| L(s) = 1 | + (−498. + 862. i)5-s + (−2.85e3 + 1.64e3i)7-s + (−1.11e4 + 6.43e3i)11-s + (−1.02e4 + 1.77e4i)13-s + 6.41e4·17-s − 1.76e5i·19-s + (−2.29e5 − 1.32e5i)23-s + (−3.01e5 − 5.21e5i)25-s + (9.65e3 + 1.67e4i)29-s + (−5.68e5 − 3.28e5i)31-s − 3.28e6i·35-s − 3.47e6·37-s + (9.83e5 − 1.70e6i)41-s + (−1.64e6 + 9.46e5i)43-s + (−6.98e6 + 4.03e6i)47-s + ⋯ |
| L(s) = 1 | + (−0.797 + 1.38i)5-s + (−1.18 + 0.685i)7-s + (−0.761 + 0.439i)11-s + (−0.359 + 0.622i)13-s + 0.768·17-s − 1.35i·19-s + (−0.819 − 0.473i)23-s + (−0.770 − 1.33i)25-s + (0.0136 + 0.0236i)29-s + (−0.615 − 0.355i)31-s − 2.18i·35-s − 1.85·37-s + (0.347 − 0.602i)41-s + (−0.479 + 0.276i)43-s + (−1.43 + 0.827i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.2990234928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2990234928\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (498. - 862. i)T + (-1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + (2.85e3 - 1.64e3i)T + (2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (1.11e4 - 6.43e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + (1.02e4 - 1.77e4i)T + (-4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 - 6.41e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.76e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (2.29e5 + 1.32e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-9.65e3 - 1.67e4i)T + (-2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (5.68e5 + 3.28e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + 3.47e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (-9.83e5 + 1.70e6i)T + (-3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (1.64e6 - 9.46e5i)T + (5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (6.98e6 - 4.03e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 - 1.33e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + (-2.37e6 - 1.37e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (1.12e7 + 1.94e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (6.00e6 + 3.46e6i)T + (2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 2.25e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.35e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (4.63e7 - 2.67e7i)T + (7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-8.20e6 + 4.73e6i)T + (1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 - 6.23e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (-3.88e7 - 6.73e7i)T + (-3.91e15 + 6.78e15i)T^{2} \) |
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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
−9.957512727720142774687164179692, −9.045324184205474608206854388589, −7.82018825433116018162146632533, −7.00118057873768690605731247211, −6.39897432274957091633924275308, −5.16014112719784789469147334218, −3.80487615638563122431257001982, −2.95508888897649789360348026329, −2.24144376353635422039987036213, −0.13808270696038522114779375588,
0.37808653981541287011510573084, 1.49920882717696948039467637903, 3.27711330806268240241277723409, 3.84996355705094813803047257052, 5.10843417791472947806540865567, 5.85876057144737240930828017269, 7.26948094505582159892573292173, 7.998730737291242103199995087006, 8.749194094315499125274315942718, 9.953544745851088226383904690137
