| L(s) = 1 | + (−0.933 − 0.250i)2-s + (0.0424 − 5.19i)3-s + (−6.11 − 3.53i)4-s + (−11.3 + 11.3i)5-s + (−1.33 + 4.84i)6-s + (−3.58 − 13.3i)7-s + (10.2 + 10.2i)8-s + (−26.9 − 0.440i)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 + (58.3 + 59.3i)15-s + (21.2 + 36.7i)16-s + (−39.8 + 69.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.330 − 0.0884i)2-s + (0.00816 − 0.999i)3-s + (−0.764 − 0.441i)4-s + (−1.01 + 1.01i)5-s + (−0.0911 + 0.329i)6-s + (−0.193 − 0.723i)7-s + (0.455 + 0.455i)8-s + (−0.999 − 0.0163i)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 + (1.00 + 1.02i)15-s + (0.331 + 0.574i)16-s + (−0.569 + 0.985i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0968733 - 0.491954i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0968733 - 0.491954i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.0424 + 5.19i)T \) |
| 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 | \( 1 + (-957. + 256. i)T + (7.90e5 - 4.56e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.94755687081149628195159973135, −13.99790970876688125647518947081, −13.04002601991951595971212383930, −11.35032049433210510852620681955, −10.56180081665455699579237399567, −8.581300052633547877211884940042, −7.57151614121239388367962487879, −6.07420759877894834988972213301, −3.54781410624785270600611703608, −0.48773154974913143950182984715,
3.98746546385687811259307950666, 4.95576541873302646573309563341, 7.65983776141093525348187366818, 9.126106217090309621339917877434, 9.449311381042476316011957212308, 11.64455573673779226220856926126, 12.41444274372738586524490107525, 14.02221045724888676690855676443, 15.52120697772573775751732747148, 16.15352879673753362714502169657
