| L(s) = 1 | + (137. − 99.5i)2-s + (−84.9 − 261. i)3-s + (6.33e3 − 1.94e4i)4-s + (−3.73e4 − 2.71e4i)5-s + (−3.76e4 − 2.73e4i)6-s + (−1.56e5 + 4.80e5i)7-s + (−6.43e5 − 1.98e6i)8-s + (1.22e6 − 8.92e5i)9-s − 7.82e6·10-s + (3.31e6 − 4.85e6i)11-s − 5.63e6·12-s + (2.33e6 − 1.69e6i)13-s + (2.64e7 + 8.14e7i)14-s + (−3.92e6 + 1.20e7i)15-s + (−1.49e8 − 1.08e8i)16-s + (−1.37e7 − 1.00e7i)17-s + ⋯ |
| L(s) = 1 | + (1.51 − 1.09i)2-s + (−0.0672 − 0.206i)3-s + (0.773 − 2.37i)4-s + (−1.07 − 0.777i)5-s + (−0.329 − 0.239i)6-s + (−0.501 + 1.54i)7-s + (−0.868 − 2.67i)8-s + (0.770 − 0.559i)9-s − 2.47·10-s + (0.564 − 0.825i)11-s − 0.544·12-s + (0.134 − 0.0974i)13-s + (0.939 + 2.89i)14-s + (−0.0889 + 0.273i)15-s + (−2.23 − 1.62i)16-s + (−0.138 − 0.100i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.502555 - 3.12445i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.502555 - 3.12445i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-3.31e6 + 4.85e6i)T \) |
| good | 2 | \( 1 + (-137. + 99.5i)T + (2.53e3 - 7.79e3i)T^{2} \) |
| 3 | \( 1 + (84.9 + 261. i)T + (-1.28e6 + 9.37e5i)T^{2} \) |
| 5 | \( 1 + (3.73e4 + 2.71e4i)T + (3.77e8 + 1.16e9i)T^{2} \) |
| 7 | \( 1 + (1.56e5 - 4.80e5i)T + (-7.83e10 - 5.69e10i)T^{2} \) |
| 13 | \( 1 + (-2.33e6 + 1.69e6i)T + (9.35e13 - 2.88e14i)T^{2} \) |
| 17 | \( 1 + (1.37e7 + 1.00e7i)T + (3.06e15 + 9.41e15i)T^{2} \) |
| 19 | \( 1 + (3.66e7 + 1.12e8i)T + (-3.40e16 + 2.47e16i)T^{2} \) |
| 23 | \( 1 - 9.77e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (-2.99e8 + 9.22e8i)T + (-8.30e18 - 6.03e18i)T^{2} \) |
| 31 | \( 1 + (-2.82e9 + 2.04e9i)T + (7.54e18 - 2.32e19i)T^{2} \) |
| 37 | \( 1 + (2.36e9 - 7.27e9i)T + (-1.97e20 - 1.43e20i)T^{2} \) |
| 41 | \( 1 + (-1.42e10 - 4.37e10i)T + (-7.48e20 + 5.43e20i)T^{2} \) |
| 43 | \( 1 - 3.98e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-1.94e10 - 5.99e10i)T + (-4.41e21 + 3.20e21i)T^{2} \) |
| 53 | \( 1 + (4.85e10 - 3.52e10i)T + (8.04e21 - 2.47e22i)T^{2} \) |
| 59 | \( 1 + (6.89e10 - 2.12e11i)T + (-8.49e22 - 6.17e22i)T^{2} \) |
| 61 | \( 1 + (1.61e11 + 1.17e11i)T + (5.00e22 + 1.53e23i)T^{2} \) |
| 67 | \( 1 + 1.09e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-5.26e11 - 3.82e11i)T + (3.60e23 + 1.10e24i)T^{2} \) |
| 73 | \( 1 + (2.71e11 - 8.34e11i)T + (-1.35e24 - 9.82e23i)T^{2} \) |
| 79 | \( 1 + (-1.97e12 + 1.43e12i)T + (1.44e24 - 4.43e24i)T^{2} \) |
| 83 | \( 1 + (3.64e12 + 2.65e12i)T + (2.74e24 + 8.43e24i)T^{2} \) |
| 89 | \( 1 - 4.87e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (1.05e13 - 7.66e12i)T + (2.07e25 - 6.40e25i)T^{2} \) |
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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
−15.84632137662868701781887796872, −15.07277199794493653596057405760, −13.10901666056557414455362821113, −12.26102410996915262479138678262, −11.42253782180439332673321826885, −9.149819158963335732633535374984, −6.14842069844237699793124488445, −4.50821133552286010171939813511, −3.00635566067444410633832006764, −0.946846283286902254406905341763,
3.60346189402924595800812295863, 4.44179303512890033715718010268, 6.90075353806527080325920575155, 7.44465005476289107811086033642, 10.78812010540669186860689854711, 12.51704582045265324064312665090, 13.82060958929809595004921473993, 15.01857293654440530102585360035, 16.04348315326873391968435446339, 17.10305771219719913519151382937
