| L(s) = 1 | + (18.7 − 25.7i)2-s + (200. − 616. i)3-s + (951. + 2.92e3i)4-s + (−2.19e4 + 1.59e4i)5-s + (−1.21e4 − 1.67e4i)6-s + (−3.99e4 + 1.29e4i)7-s + (2.17e5 + 7.06e4i)8-s + (9.03e4 + 6.56e4i)9-s + 8.66e5i·10-s + (−1.42e6 + 1.04e6i)11-s + 1.99e6·12-s + (−6.13e5 + 8.44e5i)13-s + (−4.14e5 + 1.27e6i)14-s + (5.44e6 + 1.67e7i)15-s + (−4.30e6 + 3.12e6i)16-s + (9.91e6 + 1.36e7i)17-s + ⋯ |
| L(s) = 1 | + (0.292 − 0.402i)2-s + (0.274 − 0.845i)3-s + (0.232 + 0.715i)4-s + (−1.40 + 1.02i)5-s + (−0.260 − 0.358i)6-s + (−0.339 + 0.110i)7-s + (0.829 + 0.269i)8-s + (0.170 + 0.123i)9-s + 0.866i·10-s + (−0.805 + 0.592i)11-s + 0.668·12-s + (−0.127 + 0.174i)13-s + (−0.0550 + 0.169i)14-s + (0.477 + 1.47i)15-s + (−0.256 + 0.186i)16-s + (0.410 + 0.565i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0639 - 0.997i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.0639 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.921627 + 0.864468i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.921627 + 0.864468i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (1.42e6 - 1.04e6i)T \) |
| good | 2 | \( 1 + (-18.7 + 25.7i)T + (-1.26e3 - 3.89e3i)T^{2} \) |
| 3 | \( 1 + (-200. + 616. i)T + (-4.29e5 - 3.12e5i)T^{2} \) |
| 5 | \( 1 + (2.19e4 - 1.59e4i)T + (7.54e7 - 2.32e8i)T^{2} \) |
| 7 | \( 1 + (3.99e4 - 1.29e4i)T + (1.11e10 - 8.13e9i)T^{2} \) |
| 13 | \( 1 + (6.13e5 - 8.44e5i)T + (-7.19e12 - 2.21e13i)T^{2} \) |
| 17 | \( 1 + (-9.91e6 - 1.36e7i)T + (-1.80e14 + 5.54e14i)T^{2} \) |
| 19 | \( 1 + (-1.94e6 - 6.31e5i)T + (1.79e15 + 1.30e15i)T^{2} \) |
| 23 | \( 1 + 3.65e7T + 2.19e16T^{2} \) |
| 29 | \( 1 + (1.09e9 - 3.57e8i)T + (2.86e17 - 2.07e17i)T^{2} \) |
| 31 | \( 1 + (7.31e8 + 5.31e8i)T + (2.43e17 + 7.49e17i)T^{2} \) |
| 37 | \( 1 + (-4.19e8 - 1.28e9i)T + (-5.32e18 + 3.86e18i)T^{2} \) |
| 41 | \( 1 + (-7.92e9 - 2.57e9i)T + (1.82e19 + 1.32e19i)T^{2} \) |
| 43 | \( 1 + 3.93e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (-4.21e9 + 1.29e10i)T + (-9.40e19 - 6.82e19i)T^{2} \) |
| 53 | \( 1 + (-2.02e10 - 1.47e10i)T + (1.51e20 + 4.67e20i)T^{2} \) |
| 59 | \( 1 + (-1.28e10 - 3.95e10i)T + (-1.43e21 + 1.04e21i)T^{2} \) |
| 61 | \( 1 + (-1.94e10 - 2.67e10i)T + (-8.20e20 + 2.52e21i)T^{2} \) |
| 67 | \( 1 - 7.96e10T + 8.18e21T^{2} \) |
| 71 | \( 1 + (-1.49e10 + 1.08e10i)T + (5.07e21 - 1.56e22i)T^{2} \) |
| 73 | \( 1 + (7.66e10 - 2.49e10i)T + (1.85e22 - 1.34e22i)T^{2} \) |
| 79 | \( 1 + (-1.99e11 + 2.74e11i)T + (-1.82e22 - 5.61e22i)T^{2} \) |
| 83 | \( 1 + (-3.26e11 - 4.49e11i)T + (-3.30e22 + 1.01e23i)T^{2} \) |
| 89 | \( 1 - 3.75e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (-5.52e11 - 4.01e11i)T + (2.14e23 + 6.59e23i)T^{2} \) |
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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
−18.29025079202866008936486779488, −16.28594864491598813226049512911, −14.92884017017564556627999602053, −13.11257743980484766441656202554, −12.11994170644332625444807235133, −10.78591552922668959686949819596, −7.83627452012438118733685503556, −7.19600755028753391285639977715, −3.83619707700986422375287301049, −2.38292210444545272389867685116,
0.51808555671419336756156561184, 3.85697241528454708161571843628, 5.22947478113292314696335925417, 7.63006201625786409191976638366, 9.420335587135600552658033022740, 11.05458458904728806432083668434, 12.86333681528362632243030979361, 14.74750327980748371623739880775, 15.95628007531067188971236208321, 16.20747176279062500475610340203
