| L(s) = 1 | + (4.40 − 1.43i)2-s + (4.20 + 3.05i)3-s + (4.41 − 3.20i)4-s + (12.7 − 39.2i)5-s + (22.8 + 7.43i)6-s + (37.7 + 51.9i)7-s + (−28.7 + 39.5i)8-s + (8.34 + 25.6i)9-s − 191. i·10-s + (−92.1 − 78.3i)11-s + 28.3·12-s + (−152. + 49.5i)13-s + (240. + 174. i)14-s + (173. − 126. i)15-s + (−96.8 + 298. i)16-s + (−307. − 99.9i)17-s + ⋯ |
| L(s) = 1 | + (1.10 − 0.357i)2-s + (0.467 + 0.339i)3-s + (0.275 − 0.200i)4-s + (0.510 − 1.57i)5-s + (0.635 + 0.206i)6-s + (0.770 + 1.06i)7-s + (−0.448 + 0.617i)8-s + (0.103 + 0.317i)9-s − 1.91i·10-s + (−0.761 − 0.647i)11-s + 0.196·12-s + (−0.902 + 0.293i)13-s + (1.22 + 0.892i)14-s + (0.771 − 0.560i)15-s + (−0.378 + 1.16i)16-s + (−1.06 − 0.345i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.58387 - 0.481426i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.58387 - 0.481426i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.20 - 3.05i)T \) |
| 11 | \( 1 + (92.1 + 78.3i)T \) |
| good | 2 | \( 1 + (-4.40 + 1.43i)T + (12.9 - 9.40i)T^{2} \) |
| 5 | \( 1 + (-12.7 + 39.2i)T + (-505. - 367. i)T^{2} \) |
| 7 | \( 1 + (-37.7 - 51.9i)T + (-741. + 2.28e3i)T^{2} \) |
| 13 | \( 1 + (152. - 49.5i)T + (2.31e4 - 1.67e4i)T^{2} \) |
| 17 | \( 1 + (307. + 99.9i)T + (6.75e4 + 4.90e4i)T^{2} \) |
| 19 | \( 1 + (32.3 - 44.5i)T + (-4.02e4 - 1.23e5i)T^{2} \) |
| 23 | \( 1 - 933.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-434. - 598. i)T + (-2.18e5 + 6.72e5i)T^{2} \) |
| 31 | \( 1 + (39.7 + 122. i)T + (-7.47e5 + 5.42e5i)T^{2} \) |
| 37 | \( 1 + (-1.41e3 + 1.02e3i)T + (5.79e5 - 1.78e6i)T^{2} \) |
| 41 | \( 1 + (-335. + 462. i)T + (-8.73e5 - 2.68e6i)T^{2} \) |
| 43 | \( 1 + 335. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (1.92e3 + 1.39e3i)T + (1.50e6 + 4.64e6i)T^{2} \) |
| 53 | \( 1 + (-34.1 - 104. i)T + (-6.38e6 + 4.63e6i)T^{2} \) |
| 59 | \( 1 + (-735. + 534. i)T + (3.74e6 - 1.15e7i)T^{2} \) |
| 61 | \( 1 + (664. + 215. i)T + (1.12e7 + 8.13e6i)T^{2} \) |
| 67 | \( 1 - 1.93e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (505. - 1.55e3i)T + (-2.05e7 - 1.49e7i)T^{2} \) |
| 73 | \( 1 + (-1.91e3 - 2.62e3i)T + (-8.77e6 + 2.70e7i)T^{2} \) |
| 79 | \( 1 + (2.56e3 - 832. i)T + (3.15e7 - 2.28e7i)T^{2} \) |
| 83 | \( 1 + (-7.25e3 - 2.35e3i)T + (3.83e7 + 2.78e7i)T^{2} \) |
| 89 | \( 1 + 1.45e4T + 6.27e7T^{2} \) |
| 97 | \( 1 + (1.57e3 + 4.85e3i)T + (-7.16e7 + 5.20e7i)T^{2} \) |
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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
−15.58112222162757343541630362091, −14.48293674350444544863132565185, −13.28731841741156592185426119922, −12.57605998026024471059546962031, −11.32143674134410880470755229561, −9.144782051125650612160380357861, −8.467793386972250850332382964590, −5.34761819735312201020387610593, −4.74637679967534220622316027368, −2.41372841387195210755005429878,
2.77052923453177417532643798271, 4.66095163874162458690359695371, 6.60690045496577204046409960942, 7.49700376907636485191079870342, 9.896529632377686917980610986117, 11.05324288674377033746277503453, 12.97364424626803578377323542132, 13.77251390632703118898003202530, 14.76866570460498292609299818402, 15.17408659870761278767805384359
