| L(s) = 1 | + (12.6 − 38.8i)2-s + (−482. + 350. i)3-s + (309. + 224. i)4-s + (−2.27e3 − 7.01e3i)5-s + (7.51e3 + 2.31e4i)6-s + (6.07e4 + 4.41e4i)7-s + (8.02e4 − 5.82e4i)8-s + (5.49e4 − 1.69e5i)9-s − 3.01e5·10-s + (4.72e5 − 2.48e5i)11-s − 2.27e5·12-s + (1.73e5 − 5.33e5i)13-s + (2.47e6 − 1.80e6i)14-s + (3.55e6 + 2.58e6i)15-s + (−1.00e6 − 3.10e6i)16-s + (3.21e6 + 9.89e6i)17-s + ⋯ |
| L(s) = 1 | + (0.278 − 0.857i)2-s + (−1.14 + 0.832i)3-s + (0.150 + 0.109i)4-s + (−0.326 − 1.00i)5-s + (0.394 + 1.21i)6-s + (1.36 + 0.992i)7-s + (0.865 − 0.629i)8-s + (0.310 − 0.955i)9-s − 0.952·10-s + (0.885 − 0.464i)11-s − 0.264·12-s + (0.129 − 0.398i)13-s + (1.23 − 0.895i)14-s + (1.20 + 0.878i)15-s + (−0.240 − 0.740i)16-s + (0.549 + 1.68i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.70405 - 0.357093i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70405 - 0.357093i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-4.72e5 + 2.48e5i)T \) |
| good | 2 | \( 1 + (-12.6 + 38.8i)T + (-1.65e3 - 1.20e3i)T^{2} \) |
| 3 | \( 1 + (482. - 350. i)T + (5.47e4 - 1.68e5i)T^{2} \) |
| 5 | \( 1 + (2.27e3 + 7.01e3i)T + (-3.95e7 + 2.87e7i)T^{2} \) |
| 7 | \( 1 + (-6.07e4 - 4.41e4i)T + (6.11e8 + 1.88e9i)T^{2} \) |
| 13 | \( 1 + (-1.73e5 + 5.33e5i)T + (-1.44e12 - 1.05e12i)T^{2} \) |
| 17 | \( 1 + (-3.21e6 - 9.89e6i)T + (-2.77e13 + 2.01e13i)T^{2} \) |
| 19 | \( 1 + (3.19e6 - 2.31e6i)T + (3.59e13 - 1.10e14i)T^{2} \) |
| 23 | \( 1 - 4.06e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (4.04e7 + 2.94e7i)T + (3.77e15 + 1.16e16i)T^{2} \) |
| 31 | \( 1 + (6.51e6 - 2.00e7i)T + (-2.05e16 - 1.49e16i)T^{2} \) |
| 37 | \( 1 + (4.05e8 + 2.94e8i)T + (5.49e16 + 1.69e17i)T^{2} \) |
| 41 | \( 1 + (-1.25e8 + 9.12e7i)T + (1.70e17 - 5.23e17i)T^{2} \) |
| 43 | \( 1 + 1.51e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-1.82e8 + 1.32e8i)T + (7.63e17 - 2.35e18i)T^{2} \) |
| 53 | \( 1 + (-5.64e8 + 1.73e9i)T + (-7.49e18 - 5.44e18i)T^{2} \) |
| 59 | \( 1 + (-1.78e9 - 1.29e9i)T + (9.31e18 + 2.86e19i)T^{2} \) |
| 61 | \( 1 + (-3.30e9 - 1.01e10i)T + (-3.52e19 + 2.55e19i)T^{2} \) |
| 67 | \( 1 + 1.30e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + (7.98e8 + 2.45e9i)T + (-1.86e20 + 1.35e20i)T^{2} \) |
| 73 | \( 1 + (4.57e8 + 3.32e8i)T + (9.69e19 + 2.98e20i)T^{2} \) |
| 79 | \( 1 + (4.38e9 - 1.34e10i)T + (-6.05e20 - 4.39e20i)T^{2} \) |
| 83 | \( 1 + (7.50e7 + 2.31e8i)T + (-1.04e21 + 7.56e20i)T^{2} \) |
| 89 | \( 1 + 1.05e11T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-1.16e10 + 3.58e10i)T + (-5.78e21 - 4.20e21i)T^{2} \) |
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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
−17.34104958497416179722099646154, −16.48999450795717638680927431994, −15.06137675233712265240009174296, −12.53858560029749077919207593361, −11.66418600831042396647336068789, −10.71186534353720496049008422472, −8.540658942093540276562893711091, −5.49550796733240233828982244022, −4.15825368802707150839126320169, −1.33642640649414349198264764543,
1.28206273785992018329339089245, 4.95562002677273474830105859535, 6.85094902819429622023490308900, 7.31901492099147664609229452720, 10.94252843614363001780262568342, 11.59834954792044932988903251203, 13.93216390672845375012428266891, 14.87630363617008855449630324120, 16.72724710172185715032303136527, 17.56112077681103645503842768155
