| L(s) = 1 | + (−17.0 − 9.86i)2-s + (51.0 − 62.9i)3-s + (66.7 + 115. i)4-s + (−896. + 517. i)5-s + (−1.49e3 + 571. i)6-s + (−21.9 + 37.9i)7-s + 2.41e3i·8-s + (−1.35e3 − 6.42e3i)9-s + 2.04e4·10-s + (−1.52e4 − 8.81e3i)11-s + (1.06e4 + 1.70e3i)12-s + (616. + 1.06e3i)13-s + (749. − 432. i)14-s + (−1.31e4 + 8.28e4i)15-s + (4.09e4 − 7.09e4i)16-s − 1.06e5i·17-s + ⋯ |
| L(s) = 1 | + (−1.06 − 0.616i)2-s + (0.630 − 0.776i)3-s + (0.260 + 0.451i)4-s + (−1.43 + 0.828i)5-s + (−1.15 + 0.441i)6-s + (−0.00912 + 0.0158i)7-s + 0.590i·8-s + (−0.206 − 0.978i)9-s + 2.04·10-s + (−1.04 − 0.601i)11-s + (0.515 + 0.0820i)12-s + (0.0215 + 0.0373i)13-s + (0.0194 − 0.0112i)14-s + (−0.260 + 1.63i)15-s + (0.624 − 1.08i)16-s − 1.27i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 - 0.471i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.882 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0633655 + 0.253176i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0633655 + 0.253176i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-51.0 + 62.9i)T \) |
| good | 2 | \( 1 + (17.0 + 9.86i)T + (128 + 221. i)T^{2} \) |
| 5 | \( 1 + (896. - 517. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + (21.9 - 37.9i)T + (-2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (1.52e4 + 8.81e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + (-616. - 1.06e3i)T + (-4.07e8 + 7.06e8i)T^{2} \) |
| 17 | \( 1 + 1.06e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.15e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + (9.76e4 - 5.63e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-7.20e5 - 4.16e5i)T + (2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-1.28e5 - 2.22e5i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + 2.84e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (-2.51e6 + 1.45e6i)T + (3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-7.91e5 + 1.37e6i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-1.36e6 - 7.89e5i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + 8.77e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (8.09e6 - 4.67e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (2.86e6 - 4.95e6i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-3.95e6 - 6.84e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 3.25e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.75e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-6.16e5 + 1.06e6i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (4.84e7 + 2.79e7i)T + (1.12e15 + 1.95e15i)T^{2} \) |
| 89 | \( 1 - 4.18e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-2.48e7 + 4.29e7i)T + (-3.91e15 - 6.78e15i)T^{2} \) |
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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
−18.89200333782889586432590859174, −17.98765100059290856592367517967, −15.72118680089239663948794609251, −14.14878825765288354878143629960, −12.01425983853708542365487001134, −10.70871761488426160922311788213, −8.581958206150351068275073155882, −7.41543449397807465998458711983, −2.85433045701350068535550668820, −0.23684077861348057082174797875,
4.21489295913894054357223656485, 7.83783673958618081048095724964, 8.621840052505067394793002314391, 10.38185217783966470807001029596, 12.69494430352789930654752051418, 15.26575439999466914250530384201, 15.89888317129827816303370365280, 17.12535361461676348565427998114, 19.02479834296442669621689035824, 19.94591603185630488708169957194
