| L(s) = 1 | + (11.3 + 0.284i)2-s + (33.6 − 13.9i)3-s + (127. + 6.42i)4-s + (117. − 283. i)5-s + (385. − 148. i)6-s + (−590. − 590. i)7-s + (1.44e3 + 109. i)8-s + (−606. + 606. i)9-s + (1.40e3 − 3.16e3i)10-s + (1.81e3 + 751. i)11-s + (4.39e3 − 1.56e3i)12-s + (9.99e2 + 2.41e3i)13-s + (−6.50e3 − 6.84e3i)14-s − 1.11e4i·15-s + (1.63e4 + 1.64e3i)16-s − 4.45e3i·17-s + ⋯ |
| L(s) = 1 | + (0.999 + 0.0251i)2-s + (0.720 − 0.298i)3-s + (0.998 + 0.0502i)4-s + (0.419 − 1.01i)5-s + (0.727 − 0.280i)6-s + (−0.650 − 0.650i)7-s + (0.997 + 0.0753i)8-s + (−0.277 + 0.277i)9-s + (0.444 − 1.00i)10-s + (0.410 + 0.170i)11-s + (0.734 − 0.261i)12-s + (0.126 + 0.304i)13-s + (−0.633 − 0.666i)14-s − 0.855i·15-s + (0.994 + 0.100i)16-s − 0.219i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.68046 - 1.37416i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.68046 - 1.37416i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-11.3 - 0.284i)T \) |
| good | 3 | \( 1 + (-33.6 + 13.9i)T + (1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (-117. + 283. i)T + (-5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (590. + 590. i)T + 8.23e5iT^{2} \) |
| 11 | \( 1 + (-1.81e3 - 751. i)T + (1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (-9.99e2 - 2.41e3i)T + (-4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 + 4.45e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (-1.23e4 - 2.98e4i)T + (-6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (-9.86e3 + 9.86e3i)T - 3.40e9iT^{2} \) |
| 29 | \( 1 + (-4.67e4 + 1.93e4i)T + (1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 1.77e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (2.04e5 - 4.93e5i)T + (-6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (4.79e5 - 4.79e5i)T - 1.94e11iT^{2} \) |
| 43 | \( 1 + (5.45e5 + 2.26e5i)T + (1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 + 8.38e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-9.99e5 - 4.14e5i)T + (8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (-1.02e6 + 2.47e6i)T + (-1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (-3.06e6 + 1.26e6i)T + (2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (3.07e6 - 1.27e6i)T + (4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (1.39e6 + 1.39e6i)T + 9.09e12iT^{2} \) |
| 73 | \( 1 + (1.18e6 - 1.18e6i)T - 1.10e13iT^{2} \) |
| 79 | \( 1 - 1.39e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (9.71e5 + 2.34e6i)T + (-1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (-4.99e6 - 4.99e6i)T + 4.42e13iT^{2} \) |
| 97 | \( 1 - 1.14e7T + 8.07e13T^{2} \) |
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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
−14.76141862076897517789605425568, −13.66407823302896283924109669061, −13.09726907276058238443536519073, −11.80589816483638042431469105056, −10.01167955025884360339331712223, −8.414285930959523465651238821968, −6.86141470647104462798902166983, −5.18612329174519198669844769063, −3.49608428316822485439801317019, −1.62878756414590512643202890533,
2.54979562858406402991259323577, 3.51643035169450059118613406519, 5.75865431826575592783576009909, 6.97176223418660467470917586554, 9.017926675560095905323669986455, 10.46545213888010553210264112476, 11.81287334685546926386281538525, 13.23155830094956166115643561425, 14.33292135556367396704558504685, 15.03263505154034364759195105859
