| L(s) = 1 | + (−10.7 + 3.49i)2-s + (67.1 + 48.7i)3-s + (103. − 75.2i)4-s + (202. − 623. i)5-s + (−892. − 289. i)6-s + (−159. − 219. i)7-s + (−851. + 1.17e3i)8-s + (98.3 + 302. i)9-s + 7.41e3i·10-s + (1.31e4 − 6.53e3i)11-s + 1.06e4·12-s + (2.68e4 − 8.72e3i)13-s + (2.48e3 + 1.80e3i)14-s + (4.39e4 − 3.19e4i)15-s + (5.06e3 − 1.55e4i)16-s + (1.29e5 + 4.22e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (0.828 + 0.601i)3-s + (0.404 − 0.293i)4-s + (0.324 − 0.997i)5-s + (−0.688 − 0.223i)6-s + (−0.0663 − 0.0913i)7-s + (−0.207 + 0.286i)8-s + (0.0149 + 0.0461i)9-s + 0.741i·10-s + (0.894 − 0.446i)11-s + 0.511·12-s + (0.940 − 0.305i)13-s + (0.0646 + 0.0469i)14-s + (0.868 − 0.631i)15-s + (0.0772 − 0.237i)16-s + (1.55 + 0.505i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.78124 - 0.127986i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78124 - 0.127986i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (10.7 - 3.49i)T \) |
| 11 | \( 1 + (-1.31e4 + 6.53e3i)T \) |
| good | 3 | \( 1 + (-67.1 - 48.7i)T + (2.02e3 + 6.23e3i)T^{2} \) |
| 5 | \( 1 + (-202. + 623. i)T + (-3.16e5 - 2.29e5i)T^{2} \) |
| 7 | \( 1 + (159. + 219. i)T + (-1.78e6 + 5.48e6i)T^{2} \) |
| 13 | \( 1 + (-2.68e4 + 8.72e3i)T + (6.59e8 - 4.79e8i)T^{2} \) |
| 17 | \( 1 + (-1.29e5 - 4.22e4i)T + (5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (3.57e4 - 4.92e4i)T + (-5.24e9 - 1.61e10i)T^{2} \) |
| 23 | \( 1 + 2.71e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + (-1.74e5 - 2.39e5i)T + (-1.54e11 + 4.75e11i)T^{2} \) |
| 31 | \( 1 + (2.58e5 + 7.95e5i)T + (-6.90e11 + 5.01e11i)T^{2} \) |
| 37 | \( 1 + (1.00e6 - 7.30e5i)T + (1.08e12 - 3.34e12i)T^{2} \) |
| 41 | \( 1 + (7.64e5 - 1.05e6i)T + (-2.46e12 - 7.59e12i)T^{2} \) |
| 43 | \( 1 + 3.02e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (1.52e6 + 1.10e6i)T + (7.35e12 + 2.26e13i)T^{2} \) |
| 53 | \( 1 + (-2.46e6 - 7.59e6i)T + (-5.03e13 + 3.65e13i)T^{2} \) |
| 59 | \( 1 + (1.71e7 - 1.24e7i)T + (4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (7.11e6 + 2.31e6i)T + (1.55e14 + 1.12e14i)T^{2} \) |
| 67 | \( 1 + 1.27e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + (9.45e6 - 2.91e7i)T + (-5.22e14 - 3.79e14i)T^{2} \) |
| 73 | \( 1 + (-1.62e7 - 2.23e7i)T + (-2.49e14 + 7.66e14i)T^{2} \) |
| 79 | \( 1 + (4.16e7 - 1.35e7i)T + (1.22e15 - 8.91e14i)T^{2} \) |
| 83 | \( 1 + (-8.58e7 - 2.79e7i)T + (1.82e15 + 1.32e15i)T^{2} \) |
| 89 | \( 1 - 1.03e8T + 3.93e15T^{2} \) |
| 97 | \( 1 + (4.01e7 + 1.23e8i)T + (-6.34e15 + 4.60e15i)T^{2} \) |
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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
−16.29393354495795491288150000684, −14.97452688220891601752464396038, −13.81138178918658263818498515498, −12.11796253920144146617261683604, −10.21062225859235177884692053795, −9.065331675829510207779104093546, −8.221989761810910666545214160902, −5.91531337202190091656039542392, −3.68894895854317882126954964346, −1.20205398217713687846034576246,
1.67301090049288590804025538181, 3.18280404975332237606604808488, 6.51728734628022676733562158788, 7.79402096390657135704621249298, 9.227537562070935371927466052330, 10.62785728233254076267503862982, 12.13244734884856692106122782642, 13.81722202062380637222732548831, 14.62493621482654020024755154394, 16.30159682775052956002748372458
