| L(s) = 1 | + (66.9 − 48.6i)2-s + (−196. − 604. i)3-s + (1.48e3 − 4.55e3i)4-s + (7.81e3 + 5.67e3i)5-s + (−4.25e4 − 3.08e4i)6-s + (−7.83e3 + 2.41e4i)7-s + (−7.01e4 − 2.16e5i)8-s + (−1.83e5 + 1.33e5i)9-s + 7.98e5·10-s + (−1.41e5 + 5.15e5i)11-s − 3.04e6·12-s + (1.48e6 − 1.07e6i)13-s + (6.48e5 + 1.99e6i)14-s + (1.89e6 − 5.83e6i)15-s + (−7.25e6 − 5.27e6i)16-s + (−3.49e6 − 2.53e6i)17-s + ⋯ |
| L(s) = 1 | + (1.47 − 1.07i)2-s + (−0.466 − 1.43i)3-s + (0.723 − 2.22i)4-s + (1.11 + 0.812i)5-s + (−2.23 − 1.62i)6-s + (−0.176 + 0.542i)7-s + (−0.757 − 2.33i)8-s + (−1.03 + 0.751i)9-s + 2.52·10-s + (−0.264 + 0.964i)11-s − 3.53·12-s + (1.11 − 0.806i)13-s + (0.322 + 0.991i)14-s + (0.644 − 1.98i)15-s + (−1.73 − 1.25i)16-s + (−0.596 − 0.433i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 + 0.606i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.795 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.14555 - 3.39384i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14555 - 3.39384i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (1.41e5 - 5.15e5i)T \) |
| good | 2 | \( 1 + (-66.9 + 48.6i)T + (632. - 1.94e3i)T^{2} \) |
| 3 | \( 1 + (196. + 604. i)T + (-1.43e5 + 1.04e5i)T^{2} \) |
| 5 | \( 1 + (-7.81e3 - 5.67e3i)T + (1.50e7 + 4.64e7i)T^{2} \) |
| 7 | \( 1 + (7.83e3 - 2.41e4i)T + (-1.59e9 - 1.16e9i)T^{2} \) |
| 13 | \( 1 + (-1.48e6 + 1.07e6i)T + (5.53e11 - 1.70e12i)T^{2} \) |
| 17 | \( 1 + (3.49e6 + 2.53e6i)T + (1.05e13 + 3.25e13i)T^{2} \) |
| 19 | \( 1 + (-1.71e6 - 5.27e6i)T + (-9.42e13 + 6.84e13i)T^{2} \) |
| 23 | \( 1 - 1.29e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (-5.54e6 + 1.70e7i)T + (-9.87e15 - 7.17e15i)T^{2} \) |
| 31 | \( 1 + (4.11e7 - 2.98e7i)T + (7.85e15 - 2.41e16i)T^{2} \) |
| 37 | \( 1 + (9.96e7 - 3.06e8i)T + (-1.43e17 - 1.04e17i)T^{2} \) |
| 41 | \( 1 + (-3.00e8 - 9.24e8i)T + (-4.45e17 + 3.23e17i)T^{2} \) |
| 43 | \( 1 + 8.19e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (8.83e8 + 2.71e9i)T + (-2.00e18 + 1.45e18i)T^{2} \) |
| 53 | \( 1 + (-1.49e9 + 1.08e9i)T + (2.86e18 - 8.81e18i)T^{2} \) |
| 59 | \( 1 + (3.20e9 - 9.86e9i)T + (-2.43e19 - 1.77e19i)T^{2} \) |
| 61 | \( 1 + (-2.03e9 - 1.47e9i)T + (1.34e19 + 4.13e19i)T^{2} \) |
| 67 | \( 1 + 1.25e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (-6.00e8 - 4.36e8i)T + (7.14e19 + 2.19e20i)T^{2} \) |
| 73 | \( 1 + (4.54e9 - 1.39e10i)T + (-2.53e20 - 1.84e20i)T^{2} \) |
| 79 | \( 1 + (-1.07e10 + 7.80e9i)T + (2.31e20 - 7.11e20i)T^{2} \) |
| 83 | \( 1 + (3.62e10 + 2.63e10i)T + (3.97e20 + 1.22e21i)T^{2} \) |
| 89 | \( 1 - 4.06e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-6.32e10 + 4.59e10i)T + (2.21e21 - 6.80e21i)T^{2} \) |
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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
−17.97775284058014154107403159571, −15.05160890736468953242021912733, −13.58897776561496594566599117480, −12.97266263670564969355550436083, −11.70880757776375501039730116055, −10.29320687798135520672832925415, −6.62081690224552301444142875084, −5.59160646402999524164096043253, −2.68931855934635603496201785578, −1.53974226451713644059677735876,
3.82254109320377924631433764828, 5.10195268298784493399122149715, 6.19920679442369669131724070272, 8.977903201627068747560891330088, 11.02419459381535242854169009309, 13.13186090177617184984508047180, 14.03933902915029415567483620356, 15.71063021629697350248758211636, 16.44926979727863848509423893150, 17.26014504536284081721430827247
