L(s) = 1 | + (9.88 − 30.4i)2-s + (−507. + 369. i)3-s + (−828. − 601. i)4-s + (−2.31e3 − 7.13e3i)5-s + (6.20e3 + 1.91e4i)6-s + (−5.20e4 − 3.78e4i)7-s + (−2.65e4 + 1.92e4i)8-s + (6.70e4 − 2.06e5i)9-s − 2.40e5·10-s + (4.77e5 + 2.39e5i)11-s + 6.42e5·12-s + (−5.94e5 + 1.82e6i)13-s + (−1.66e6 + 1.20e6i)14-s + (3.81e6 + 2.76e6i)15-s + (3.24e5 + 9.97e5i)16-s + (8.65e5 + 2.66e6i)17-s + ⋯ |
L(s) = 1 | + (0.218 − 0.672i)2-s + (−1.20 + 0.876i)3-s + (−0.404 − 0.293i)4-s + (−0.331 − 1.02i)5-s + (0.325 + 1.00i)6-s + (−1.17 − 0.850i)7-s + (−0.286 + 0.207i)8-s + (0.378 − 1.16i)9-s − 0.759·10-s + (0.893 + 0.448i)11-s + 0.745·12-s + (−0.443 + 1.36i)13-s + (−0.827 + 0.601i)14-s + (1.29 + 0.941i)15-s + (0.0772 + 0.237i)16-s + (0.147 + 0.455i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.652921 + 0.174792i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.652921 + 0.174792i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-9.88 + 30.4i)T \) |
| 11 | \( 1 + (-4.77e5 - 2.39e5i)T \) |
good | 3 | \( 1 + (507. - 369. i)T + (5.47e4 - 1.68e5i)T^{2} \) |
| 5 | \( 1 + (2.31e3 + 7.13e3i)T + (-3.95e7 + 2.87e7i)T^{2} \) |
| 7 | \( 1 + (5.20e4 + 3.78e4i)T + (6.11e8 + 1.88e9i)T^{2} \) |
| 13 | \( 1 + (5.94e5 - 1.82e6i)T + (-1.44e12 - 1.05e12i)T^{2} \) |
| 17 | \( 1 + (-8.65e5 - 2.66e6i)T + (-2.77e13 + 2.01e13i)T^{2} \) |
| 19 | \( 1 + (-1.34e7 + 9.76e6i)T + (3.59e13 - 1.10e14i)T^{2} \) |
| 23 | \( 1 + 5.10e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (-1.71e8 - 1.24e8i)T + (3.77e15 + 1.16e16i)T^{2} \) |
| 31 | \( 1 + (5.45e6 - 1.67e7i)T + (-2.05e16 - 1.49e16i)T^{2} \) |
| 37 | \( 1 + (-1.07e8 - 7.83e7i)T + (5.49e16 + 1.69e17i)T^{2} \) |
| 41 | \( 1 + (-1.63e8 + 1.18e8i)T + (1.70e17 - 5.23e17i)T^{2} \) |
| 43 | \( 1 + 7.57e7T + 9.29e17T^{2} \) |
| 47 | \( 1 + (5.46e8 - 3.97e8i)T + (7.63e17 - 2.35e18i)T^{2} \) |
| 53 | \( 1 + (-2.29e8 + 7.05e8i)T + (-7.49e18 - 5.44e18i)T^{2} \) |
| 59 | \( 1 + (3.91e9 + 2.84e9i)T + (9.31e18 + 2.86e19i)T^{2} \) |
| 61 | \( 1 + (-1.69e9 - 5.22e9i)T + (-3.52e19 + 2.55e19i)T^{2} \) |
| 67 | \( 1 + 2.59e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (-5.78e9 - 1.78e10i)T + (-1.86e20 + 1.35e20i)T^{2} \) |
| 73 | \( 1 + (-2.10e9 - 1.53e9i)T + (9.69e19 + 2.98e20i)T^{2} \) |
| 79 | \( 1 + (-7.35e9 + 2.26e10i)T + (-6.05e20 - 4.39e20i)T^{2} \) |
| 83 | \( 1 + (-1.16e10 - 3.59e10i)T + (-1.04e21 + 7.56e20i)T^{2} \) |
| 89 | \( 1 - 2.38e8T + 2.77e21T^{2} \) |
| 97 | \( 1 + (3.47e10 - 1.07e11i)T + (-5.78e21 - 4.20e21i)T^{2} \) |
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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
−16.04572464109136900368181789739, −13.98593431963521645862379781180, −12.40149259720133572434953112752, −11.67758100399682451123532193116, −10.15893643583028677684597142588, −9.303039888190596972209303616865, −6.57088109002260735181221243564, −4.79430934806051813499109307739, −3.92933861540801043824957613808, −0.928912725212554537777961761763,
0.39268009689155710571394071046, 3.14498807741844225831053045842, 5.76490848035555312747579335580, 6.42337639217353319901634949694, 7.71942412247060939447931355873, 9.955399395823884011609221943233, 11.75630866900635482474722862568, 12.45474066388659209431310837823, 13.98387956776093946536973511072, 15.47743264057990625524893788235