| L(s) = 1 | + (25.4 + 14.6i)2-s + (304 + 526. i)4-s + (712. − 411. i)5-s + (−983.5 + 1.70e3i)7-s + 1.03e4i·8-s + 2.41e4·10-s + (−1.08e4 − 6.29e3i)11-s + (2.27e4 + 3.94e4i)13-s + (−5.00e4 + 2.89e4i)14-s + (−7.42e4 + 1.28e5i)16-s + 5.96e4i·17-s + 1.52e5·19-s + (4.33e5 + 2.50e5i)20-s + (−1.84e5 − 3.20e5i)22-s + (1.13e5 − 6.56e4i)23-s + ⋯ |
| L(s) = 1 | + (1.59 + 0.918i)2-s + (1.18 + 2.05i)4-s + (1.14 − 0.658i)5-s + (−0.409 + 0.709i)7-s + 2.52i·8-s + 2.41·10-s + (−0.744 − 0.429i)11-s + (0.796 + 1.37i)13-s + (−1.30 + 0.752i)14-s + (−1.13 + 1.96i)16-s + 0.713i·17-s + 1.16·19-s + (2.70 + 1.56i)20-s + (−0.789 − 1.36i)22-s + (0.406 − 0.234i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.40355 + 4.86078i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.40355 + 4.86078i\) |
| \(L(5)\) |
|
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 | \( 1 + (-25.4 - 14.6i)T + (128 + 221. i)T^{2} \) |
| 5 | \( 1 + (-712. + 411. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + (983.5 - 1.70e3i)T + (-2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (1.08e4 + 6.29e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + (-2.27e4 - 3.94e4i)T + (-4.07e8 + 7.06e8i)T^{2} \) |
| 17 | \( 1 - 5.96e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.52e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + (-1.13e5 + 6.56e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (5.09e5 + 2.94e5i)T + (2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-8.21e4 - 1.42e5i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + 6.63e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + (-8.12e5 + 4.69e5i)T + (3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (2.87e5 - 4.98e5i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (7.99e6 + 4.61e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + 1.03e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-4.36e6 + 2.51e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-9.60e6 + 1.66e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-2.99e5 - 5.17e5i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 2.92e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.28e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-1.17e7 + 2.04e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-2.89e7 - 1.67e7i)T + (1.12e15 + 1.95e15i)T^{2} \) |
| 89 | \( 1 - 2.82e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (6.82e7 - 1.18e8i)T + (-3.91e15 - 6.78e15i)T^{2} \) |
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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
−13.34450120501815180417643526288, −12.49481343785249503135280411659, −11.35316681000568789343291472602, −9.467241754996192991242606249924, −8.298272524065041022044410712408, −6.66325230025739260424999353779, −5.79737468239884109229724063311, −5.02203283637672483115106277881, −3.47718258096914453659246642738, −1.99399241660350366866713959910,
1.11637042076213741673761342588, 2.61336520083607999854862484340, 3.43524444481153067024501430182, 5.13146208090468048727092180639, 5.95775917723816383732082545175, 7.24386836750511443779637780404, 9.798896288008967586185600070921, 10.41571170682040141510629263326, 11.30431461026446242842591964433, 12.79050577280421776201627179869
