| L(s) = 1 | + (5.84 − 9.68i)2-s + (3.37 − 8.14i)3-s + (−59.6 − 113. i)4-s + (482. − 200. i)5-s + (−59.2 − 80.3i)6-s + (−141. + 141. i)7-s + (−1.44e3 − 84.0i)8-s + (1.49e3 + 1.49e3i)9-s + (885. − 5.84e3i)10-s + (−2.97e3 − 7.19e3i)11-s + (−1.12e3 + 103. i)12-s + (−2.56e3 − 1.06e3i)13-s + (542. + 2.19e3i)14-s − 4.61e3i·15-s + (−9.26e3 + 1.35e4i)16-s + 6.72e3i·17-s + ⋯ |
| L(s) = 1 | + (0.516 − 0.856i)2-s + (0.0721 − 0.174i)3-s + (−0.466 − 0.884i)4-s + (1.72 − 0.715i)5-s + (−0.111 − 0.151i)6-s + (−0.155 + 0.155i)7-s + (−0.998 − 0.0580i)8-s + (0.681 + 0.681i)9-s + (0.279 − 1.84i)10-s + (−0.674 − 1.62i)11-s + (−0.187 + 0.0173i)12-s + (−0.324 − 0.134i)13-s + (0.0528 + 0.213i)14-s − 0.352i·15-s + (−0.565 + 0.824i)16-s + 0.332i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 + 0.849i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.526 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.29756 - 2.33093i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29756 - 2.33093i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-5.84 + 9.68i)T \) |
| good | 3 | \( 1 + (-3.37 + 8.14i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (-482. + 200. i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (141. - 141. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (2.97e3 + 7.19e3i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (2.56e3 + 1.06e3i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 - 6.72e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (1.95e4 + 8.08e3i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (-4.64e4 - 4.64e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (5.58e4 - 1.34e5i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 - 1.84e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-2.03e5 + 8.43e4i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (-4.91e5 - 4.91e5i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (1.89e5 + 4.57e5i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 3.09e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (5.25e5 + 1.26e6i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (-7.97e5 + 3.30e5i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (-2.04e5 + 4.94e5i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (1.18e6 - 2.85e6i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (2.14e6 - 2.14e6i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (-1.27e6 - 1.27e6i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 + 1.67e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (5.03e6 + 2.08e6i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (-2.21e5 + 2.21e5i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 - 9.62e6T + 8.07e13T^{2} \) |
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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
−14.29099241071790261823046444695, −13.19430367165665121874123544735, −12.97310552075066130105284645100, −10.95405607711417662170524062674, −9.916682521950009417278768906387, −8.708863843662902814807614253702, −6.04004395777687252965274686110, −5.01642674440778285715564155442, −2.61685192574315732766047719121, −1.18689113729211784265173935635,
2.46919024429968959352757057281, 4.65676105828264167431272890017, 6.24867518023419363189007167006, 7.21337995084534661561681618933, 9.400175361812453219085184552571, 10.19402374580282794819685098112, 12.53651597574404954499254381908, 13.39138093384338195851420754897, 14.63130810723593532183570145806, 15.29924789346274794758760887068
