| L(s) = 1 | + (7.44 − 22.9i)2-s + (273. − 198. i)3-s + (1.18e3 + 862. i)4-s + (2.51e3 + 7.74e3i)5-s + (−2.51e3 − 7.73e3i)6-s + (5.41e3 + 3.93e3i)7-s + (6.84e4 − 4.97e4i)8-s + (−1.95e4 + 6.00e4i)9-s + 1.96e5·10-s + (4.23e5 − 3.25e5i)11-s + 4.95e5·12-s + (2.66e5 − 8.21e5i)13-s + (1.30e5 − 9.46e4i)14-s + (2.22e6 + 1.61e6i)15-s + (2.99e5 + 9.20e5i)16-s + (−1.30e6 − 4.01e6i)17-s + ⋯ |
| L(s) = 1 | + (0.164 − 0.506i)2-s + (0.648 − 0.471i)3-s + (0.579 + 0.421i)4-s + (0.360 + 1.10i)5-s + (−0.131 − 0.405i)6-s + (0.121 + 0.0884i)7-s + (0.739 − 0.536i)8-s + (−0.110 + 0.339i)9-s + 0.620·10-s + (0.793 − 0.608i)11-s + 0.575·12-s + (0.199 − 0.613i)13-s + (0.0647 − 0.0470i)14-s + (0.756 + 0.549i)15-s + (0.0713 + 0.219i)16-s + (−0.222 − 0.686i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.74198 - 0.333451i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.74198 - 0.333451i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-4.23e5 + 3.25e5i)T \) |
| good | 2 | \( 1 + (-7.44 + 22.9i)T + (-1.65e3 - 1.20e3i)T^{2} \) |
| 3 | \( 1 + (-273. + 198. i)T + (5.47e4 - 1.68e5i)T^{2} \) |
| 5 | \( 1 + (-2.51e3 - 7.74e3i)T + (-3.95e7 + 2.87e7i)T^{2} \) |
| 7 | \( 1 + (-5.41e3 - 3.93e3i)T + (6.11e8 + 1.88e9i)T^{2} \) |
| 13 | \( 1 + (-2.66e5 + 8.21e5i)T + (-1.44e12 - 1.05e12i)T^{2} \) |
| 17 | \( 1 + (1.30e6 + 4.01e6i)T + (-2.77e13 + 2.01e13i)T^{2} \) |
| 19 | \( 1 + (3.78e6 - 2.74e6i)T + (3.59e13 - 1.10e14i)T^{2} \) |
| 23 | \( 1 + 5.87e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (-7.94e7 - 5.77e7i)T + (3.77e15 + 1.16e16i)T^{2} \) |
| 31 | \( 1 + (-2.74e7 + 8.46e7i)T + (-2.05e16 - 1.49e16i)T^{2} \) |
| 37 | \( 1 + (1.17e8 + 8.50e7i)T + (5.49e16 + 1.69e17i)T^{2} \) |
| 41 | \( 1 + (-1.62e8 + 1.18e8i)T + (1.70e17 - 5.23e17i)T^{2} \) |
| 43 | \( 1 + 1.66e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (1.32e9 - 9.63e8i)T + (7.63e17 - 2.35e18i)T^{2} \) |
| 53 | \( 1 + (-9.40e8 + 2.89e9i)T + (-7.49e18 - 5.44e18i)T^{2} \) |
| 59 | \( 1 + (-1.97e8 - 1.43e8i)T + (9.31e18 + 2.86e19i)T^{2} \) |
| 61 | \( 1 + (5.11e8 + 1.57e9i)T + (-3.52e19 + 2.55e19i)T^{2} \) |
| 67 | \( 1 + 2.36e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (-3.10e9 - 9.54e9i)T + (-1.86e20 + 1.35e20i)T^{2} \) |
| 73 | \( 1 + (2.21e10 + 1.60e10i)T + (9.69e19 + 2.98e20i)T^{2} \) |
| 79 | \( 1 + (-1.10e10 + 3.41e10i)T + (-6.05e20 - 4.39e20i)T^{2} \) |
| 83 | \( 1 + (-1.32e10 - 4.07e10i)T + (-1.04e21 + 7.56e20i)T^{2} \) |
| 89 | \( 1 - 6.78e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (1.75e9 - 5.41e9i)T + (-5.78e21 - 4.20e21i)T^{2} \) |
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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
−18.03932023555388601069539497437, −16.27717836090073694522503800895, −14.50638946890054237714409834596, −13.43885442761955106885550673919, −11.71845133810987708520986552010, −10.41215948743220975477927603282, −8.087700845408784396561432554848, −6.57070182485366073022237695910, −3.27680425261102447989646824440, −2.02605980383798468626671036738,
1.66526460049834906386256311965, 4.41653368994935386805539121317, 6.33228636279105276061068691694, 8.504151086006411863738157160272, 9.900686455745563203918670037710, 11.98971580423993187658081480340, 13.90001010810273112752045893925, 15.02132335913611640093963171458, 16.23372657402598096663514148829, 17.41947264317926895363033383508
