| L(s) = 1 | + (23.3 − 71.9i)2-s + (544. − 395. i)3-s + (−2.97e3 − 2.16e3i)4-s + (−1.02e3 − 3.14e3i)5-s + (−1.57e4 − 4.84e4i)6-s + (6.06e4 + 4.40e4i)7-s + (−1.00e5 + 7.27e4i)8-s + (8.50e4 − 2.61e5i)9-s − 2.50e5·10-s + (−1.74e5 + 5.04e5i)11-s − 2.47e6·12-s + (−4.46e5 + 1.37e6i)13-s + (4.59e6 − 3.33e6i)14-s + (−1.80e6 − 1.30e6i)15-s + (5.64e5 + 1.73e6i)16-s + (5.51e5 + 1.69e6i)17-s + ⋯ |
| L(s) = 1 | + (0.516 − 1.59i)2-s + (1.29 − 0.939i)3-s + (−1.45 − 1.05i)4-s + (−0.146 − 0.450i)5-s + (−0.826 − 2.54i)6-s + (1.36 + 0.991i)7-s + (−1.08 + 0.784i)8-s + (0.480 − 1.47i)9-s − 0.792·10-s + (−0.326 + 0.945i)11-s − 2.87·12-s + (−0.333 + 1.02i)13-s + (2.28 − 1.65i)14-s + (−0.612 − 0.445i)15-s + (0.134 + 0.414i)16-s + (0.0942 + 0.289i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.334i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.539322 - 3.13135i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.539322 - 3.13135i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (1.74e5 - 5.04e5i)T \) |
| good | 2 | \( 1 + (-23.3 + 71.9i)T + (-1.65e3 - 1.20e3i)T^{2} \) |
| 3 | \( 1 + (-544. + 395. i)T + (5.47e4 - 1.68e5i)T^{2} \) |
| 5 | \( 1 + (1.02e3 + 3.14e3i)T + (-3.95e7 + 2.87e7i)T^{2} \) |
| 7 | \( 1 + (-6.06e4 - 4.40e4i)T + (6.11e8 + 1.88e9i)T^{2} \) |
| 13 | \( 1 + (4.46e5 - 1.37e6i)T + (-1.44e12 - 1.05e12i)T^{2} \) |
| 17 | \( 1 + (-5.51e5 - 1.69e6i)T + (-2.77e13 + 2.01e13i)T^{2} \) |
| 19 | \( 1 + (-4.86e6 + 3.53e6i)T + (3.59e13 - 1.10e14i)T^{2} \) |
| 23 | \( 1 + 4.06e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (3.35e7 + 2.43e7i)T + (3.77e15 + 1.16e16i)T^{2} \) |
| 31 | \( 1 + (-6.62e7 + 2.03e8i)T + (-2.05e16 - 1.49e16i)T^{2} \) |
| 37 | \( 1 + (-1.04e8 - 7.59e7i)T + (5.49e16 + 1.69e17i)T^{2} \) |
| 41 | \( 1 + (7.40e8 - 5.38e8i)T + (1.70e17 - 5.23e17i)T^{2} \) |
| 43 | \( 1 - 6.64e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (1.08e9 - 7.85e8i)T + (7.63e17 - 2.35e18i)T^{2} \) |
| 53 | \( 1 + (5.03e8 - 1.54e9i)T + (-7.49e18 - 5.44e18i)T^{2} \) |
| 59 | \( 1 + (1.74e9 + 1.26e9i)T + (9.31e18 + 2.86e19i)T^{2} \) |
| 61 | \( 1 + (1.41e9 + 4.35e9i)T + (-3.52e19 + 2.55e19i)T^{2} \) |
| 67 | \( 1 + 5.54e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (-2.57e9 - 7.91e9i)T + (-1.86e20 + 1.35e20i)T^{2} \) |
| 73 | \( 1 + (-1.48e10 - 1.07e10i)T + (9.69e19 + 2.98e20i)T^{2} \) |
| 79 | \( 1 + (-1.43e10 + 4.41e10i)T + (-6.05e20 - 4.39e20i)T^{2} \) |
| 83 | \( 1 + (-6.21e9 - 1.91e10i)T + (-1.04e21 + 7.56e20i)T^{2} \) |
| 89 | \( 1 - 1.37e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (3.12e10 - 9.60e10i)T + (-5.78e21 - 4.20e21i)T^{2} \) |
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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
−18.08090791366470283911601634508, −14.88341900254775021653595380720, −13.89077528099770454762684929654, −12.54976202080664445313236887808, −11.69471434938392708170317511630, −9.411707706496277260751042469390, −8.013284058730463705654293814800, −4.55601694582420964000282583637, −2.39205872516842631975096451695, −1.62607397854640524672173944170,
3.56606023085869627359501112236, 5.05083184635557354755034984174, 7.58924187521050960836926826683, 8.395692756993100614512173432271, 10.53528168943688978779057724086, 13.76970902517005077313993942148, 14.34768531214458776518013126623, 15.29936071304899168752277056711, 16.44944618292844130736085527840, 17.94506199416069616564003869808
