L(s) = 1 | + (−5.45 − 1.48i)2-s + (3.65 + 3.65i)3-s + (27.6 + 16.1i)4-s + (−18.9 − 52.5i)5-s + (−14.5 − 25.3i)6-s + (162. − 162. i)7-s + (−126. − 129. i)8-s − 216. i·9-s + (25.7 + 315. i)10-s + 539. i·11-s + (41.8 + 160. i)12-s + (297. − 297. i)13-s + (−1.13e3 + 648. i)14-s + (122. − 261. i)15-s + (501. + 892. i)16-s + (−350. − 350. i)17-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.261i)2-s + (0.234 + 0.234i)3-s + (0.863 + 0.505i)4-s + (−0.339 − 0.940i)5-s + (−0.165 − 0.287i)6-s + (1.25 − 1.25i)7-s + (−0.700 − 0.713i)8-s − 0.889i·9-s + (0.0814 + 0.996i)10-s + 1.34i·11-s + (0.0839 + 0.320i)12-s + (0.488 − 0.488i)13-s + (−1.54 + 0.884i)14-s + (0.141 − 0.300i)15-s + (0.489 + 0.871i)16-s + (−0.293 − 0.293i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.829665 - 0.567888i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.829665 - 0.567888i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (5.45 + 1.48i)T \) |
| 5 | \( 1 + (18.9 + 52.5i)T \) |
good | 3 | \( 1 + (-3.65 - 3.65i)T + 243iT^{2} \) |
| 7 | \( 1 + (-162. + 162. i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 - 539. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-297. + 297. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (350. + 350. i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 + 220.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (314. + 314. i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 - 1.57e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.10e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (-3.96e3 - 3.96e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 8.30e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-8.38e3 - 8.38e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (3.66e3 - 3.66e3i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (657. - 657. i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 - 2.99e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.20e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (2.90e4 - 2.90e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 3.94e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (3.44e4 - 3.44e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 + 1.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (7.49e3 + 7.49e3i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + 6.07e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (7.40e4 + 7.40e4i)T + 8.58e9iT^{2} \) |
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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
−17.39629464036593457781967126022, −15.99167310503911369592003253744, −14.72830682465063910727411629641, −12.75232953415127353107661235948, −11.42501152920982009915483097529, −9.995398200979599892615337665436, −8.567335076918236624365247854116, −7.33003349708165727445600367166, −4.26834529643224497079728503954, −1.12671489703964996035028392121,
2.26099711757002984713237222359, 5.91432525089648554068990198733, 7.81367365436285104640636899527, 8.727216473105348867146382020190, 10.89000357815974607524934784984, 11.54420416186349848357380179026, 14.04387039620281849518631637614, 15.09325388391720766875130246817, 16.25405741588500963237923384240, 17.85641553777543844722039813139