| L(s) = 1 | + (25.3 + 78.1i)2-s + (−336. − 244. i)3-s + (−3.79e3 + 2.76e3i)4-s + (3.27e3 − 1.00e4i)5-s + (1.05e4 − 3.25e4i)6-s + (−1.52e4 + 1.10e4i)7-s + (−1.75e5 − 1.27e5i)8-s + (−1.19e3 − 3.67e3i)9-s + 8.69e5·10-s + (−1.92e5 + 4.98e5i)11-s + 1.95e6·12-s + (−6.56e5 − 2.02e6i)13-s + (−1.25e6 − 9.10e5i)14-s + (−3.56e6 + 2.58e6i)15-s + (2.54e6 − 7.84e6i)16-s + (1.07e6 − 3.31e6i)17-s + ⋯ |
| L(s) = 1 | + (0.560 + 1.72i)2-s + (−0.800 − 0.581i)3-s + (−1.85 + 1.34i)4-s + (0.467 − 1.44i)5-s + (0.554 − 1.70i)6-s + (−0.343 + 0.249i)7-s + (−1.89 − 1.37i)8-s + (−0.00673 − 0.0207i)9-s + 2.74·10-s + (−0.359 + 0.932i)11-s + 2.26·12-s + (−0.490 − 1.51i)13-s + (−0.622 − 0.452i)14-s + (−1.21 + 0.880i)15-s + (0.607 − 1.86i)16-s + (0.183 − 0.565i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.504 + 0.863i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.551198 - 0.316445i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.551198 - 0.316445i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (1.92e5 - 4.98e5i)T \) |
| good | 2 | \( 1 + (-25.3 - 78.1i)T + (-1.65e3 + 1.20e3i)T^{2} \) |
| 3 | \( 1 + (336. + 244. i)T + (5.47e4 + 1.68e5i)T^{2} \) |
| 5 | \( 1 + (-3.27e3 + 1.00e4i)T + (-3.95e7 - 2.87e7i)T^{2} \) |
| 7 | \( 1 + (1.52e4 - 1.10e4i)T + (6.11e8 - 1.88e9i)T^{2} \) |
| 13 | \( 1 + (6.56e5 + 2.02e6i)T + (-1.44e12 + 1.05e12i)T^{2} \) |
| 17 | \( 1 + (-1.07e6 + 3.31e6i)T + (-2.77e13 - 2.01e13i)T^{2} \) |
| 19 | \( 1 + (8.93e6 + 6.49e6i)T + (3.59e13 + 1.10e14i)T^{2} \) |
| 23 | \( 1 + 4.13e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + (1.50e8 - 1.09e8i)T + (3.77e15 - 1.16e16i)T^{2} \) |
| 31 | \( 1 + (-2.90e6 - 8.95e6i)T + (-2.05e16 + 1.49e16i)T^{2} \) |
| 37 | \( 1 + (-4.90e8 + 3.56e8i)T + (5.49e16 - 1.69e17i)T^{2} \) |
| 41 | \( 1 + (-1.14e8 - 8.35e7i)T + (1.70e17 + 5.23e17i)T^{2} \) |
| 43 | \( 1 - 4.62e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-6.37e8 - 4.63e8i)T + (7.63e17 + 2.35e18i)T^{2} \) |
| 53 | \( 1 + (6.62e8 + 2.03e9i)T + (-7.49e18 + 5.44e18i)T^{2} \) |
| 59 | \( 1 + (-3.64e9 + 2.64e9i)T + (9.31e18 - 2.86e19i)T^{2} \) |
| 61 | \( 1 + (7.87e7 - 2.42e8i)T + (-3.52e19 - 2.55e19i)T^{2} \) |
| 67 | \( 1 + 5.42e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (-5.68e9 + 1.75e10i)T + (-1.86e20 - 1.35e20i)T^{2} \) |
| 73 | \( 1 + (1.46e10 - 1.06e10i)T + (9.69e19 - 2.98e20i)T^{2} \) |
| 79 | \( 1 + (9.44e9 + 2.90e10i)T + (-6.05e20 + 4.39e20i)T^{2} \) |
| 83 | \( 1 + (-6.77e9 + 2.08e10i)T + (-1.04e21 - 7.56e20i)T^{2} \) |
| 89 | \( 1 - 3.47e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-1.66e10 - 5.11e10i)T + (-5.78e21 + 4.20e21i)T^{2} \) |
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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
−17.32157128110482879558901768526, −16.25152881268626615412647778665, −14.96195335365320906407716214545, −12.83506649581840147076563640251, −12.74934655900947197825099063598, −9.180145333360043694475985793435, −7.50736504156792992614327212812, −5.86173482058351014303158414919, −4.94570635985681554036662624096, −0.28523459330874338058580900439,
2.32465156261960195105286974427, 4.00046178281975440757360700892, 5.97955648550218033129860265733, 9.848344590469649974983000933824, 10.75904324416173124482512052848, 11.57830063027182180574059049083, 13.47109013931493877840212028571, 14.58361138716762412039401692283, 16.85636441499702822266566871621, 18.64198601142338049475186355668
