| L(s) = 1 | + (16.0 − 49.5i)2-s + (−0.0680 + 0.0494i)3-s + (−536. − 389. i)4-s + (−1.92e3 − 5.91e3i)5-s + (1.35 + 4.16i)6-s + (−6.03e4 − 4.38e4i)7-s + (5.83e4 − 4.23e4i)8-s + (−5.47e4 + 1.68e5i)9-s − 3.23e5·10-s + (−3.34e5 + 4.16e5i)11-s + 55.7·12-s + (3.28e5 − 1.01e6i)13-s + (−3.14e6 + 2.28e6i)14-s + (423. + 307. i)15-s + (−1.57e6 − 4.86e6i)16-s + (−1.81e6 − 5.57e6i)17-s + ⋯ |
| L(s) = 1 | + (0.355 − 1.09i)2-s + (−0.000161 + 0.000117i)3-s + (−0.261 − 0.190i)4-s + (−0.275 − 0.846i)5-s + (7.10e−5 + 0.000218i)6-s + (−1.35 − 0.986i)7-s + (0.629 − 0.457i)8-s + (−0.309 + 0.951i)9-s − 1.02·10-s + (−0.626 + 0.779i)11-s + 6.47e−5·12-s + (0.245 − 0.756i)13-s + (−1.56 + 1.13i)14-s + (0.000144 + 0.000104i)15-s + (−0.376 − 1.15i)16-s + (−0.309 − 0.952i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00256i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.00177299 + 1.38102i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00177299 + 1.38102i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (3.34e5 - 4.16e5i)T \) |
| good | 2 | \( 1 + (-16.0 + 49.5i)T + (-1.65e3 - 1.20e3i)T^{2} \) |
| 3 | \( 1 + (0.0680 - 0.0494i)T + (5.47e4 - 1.68e5i)T^{2} \) |
| 5 | \( 1 + (1.92e3 + 5.91e3i)T + (-3.95e7 + 2.87e7i)T^{2} \) |
| 7 | \( 1 + (6.03e4 + 4.38e4i)T + (6.11e8 + 1.88e9i)T^{2} \) |
| 13 | \( 1 + (-3.28e5 + 1.01e6i)T + (-1.44e12 - 1.05e12i)T^{2} \) |
| 17 | \( 1 + (1.81e6 + 5.57e6i)T + (-2.77e13 + 2.01e13i)T^{2} \) |
| 19 | \( 1 + (-1.47e6 + 1.07e6i)T + (3.59e13 - 1.10e14i)T^{2} \) |
| 23 | \( 1 - 5.29e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (9.34e7 + 6.79e7i)T + (3.77e15 + 1.16e16i)T^{2} \) |
| 31 | \( 1 + (-3.91e7 + 1.20e8i)T + (-2.05e16 - 1.49e16i)T^{2} \) |
| 37 | \( 1 + (-3.98e8 - 2.89e8i)T + (5.49e16 + 1.69e17i)T^{2} \) |
| 41 | \( 1 + (-4.07e8 + 2.96e8i)T + (1.70e17 - 5.23e17i)T^{2} \) |
| 43 | \( 1 - 2.89e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (1.70e9 - 1.23e9i)T + (7.63e17 - 2.35e18i)T^{2} \) |
| 53 | \( 1 + (-4.87e8 + 1.49e9i)T + (-7.49e18 - 5.44e18i)T^{2} \) |
| 59 | \( 1 + (5.67e9 + 4.12e9i)T + (9.31e18 + 2.86e19i)T^{2} \) |
| 61 | \( 1 + (-1.05e9 - 3.25e9i)T + (-3.52e19 + 2.55e19i)T^{2} \) |
| 67 | \( 1 - 5.93e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (-6.45e9 - 1.98e10i)T + (-1.86e20 + 1.35e20i)T^{2} \) |
| 73 | \( 1 + (2.46e10 + 1.78e10i)T + (9.69e19 + 2.98e20i)T^{2} \) |
| 79 | \( 1 + (3.34e9 - 1.03e10i)T + (-6.05e20 - 4.39e20i)T^{2} \) |
| 83 | \( 1 + (6.92e9 + 2.13e10i)T + (-1.04e21 + 7.56e20i)T^{2} \) |
| 89 | \( 1 + 3.17e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + (1.47e10 - 4.55e10i)T + (-5.78e21 - 4.20e21i)T^{2} \) |
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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
−16.86744897829340307477315922079, −15.91677109340766006024836458498, −13.28580798181598947246727515188, −12.88979242830127790655520009618, −11.07850816408559241756912126237, −9.786301433250477048797611115108, −7.44195881103092700640278714837, −4.65269721881619238211253588574, −2.87234182964088365703611595080, −0.61214089937042505873541871109,
3.15551692247625742785098467976, 5.90168683739995552894856889578, 6.83385404284948026871137047872, 8.931092961666225201425545489872, 11.05842675525210285185263048318, 12.95995878659479277284951411981, 14.67504529761865569116337195381, 15.50152695466408561470015304133, 16.61820526344048408182379376550, 18.46185174977691278001413468651
