| L(s) = 1 | + (12.1 + 8.81i)2-s + (29.9 + 92.0i)4-s + (3.10 − 2.25i)5-s + (300. + 925. i)7-s + (144. − 445. i)8-s + 57.5·10-s + (3.92e3 − 2.01e3i)11-s + (7.88e3 + 5.72e3i)13-s + (−4.50e3 + 1.38e4i)14-s + (1.56e4 − 1.14e4i)16-s + (−2.98e4 + 2.16e4i)17-s + (−3.26e3 + 1.00e4i)19-s + (300. + 218. i)20-s + (6.53e4 + 1.02e4i)22-s + 1.34e4·23-s + ⋯ |
| L(s) = 1 | + (1.07 + 0.778i)2-s + (0.233 + 0.719i)4-s + (0.0111 − 0.00807i)5-s + (0.331 + 1.01i)7-s + (0.0998 − 0.307i)8-s + 0.0182·10-s + (0.890 − 0.455i)11-s + (0.994 + 0.722i)13-s + (−0.438 + 1.35i)14-s + (0.958 − 0.696i)16-s + (−1.47 + 1.06i)17-s + (−0.109 + 0.335i)19-s + (0.00840 + 0.00610i)20-s + (1.30 + 0.204i)22-s + 0.229·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.44957 + 2.94453i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.44957 + 2.94453i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-3.92e3 + 2.01e3i)T \) |
| good | 2 | \( 1 + (-12.1 - 8.81i)T + (39.5 + 121. i)T^{2} \) |
| 5 | \( 1 + (-3.10 + 2.25i)T + (2.41e4 - 7.43e4i)T^{2} \) |
| 7 | \( 1 + (-300. - 925. i)T + (-6.66e5 + 4.84e5i)T^{2} \) |
| 13 | \( 1 + (-7.88e3 - 5.72e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (2.98e4 - 2.16e4i)T + (1.26e8 - 3.90e8i)T^{2} \) |
| 19 | \( 1 + (3.26e3 - 1.00e4i)T + (-7.23e8 - 5.25e8i)T^{2} \) |
| 23 | \( 1 - 1.34e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-1.27e4 - 3.93e4i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-7.63e4 - 5.54e4i)T + (8.50e9 + 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-1.45e5 - 4.47e5i)T + (-7.68e10 + 5.57e10i)T^{2} \) |
| 41 | \( 1 + (1.36e4 - 4.19e4i)T + (-1.57e11 - 1.14e11i)T^{2} \) |
| 43 | \( 1 - 3.21e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-5.93e4 + 1.82e5i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (7.01e5 + 5.09e5i)T + (3.63e11 + 1.11e12i)T^{2} \) |
| 59 | \( 1 + (9.53e5 + 2.93e6i)T + (-2.01e12 + 1.46e12i)T^{2} \) |
| 61 | \( 1 + (3.70e5 - 2.69e5i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + 1.21e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (7.21e5 - 5.23e5i)T + (2.81e12 - 8.64e12i)T^{2} \) |
| 73 | \( 1 + (9.10e5 + 2.80e6i)T + (-8.93e12 + 6.49e12i)T^{2} \) |
| 79 | \( 1 + (-5.31e6 - 3.86e6i)T + (5.93e12 + 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-6.31e6 + 4.58e6i)T + (8.38e12 - 2.58e13i)T^{2} \) |
| 89 | \( 1 - 1.00e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-8.21e5 - 5.96e5i)T + (2.49e13 + 7.68e13i)T^{2} \) |
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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
−13.09080323596796057152846470963, −11.96318060752641342778025464834, −10.98759974114324621750967573168, −9.245303323431978558127994921819, −8.323461330562402863451709711703, −6.59730731483961905511695475824, −6.03335616424162768681123153685, −4.69376650076404697697252185476, −3.56673556837880326068581016649, −1.60332789899150182899916803180,
0.906293865235846075433940659654, 2.44035389666899899980084869599, 3.93478231405937375005426715208, 4.62972113784591174878135681972, 6.23079234051721773332570790615, 7.60926214656589597352705420064, 9.045015932635681173672872987440, 10.58966890417125406373771776581, 11.22972531714248773773487411409, 12.24743362562839791438731996433
