| L(s) = 1 | + (23.8 + 7.75i)2-s + (59.7 − 43.4i)3-s + (302. + 219. i)4-s + (−128. − 396. i)5-s + (1.76e3 − 572. i)6-s + (−1.69e3 + 2.32e3i)7-s + (1.73e3 + 2.39e3i)8-s + (−342. + 1.05e3i)9-s − 1.04e4i·10-s + (1.45e4 + 1.30e3i)11-s + 2.76e4·12-s + (−4.03e4 − 1.31e4i)13-s + (−5.84e4 + 4.24e4i)14-s + (−2.49e4 − 1.80e4i)15-s + (−6.62e3 − 2.03e4i)16-s + (402. − 130. i)17-s + ⋯ |
| L(s) = 1 | + (1.49 + 0.484i)2-s + (0.737 − 0.535i)3-s + (1.18 + 0.858i)4-s + (−0.206 − 0.634i)5-s + (1.36 − 0.441i)6-s + (−0.704 + 0.969i)7-s + (0.424 + 0.584i)8-s + (−0.0521 + 0.160i)9-s − 1.04i·10-s + (0.996 + 0.0888i)11-s + 1.33·12-s + (−1.41 − 0.458i)13-s + (−1.52 + 1.10i)14-s + (−0.492 − 0.357i)15-s + (−0.101 − 0.311i)16-s + (0.00481 − 0.00156i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.38130 + 0.344820i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.38130 + 0.344820i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-1.45e4 - 1.30e3i)T \) |
| good | 2 | \( 1 + (-23.8 - 7.75i)T + (207. + 150. i)T^{2} \) |
| 3 | \( 1 + (-59.7 + 43.4i)T + (2.02e3 - 6.23e3i)T^{2} \) |
| 5 | \( 1 + (128. + 396. i)T + (-3.16e5 + 2.29e5i)T^{2} \) |
| 7 | \( 1 + (1.69e3 - 2.32e3i)T + (-1.78e6 - 5.48e6i)T^{2} \) |
| 13 | \( 1 + (4.03e4 + 1.31e4i)T + (6.59e8 + 4.79e8i)T^{2} \) |
| 17 | \( 1 + (-402. + 130. i)T + (5.64e9 - 4.10e9i)T^{2} \) |
| 19 | \( 1 + (3.60e4 + 4.96e4i)T + (-5.24e9 + 1.61e10i)T^{2} \) |
| 23 | \( 1 - 4.04e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + (-1.35e5 + 1.86e5i)T + (-1.54e11 - 4.75e11i)T^{2} \) |
| 31 | \( 1 + (4.88e5 - 1.50e6i)T + (-6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (-1.49e6 - 1.08e6i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (1.23e6 + 1.70e6i)T + (-2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 + 6.01e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (3.16e6 - 2.29e6i)T + (7.35e12 - 2.26e13i)T^{2} \) |
| 53 | \( 1 + (1.22e6 - 3.77e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (-2.94e6 - 2.13e6i)T + (4.53e13 + 1.39e14i)T^{2} \) |
| 61 | \( 1 + (4.45e6 - 1.44e6i)T + (1.55e14 - 1.12e14i)T^{2} \) |
| 67 | \( 1 - 1.98e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + (7.71e6 + 2.37e7i)T + (-5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (7.58e6 - 1.04e7i)T + (-2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (2.32e7 + 7.54e6i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (1.37e7 - 4.46e6i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + 8.22e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (7.27e6 - 2.23e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−19.10868073688628356318894462401, −16.80237401877210898567825888604, −15.34735935701062973559686127676, −14.33210583612427267991489411767, −12.90471899502064384528852004043, −12.20710360467327650527801306625, −8.950448696420752879643680865031, −6.96170268691474833810733501090, −5.05518324034266631281083022649, −2.84989860434050258203097824232,
3.03708345094189058389521787601, 4.20793739736565747024335477752, 6.76110791551603748321160454018, 9.647793139125305172999952989489, 11.36335215995211254486245022489, 12.95973807528151146647215634927, 14.46678467565725127505504097895, 14.85615670725967228894068613253, 16.83017220984524604684236748312, 19.39979797455634593208045594841
