| L(s) = 1 | − 42.2i·2-s + (235. − 135. i)3-s − 759.·4-s + (1.17e3 − 677. i)5-s + (−5.73e3 − 9.92e3i)6-s + (2.61e4 + 1.50e4i)7-s − 1.11e4i·8-s + (7.30e3 − 1.26e4i)9-s + (−2.86e4 − 4.95e4i)10-s + 2.29e5·11-s + (−1.78e5 + 1.03e5i)12-s + (−6.59e4 + 1.14e5i)13-s + (6.37e5 − 1.10e6i)14-s + (1.83e5 − 3.18e5i)15-s − 1.24e6·16-s + (5.29e5 − 9.17e5i)17-s + ⋯ |
| L(s) = 1 | − 1.31i·2-s + (0.967 − 0.558i)3-s − 0.741·4-s + (0.375 − 0.216i)5-s + (−0.736 − 1.27i)6-s + (1.55 + 0.898i)7-s − 0.340i·8-s + (0.123 − 0.214i)9-s + (−0.286 − 0.495i)10-s + 1.42·11-s + (−0.717 + 0.414i)12-s + (−0.177 + 0.307i)13-s + (1.18 − 2.05i)14-s + (0.242 − 0.419i)15-s − 1.19·16-s + (0.373 − 0.646i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 + 0.856i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.86255 - 3.30002i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86255 - 3.30002i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.18e8 + 8.65e7i)T \) |
| good | 2 | \( 1 + 42.2iT - 1.02e3T^{2} \) |
| 3 | \( 1 + (-235. + 135. i)T + (2.95e4 - 5.11e4i)T^{2} \) |
| 5 | \( 1 + (-1.17e3 + 677. i)T + (4.88e6 - 8.45e6i)T^{2} \) |
| 7 | \( 1 + (-2.61e4 - 1.50e4i)T + (1.41e8 + 2.44e8i)T^{2} \) |
| 11 | \( 1 - 2.29e5T + 2.59e10T^{2} \) |
| 13 | \( 1 + (6.59e4 - 1.14e5i)T + (-6.89e10 - 1.19e11i)T^{2} \) |
| 17 | \( 1 + (-5.29e5 + 9.17e5i)T + (-1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (-2.71e6 + 1.56e6i)T + (3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (-2.85e6 - 4.94e6i)T + (-2.07e13 + 3.58e13i)T^{2} \) |
| 29 | \( 1 + (3.41e6 + 1.97e6i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (1.98e7 + 3.44e7i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + (7.59e7 - 4.38e7i)T + (2.40e15 - 4.16e15i)T^{2} \) |
| 41 | \( 1 + 1.46e8T + 1.34e16T^{2} \) |
| 47 | \( 1 + 3.17e8T + 5.25e16T^{2} \) |
| 53 | \( 1 + (-1.12e8 - 1.94e8i)T + (-8.74e16 + 1.51e17i)T^{2} \) |
| 59 | \( 1 - 4.85e7T + 5.11e17T^{2} \) |
| 61 | \( 1 + (-4.04e8 - 2.33e8i)T + (3.56e17 + 6.17e17i)T^{2} \) |
| 67 | \( 1 + (7.18e8 + 1.24e9i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 + (-1.52e9 - 8.82e8i)T + (1.62e18 + 2.81e18i)T^{2} \) |
| 73 | \( 1 + (1.34e9 + 7.78e8i)T + (2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (9.09e8 - 1.57e9i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + (2.84e9 + 4.91e9i)T + (-7.75e18 + 1.34e19i)T^{2} \) |
| 89 | \( 1 + (-6.84e9 + 3.95e9i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 - 5.53e9T + 7.37e19T^{2} \) |
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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
−13.33422359917238775243672158139, −11.74494372753135925812434062143, −11.52695489963643506379782301084, −9.511754939214272810726731112363, −8.787685977327681068613927007096, −7.30917591240559102157373591992, −5.09698509624971474227791024237, −3.30369604660417533049373330610, −1.91216149929544048239989367113, −1.42400380215240351715673373606,
1.59923809728615515028682283691, 3.72689180363797022053213689560, 5.08858549067710345322930720033, 6.71668334990761379927795924218, 7.951440853986530106741201659632, 8.768791024111175631972577930811, 10.22617991815724256000331853703, 11.68192215274559483230587860981, 13.94127424058769745984298382535, 14.45401523138240300572070482197
