12-s + (7.32e5 − 2.25e6i)13-s + (1.20e6 − 8.72e5i)14-s + (−3.20e5 − 2.32e5i)15-s + (3.24e5 + 9.97e5i)16-s + (−2.50e6 − 7.70e6i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.937 + 0.681i)3-s + (−0.404 − 0.293i)4-s + (0.0358 + 0.110i)5-s + (0.253 + 0.779i)6-s + (0.844 + 0.613i)7-s + (−0.286 + 0.207i)8-s + (0.106 − 0.326i)9-s + 0.0820·10-s + (−0.982 + 0.188i)11-s + 0.579·12-s + (0.547 − 1.68i)13-s + (0.597 − 0.433i)14-s + (−0.108 − 0.0791i)15-s + (0.0772 + 0.237i)16-s + (−0.427 − 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 + 0.923i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.520320 - 0.779161i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.520320 - 0.779161i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-9.88 + 30.4i)T \) |
| 11 | \( 1 + (5.24e5 - 1.00e5i)T \) |
| good | 3 | \( 1 + (394. - 286. i)T + (5.47e4 - 1.68e5i)T^{2} \) |
| 5 | \( 1 + (-250. - 771. i)T + (-3.95e7 + 2.87e7i)T^{2} \) |
| 7 | \( 1 + (-3.75e4 - 2.72e4i)T + (6.11e8 + 1.88e9i)T^{2} \) |
| 13 | \( 1 + (-7.32e5 + 2.25e6i)T + (-1.44e12 - 1.05e12i)T^{2} \) |
| 17 | \( 1 + (2.50e6 + 7.70e6i)T + (-2.77e13 + 2.01e13i)T^{2} \) |
| 19 | \( 1 + (-1.91e6 + 1.39e6i)T + (3.59e13 - 1.10e14i)T^{2} \) |
| 23 | \( 1 + 3.24e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (-6.58e7 - 4.78e7i)T + (3.77e15 + 1.16e16i)T^{2} \) |
| 31 | \( 1 + (-5.63e7 + 1.73e8i)T + (-2.05e16 - 1.49e16i)T^{2} \) |
| 37 | \( 1 + (-9.26e7 - 6.73e7i)T + (5.49e16 + 1.69e17i)T^{2} \) |
| 41 | \( 1 + (3.48e8 - 2.53e8i)T + (1.70e17 - 5.23e17i)T^{2} \) |
| 43 | \( 1 - 1.72e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-8.17e8 + 5.93e8i)T + (7.63e17 - 2.35e18i)T^{2} \) |
| 53 | \( 1 + (-7.85e8 + 2.41e9i)T + (-7.49e18 - 5.44e18i)T^{2} \) |
| 59 | \( 1 + (-5.26e9 - 3.82e9i)T + (9.31e18 + 2.86e19i)T^{2} \) |
| 61 | \( 1 + (2.10e9 + 6.48e9i)T + (-3.52e19 + 2.55e19i)T^{2} \) |
| 67 | \( 1 + 9.75e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (5.07e9 + 1.56e10i)T + (-1.86e20 + 1.35e20i)T^{2} \) |
| 73 | \( 1 + (2.37e10 + 1.72e10i)T + (9.69e19 + 2.98e20i)T^{2} \) |
| 79 | \( 1 + (5.11e9 - 1.57e10i)T + (-6.05e20 - 4.39e20i)T^{2} \) |
| 83 | \( 1 + (-1.54e10 - 4.74e10i)T + (-1.04e21 + 7.56e20i)T^{2} \) |
| 89 | \( 1 + 7.22e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-1.63e10 + 5.04e10i)T + (-5.78e21 - 4.20e21i)T^{2} \) |
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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
−15.14413005374139850387550057847, −13.48770815520115778644796746602, −12.02069380592423446883131672808, −10.97743504176591470261179516613, −10.09302523321882253552803551925, −8.185009473881369405981915951773, −5.63132847023946085849191644535, −4.77516259723104486106149677223, −2.63275725787941893874361717320, −0.40473062924292375728772248349,
1.39468455097064076061739330151, 4.35859487673300373675952420769, 5.89319442695289228563242430662, 7.08864113710604313007187145443, 8.515999840275941718692567692609, 10.73461348801195346017415480501, 11.95119106612795316871331124883, 13.27579575352642536802242879668, 14.38515203091848450126905572351, 16.00759463051389714981536567288
