| L(s) = 1 | + (21.8 − 5.85i)2-s + (−161. + 181. i)3-s + (443. − 256i)4-s + (975. − 2.96e3i)5-s + (−2.45e3 + 4.91e3i)6-s + (−526. + 141. i)7-s + (8.19e3 − 8.19e3i)8-s + (−7.04e3 − 5.86e4i)9-s + (3.93e3 − 7.06e4i)10-s + (−1.24e5 + 2.14e5i)11-s + (−2.49e4 + 1.21e5i)12-s + (−4.16e5 − 1.11e5i)13-s + (−1.06e4 + 6.16e3i)14-s + (3.82e5 + 6.56e5i)15-s + (1.31e5 − 2.27e5i)16-s + (1.52e6 + 1.52e6i)17-s + ⋯ |
| L(s) = 1 | + (0.683 − 0.183i)2-s + (−0.663 + 0.748i)3-s + (0.433 − 0.250i)4-s + (0.312 − 0.950i)5-s + (−0.316 + 0.632i)6-s + (−0.0313 + 0.00839i)7-s + (0.249 − 0.250i)8-s + (−0.119 − 0.992i)9-s + (0.0393 − 0.706i)10-s + (−0.770 + 1.33i)11-s + (−0.100 + 0.489i)12-s + (−1.12 − 0.300i)13-s + (−0.0198 + 0.0114i)14-s + (0.503 + 0.863i)15-s + (0.124 − 0.216i)16-s + (1.07 + 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0805 - 0.996i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.0805 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.27335 + 1.17462i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27335 + 1.17462i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-21.8 + 5.85i)T \) |
| 3 | \( 1 + (161. - 181. i)T \) |
| 5 | \( 1 + (-975. + 2.96e3i)T \) |
| good | 7 | \( 1 + (526. - 141. i)T + (2.44e8 - 1.41e8i)T^{2} \) |
| 11 | \( 1 + (1.24e5 - 2.14e5i)T + (-1.29e10 - 2.24e10i)T^{2} \) |
| 13 | \( 1 + (4.16e5 + 1.11e5i)T + (1.19e11 + 6.89e10i)T^{2} \) |
| 17 | \( 1 + (-1.52e6 - 1.52e6i)T + 2.01e12iT^{2} \) |
| 19 | \( 1 + 3.11e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + (-3.17e6 - 8.51e5i)T + (3.58e13 + 2.07e13i)T^{2} \) |
| 29 | \( 1 + (-9.18e6 - 5.30e6i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (4.73e5 + 8.19e5i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + (-6.90e7 - 6.90e7i)T + 4.80e15iT^{2} \) |
| 41 | \( 1 + (-8.85e7 - 1.53e8i)T + (-6.71e15 + 1.16e16i)T^{2} \) |
| 43 | \( 1 + (-4.41e7 - 1.64e8i)T + (-1.87e16 + 1.08e16i)T^{2} \) |
| 47 | \( 1 + (3.21e8 - 8.61e7i)T + (4.55e16 - 2.62e16i)T^{2} \) |
| 53 | \( 1 + (4.44e8 - 4.44e8i)T - 1.74e17iT^{2} \) |
| 59 | \( 1 + (-6.93e8 + 4.00e8i)T + (2.55e17 - 4.42e17i)T^{2} \) |
| 61 | \( 1 + (5.70e8 - 9.88e8i)T + (-3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (-3.09e8 + 1.15e9i)T + (-1.57e18 - 9.11e17i)T^{2} \) |
| 71 | \( 1 - 1.81e9T + 3.25e18T^{2} \) |
| 73 | \( 1 + (1.83e9 - 1.83e9i)T - 4.29e18iT^{2} \) |
| 79 | \( 1 + (-1.59e8 - 9.22e7i)T + (4.73e18 + 8.19e18i)T^{2} \) |
| 83 | \( 1 + (4.38e8 + 1.63e9i)T + (-1.34e19 + 7.75e18i)T^{2} \) |
| 89 | \( 1 - 4.15e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (-1.26e10 + 3.39e9i)T + (6.38e19 - 3.68e19i)T^{2} \) |
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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
−12.60766592477478941438803484034, −11.39102870392730916218375070197, −10.09866243992819775958518921874, −9.540099082050216382125089322134, −7.77808712767740996021824238139, −6.22603145344592717432731023698, −4.93008642948371070669061463733, −4.63867097854195354232925317814, −2.82506066283577867445599139372, −1.17241977075242863785616010740,
0.39704244023156241960652442995, 2.18181708800229810040679317061, 3.24432935320111360684118697261, 5.20778625942768315547760550895, 5.97456480840270193248892187999, 7.12195962201109046404350397285, 7.941382179701597782286124847587, 9.993318619851952930386677251856, 11.03045145965292233018693758471, 11.88860055880854370148595949797
