| L(s) = 1 | + (−16.3 − 28.4i)2-s + (−281. + 488. i)4-s + (−657. + 1.13e3i)5-s + (−2.57e3 − 4.46e3i)7-s + 1.69e3·8-s + 4.31e4·10-s + (7.56e3 + 1.31e4i)11-s + (−9.02e4 + 1.56e5i)13-s + (−8.45e4 + 1.46e5i)14-s + (1.16e5 + 2.01e5i)16-s + 5.95e5·17-s − 7.85e5·19-s + (−3.70e5 − 6.42e5i)20-s + (2.48e5 − 4.29e5i)22-s + (5.87e5 − 1.01e6i)23-s + ⋯ |
| L(s) = 1 | + (−0.724 − 1.25i)2-s + (−0.550 + 0.953i)4-s + (−0.470 + 0.815i)5-s + (−0.406 − 0.703i)7-s + 0.146·8-s + 1.36·10-s + (0.155 + 0.269i)11-s + (−0.876 + 1.51i)13-s + (−0.588 + 1.01i)14-s + (0.444 + 0.769i)16-s + 1.73·17-s − 1.38·19-s + (−0.518 − 0.897i)20-s + (0.225 − 0.391i)22-s + (0.437 − 0.758i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.211351 - 0.580683i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.211351 - 0.580683i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (16.3 + 28.4i)T + (-256 + 443. i)T^{2} \) |
| 5 | \( 1 + (657. - 1.13e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 + (2.57e3 + 4.46e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-7.56e3 - 1.31e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (9.02e4 - 1.56e5i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 - 5.95e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.85e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + (-5.87e5 + 1.01e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (7.71e5 + 1.33e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + (-1.13e6 + 1.96e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + 1.12e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-8.36e6 + 1.44e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-1.55e7 - 2.70e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (9.67e6 + 1.67e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + 3.77e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (6.88e7 - 1.19e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (9.22e7 + 1.59e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-7.19e7 + 1.24e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 2.34e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.57e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (2.34e7 + 4.06e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (1.22e8 + 2.12e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 - 1.10e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + (1.48e8 + 2.57e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−11.86083578756564431001024469726, −10.83805381560855583872294308934, −10.08959201178591742277153735279, −9.124128604239703772276189638980, −7.61487703813588686059904269660, −6.52344108931249319882192299693, −4.22769010719140434162750189650, −3.11111356171392011906069762585, −1.84134361522048391132144640682, −0.32106350409031836135293643095,
0.800586134874715879831539781135, 3.11089624055602592711906574142, 5.12687328243705742334798788586, 5.98382788295274525885532946983, 7.48055736920411863636609849389, 8.279562266213614368794160568693, 9.171155911402733431205675618689, 10.30112383207799656962035265251, 12.17939384521249419364364239672, 12.69061894068133089005077413955
