L(s) = 1 | − 14.5i·2-s − 66.9·3-s + 43.4·4-s − 870.·5-s + 976. i·6-s + 4.07e3i·7-s − 4.36e3i·8-s − 2.07e3·9-s + 1.26e4i·10-s + (−1.07e4 − 9.94e3i)11-s − 2.91e3·12-s − 3.71e4i·13-s + 5.94e4·14-s + 5.82e4·15-s − 5.25e4·16-s − 2.00e3i·17-s + ⋯ |
L(s) = 1 | − 0.911i·2-s − 0.827·3-s + 0.169·4-s − 1.39·5-s + 0.753i·6-s + 1.69i·7-s − 1.06i·8-s − 0.315·9-s + 1.26i·10-s + (−0.734 − 0.679i)11-s − 0.140·12-s − 1.29i·13-s + 1.54·14-s + 1.15·15-s − 0.801·16-s − 0.0239i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.679i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.734 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0387760 + 0.0990321i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0387760 + 0.0990321i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (1.07e4 + 9.94e3i)T \) |
good | 2 | \( 1 + 14.5iT - 256T^{2} \) |
| 3 | \( 1 + 66.9T + 6.56e3T^{2} \) |
| 5 | \( 1 + 870.T + 3.90e5T^{2} \) |
| 7 | \( 1 - 4.07e3iT - 5.76e6T^{2} \) |
| 13 | \( 1 + 3.71e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 2.00e3iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 8.95e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 2.77e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 3.25e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 4.44e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.48e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 1.84e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 3.67e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 3.89e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 6.99e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 4.71e6T + 1.46e14T^{2} \) |
| 61 | \( 1 - 7.61e6iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 1.80e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 4.47e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + 1.06e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 4.97e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 6.38e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 1.01e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 1.28e8T + 7.83e15T^{2} \) |
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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
−18.23299238857303584888674060071, −16.07884733010001424020066163746, −15.36213277962131614797666133382, −12.40646716794620736051183507494, −11.83031250380549992946213567459, −10.68172286464354837595652189654, −8.233155449778884049974857026577, −5.71929853712523623839858189954, −3.02096181626994876041925002622, −0.07219999791047698356237564162,
4.49043660283278153192048998246, 6.79246458963393192223536680385, 7.82506140638232465592995246767, 10.81778528999316642037089821459, 11.83555732644907705959175148161, 14.08457782539895700814437892284, 15.67821881314275487116669255784, 16.56626244504167105256625448072, 17.52350933248890116121992664240, 19.59062115631709699410581356251