| 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
−17.10305771219719913519151382937, −16.04348315326873391968435446339, −15.01857293654440530102585360035, −13.82060958929809595004921473993, −12.51704582045265324064312665090, −10.78812010540669186860689854711, −7.44465005476289107811086033642, −6.90075353806527080325920575155, −4.44179303512890033715718010268, −3.60346189402924595800812295863,
0.946846283286902254406905341763, 3.00635566067444410633832006764, 4.50821133552286010171939813511, 6.14842069844237699793124488445, 9.149819158963335732633535374984, 11.42253782180439332673321826885, 12.26102410996915262479138678262, 13.10901666056557414455362821113, 15.07277199794493653596057405760, 15.84632137662868701781887796872
