| L(s) = 1 | + (14.9 − 4.57i)3-s + (−45.2 + 78.3i)5-s + (−34.5 − 59.8i)7-s + (201. − 136. i)9-s + (198. + 344. i)11-s + (47.8 − 82.8i)13-s + (−315. + 1.37e3i)15-s − 2.20e3·17-s − 849.·19-s + (−788. − 733. i)21-s + (−2.31e3 + 4.00e3i)23-s + (−2.53e3 − 4.38e3i)25-s + (2.37e3 − 2.95e3i)27-s + (1.71e3 + 2.97e3i)29-s + (−2.91e3 + 5.05e3i)31-s + ⋯ |
| L(s) = 1 | + (0.955 − 0.293i)3-s + (−0.809 + 1.40i)5-s + (−0.266 − 0.461i)7-s + (0.827 − 0.561i)9-s + (0.495 + 0.858i)11-s + (0.0784 − 0.135i)13-s + (−0.362 + 1.57i)15-s − 1.85·17-s − 0.539·19-s + (−0.390 − 0.362i)21-s + (−0.910 + 1.57i)23-s + (−0.810 − 1.40i)25-s + (0.626 − 0.779i)27-s + (0.379 + 0.656i)29-s + (−0.545 + 0.944i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.221999619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.221999619\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-14.9 + 4.57i)T \) |
| good | 5 | \( 1 + (45.2 - 78.3i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (34.5 + 59.8i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-198. - 344. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-47.8 + 82.8i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + 2.20e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 849.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (2.31e3 - 4.00e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-1.71e3 - 2.97e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (2.91e3 - 5.05e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 2.64e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (2.27e3 - 3.94e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-1.18e3 - 2.04e3i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-1.20e3 - 2.08e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + 3.38e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-1.74e4 + 3.01e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.75e4 - 3.03e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (9.21e3 - 1.59e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 5.93e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.76e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-3.49e4 - 6.04e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-3.42e4 - 5.92e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 - 9.15e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (6.68e4 + 1.15e5i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−12.66014936811604388356735338594, −11.48350469206947341887396366041, −10.51975198033176874855235767701, −9.469942821546360980528963178958, −8.217536346975198244613578136788, −7.08197384020273975265055640977, −6.70919903665791542436527104768, −4.21090917531617279390675135678, −3.33246705321111313358492797231, −1.97918925189531202139611275787,
0.34936702178395212050345625283, 2.19049274860112301809661828453, 3.90730991233748880093358830723, 4.64588684406194459282802300688, 6.37074016066203417150981183586, 8.026647790022924705934774015795, 8.720014129005331926060127422215, 9.247647254932636348776766806728, 10.79957593788579292003603357401, 11.98127270710495994508135677360
