| L(s) = 1 | + (0.933 + 0.250i)2-s + (−4.52 − 2.56i)3-s + (−6.11 − 3.53i)4-s + (11.3 − 11.3i)5-s + (−3.58 − 3.52i)6-s + (−3.58 − 13.3i)7-s + (−10.2 − 10.2i)8-s + (13.8 + 23.1i)9-s + (13.4 − 7.73i)10-s + (−13.6 + 51.0i)11-s + (18.6 + 31.6i)12-s + (7.70 − 46.2i)13-s − 13.4i·14-s + (−80.1 + 22.1i)15-s + (21.2 + 36.7i)16-s + (39.8 − 69.1i)17-s + ⋯ |
| L(s) = 1 | + (0.330 + 0.0884i)2-s + (−0.870 − 0.492i)3-s + (−0.764 − 0.441i)4-s + (1.01 − 1.01i)5-s + (−0.243 − 0.239i)6-s + (−0.193 − 0.723i)7-s + (−0.455 − 0.455i)8-s + (0.514 + 0.857i)9-s + (0.423 − 0.244i)10-s + (−0.374 + 1.39i)11-s + (0.447 + 0.761i)12-s + (0.164 − 0.986i)13-s − 0.255i·14-s + (−1.38 + 0.381i)15-s + (0.331 + 0.574i)16-s + (0.569 − 0.985i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 + 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.697588 - 0.785506i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.697588 - 0.785506i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.52 + 2.56i)T \) |
| 13 | \( 1 + (-7.70 + 46.2i)T \) |
| good | 2 | \( 1 + (-0.933 - 0.250i)T + (6.92 + 4i)T^{2} \) |
| 5 | \( 1 + (-11.3 + 11.3i)T - 125iT^{2} \) |
| 7 | \( 1 + (3.58 + 13.3i)T + (-297. + 171.5i)T^{2} \) |
| 11 | \( 1 + (13.6 - 51.0i)T + (-1.15e3 - 665.5i)T^{2} \) |
| 17 | \( 1 + (-39.8 + 69.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-77.1 + 20.6i)T + (5.94e3 - 3.42e3i)T^{2} \) |
| 23 | \( 1 + (-59.2 - 102. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-84.1 + 48.5i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (86.1 + 86.1i)T + 2.97e4iT^{2} \) |
| 37 | \( 1 + (-24.6 - 6.60i)T + (4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (220. + 59.0i)T + (5.96e4 + 3.44e4i)T^{2} \) |
| 43 | \( 1 + (34.1 + 19.7i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-79.3 - 79.3i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 - 509. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-134. + 35.9i)T + (1.77e5 - 1.02e5i)T^{2} \) |
| 61 | \( 1 + (352. - 609. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (32.5 - 121. i)T + (-2.60e5 - 1.50e5i)T^{2} \) |
| 71 | \( 1 + (-4.46 - 16.6i)T + (-3.09e5 + 1.78e5i)T^{2} \) |
| 73 | \( 1 + (-644. + 644. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + 106.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (136. - 136. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + (221. - 826. i)T + (-6.10e5 - 3.52e5i)T^{2} \) |
| 97 | \( 1 + (-957. + 256. i)T + (7.90e5 - 4.56e5i)T^{2} \) |
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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
−15.49023876886006394504254717886, −13.71506505206968305271485957443, −13.20789355470864568276684506653, −12.24186201647953932101803570892, −10.26714202864012539893449345772, −9.485068140747087076854676025462, −7.40328663215423212106943599297, −5.61014856153357939702452577282, −4.85538446872717387022122976391, −0.999455512335164618434341108337,
3.31690675104111134426711276895, 5.37189894297874860566505369655, 6.37131007623750463110266696223, 8.770415348964985629958410332986, 10.01719176619326603493289080187, 11.21427302380230908580440162437, 12.47041671513335055928796313659, 13.78097720648744380003413622021, 14.66294103430330776514873611670, 16.23608825144547425058474050495
