| L(s) = 1 | + (−1.23 + 3.80i)2-s + (−14.1 − 10.2i)3-s + (−12.9 − 9.40i)4-s + (55.6 + 5.53i)5-s + (56.6 − 41.1i)6-s − 46.1·7-s + (51.7 − 37.6i)8-s + (19.6 + 60.3i)9-s + (−89.8 + 204. i)10-s + (−156. + 482. i)11-s + (86.5 + 266. i)12-s + (337. + 1.03e3i)13-s + (57.0 − 175. i)14-s + (−730. − 650. i)15-s + (79.1 + 243. i)16-s + (−1.65e3 + 1.20e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.908 − 0.660i)3-s + (−0.404 − 0.293i)4-s + (0.995 + 0.0990i)5-s + (0.642 − 0.466i)6-s − 0.356·7-s + (0.286 − 0.207i)8-s + (0.0806 + 0.248i)9-s + (−0.284 + 0.647i)10-s + (−0.390 + 1.20i)11-s + (0.173 + 0.534i)12-s + (0.554 + 1.70i)13-s + (0.0778 − 0.239i)14-s + (−0.838 − 0.746i)15-s + (0.0772 + 0.237i)16-s + (−1.38 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.482587 + 0.702562i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.482587 + 0.702562i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 3.80i)T \) |
| 5 | \( 1 + (-55.6 - 5.53i)T \) |
| good | 3 | \( 1 + (14.1 + 10.2i)T + (75.0 + 231. i)T^{2} \) |
| 7 | \( 1 + 46.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + (156. - 482. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-337. - 1.03e3i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (1.65e3 - 1.20e3i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.12e3 + 813. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-99.7 + 306. i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-4.83e3 - 3.51e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-1.78e3 + 1.29e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-1.69e3 - 5.22e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-646. - 1.98e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.66e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.67e4 + 1.21e4i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.28e4 - 9.30e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.37e3 - 4.22e3i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.10e4 + 3.41e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (2.20e4 - 1.59e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-2.20e4 - 1.60e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.93e3 - 5.94e3i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (3.63e4 + 2.64e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-1.31e4 + 9.55e3i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (1.17e4 - 3.63e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (5.04e3 + 3.66e3i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−14.95944585796099173209028646908, −13.63852608043441813210518244117, −12.80859545498476572633816443328, −11.43674906548569748034567820487, −10.02109345318322263175702175842, −8.869891567796969024587485905142, −6.77820960055972268311084447797, −6.46760636172901534735430097309, −4.84461678428364103781183727083, −1.67670004639608273850661442271,
0.53346491688014204711325681767, 2.95726849338653901872476427579, 5.05950490432610958441080935386, 6.06864850782711712996847058006, 8.387272066911358059577440143490, 9.819173074097067294133593583167, 10.60659034005206706075756095582, 11.50387585181146401913438746347, 13.08033245949933177965581617600, 13.76524720989397936768279042511
