| L(s) = 1 | + (−55.1 − 40.0i)2-s + (−247. + 761. i)3-s + (805. + 2.47e3i)4-s + (−3.14e3 + 2.28e3i)5-s + (4.41e4 − 3.20e4i)6-s + (1.58e4 + 4.89e4i)7-s + (1.17e4 − 3.61e4i)8-s + (−3.74e5 − 2.72e5i)9-s + 2.65e5·10-s + (−5.25e5 + 9.41e4i)11-s − 2.08e6·12-s + (5.21e5 + 3.78e5i)13-s + (1.08e6 − 3.33e6i)14-s + (−9.60e5 − 2.95e6i)15-s + (2.21e6 − 1.61e6i)16-s + (3.69e5 − 2.68e5i)17-s + ⋯ |
| L(s) = 1 | + (−1.21 − 0.886i)2-s + (−0.587 + 1.80i)3-s + (0.393 + 1.21i)4-s + (−0.449 + 0.326i)5-s + (2.31 − 1.68i)6-s + (0.357 + 1.09i)7-s + (0.126 − 0.390i)8-s + (−2.11 − 1.53i)9-s + 0.838·10-s + (−0.984 + 0.176i)11-s − 2.41·12-s + (0.389 + 0.283i)13-s + (0.538 − 1.65i)14-s + (−0.326 − 1.00i)15-s + (0.528 − 0.384i)16-s + (0.0631 − 0.0458i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 + 0.737i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.0509990 - 0.115856i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0509990 - 0.115856i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (5.25e5 - 9.41e4i)T \) |
| good | 2 | \( 1 + (55.1 + 40.0i)T + (632. + 1.94e3i)T^{2} \) |
| 3 | \( 1 + (247. - 761. i)T + (-1.43e5 - 1.04e5i)T^{2} \) |
| 5 | \( 1 + (3.14e3 - 2.28e3i)T + (1.50e7 - 4.64e7i)T^{2} \) |
| 7 | \( 1 + (-1.58e4 - 4.89e4i)T + (-1.59e9 + 1.16e9i)T^{2} \) |
| 13 | \( 1 + (-5.21e5 - 3.78e5i)T + (5.53e11 + 1.70e12i)T^{2} \) |
| 17 | \( 1 + (-3.69e5 + 2.68e5i)T + (1.05e13 - 3.25e13i)T^{2} \) |
| 19 | \( 1 + (-5.09e6 + 1.56e7i)T + (-9.42e13 - 6.84e13i)T^{2} \) |
| 23 | \( 1 + 4.66e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + (-1.21e7 - 3.74e7i)T + (-9.87e15 + 7.17e15i)T^{2} \) |
| 31 | \( 1 + (-3.97e7 - 2.88e7i)T + (7.85e15 + 2.41e16i)T^{2} \) |
| 37 | \( 1 + (-1.09e8 - 3.37e8i)T + (-1.43e17 + 1.04e17i)T^{2} \) |
| 41 | \( 1 + (8.96e7 - 2.76e8i)T + (-4.45e17 - 3.23e17i)T^{2} \) |
| 43 | \( 1 + 3.00e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-5.37e8 + 1.65e9i)T + (-2.00e18 - 1.45e18i)T^{2} \) |
| 53 | \( 1 + (4.32e9 + 3.14e9i)T + (2.86e18 + 8.81e18i)T^{2} \) |
| 59 | \( 1 + (1.25e9 + 3.85e9i)T + (-2.43e19 + 1.77e19i)T^{2} \) |
| 61 | \( 1 + (6.74e9 - 4.89e9i)T + (1.34e19 - 4.13e19i)T^{2} \) |
| 67 | \( 1 - 8.13e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (9.77e9 - 7.10e9i)T + (7.14e19 - 2.19e20i)T^{2} \) |
| 73 | \( 1 + (-6.42e9 - 1.97e10i)T + (-2.53e20 + 1.84e20i)T^{2} \) |
| 79 | \( 1 + (2.66e10 + 1.93e10i)T + (2.31e20 + 7.11e20i)T^{2} \) |
| 83 | \( 1 + (4.95e9 - 3.60e9i)T + (3.97e20 - 1.22e21i)T^{2} \) |
| 89 | \( 1 - 4.60e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (1.04e11 + 7.58e10i)T + (2.21e21 + 6.80e21i)T^{2} \) |
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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
−18.53582900704162017612558609002, −17.46483796413622957272965234264, −15.96305286086818204975488827295, −15.08350510282706943167582180838, −11.70872359959233530973207943258, −10.99882705333151047242916999803, −9.738977896663037218471126343255, −8.607948547818886823775223511355, −5.16016639109761438454062137243, −2.96792091495255602921651017039,
0.11684425833418329278561636792, 1.20427140801200604742959889114, 6.02890980772554351113863685716, 7.61284525754964664568437026430, 8.026542227339090274256205468245, 10.69663986418003421677301411433, 12.45276096019302552679591865225, 13.88586004990992593441140308293, 16.17093540143304495927819471401, 17.19463015694951561188780461052
