| L(s) = 1 | + (−6.65 + 9.15i)2-s + (15.7 − 48.6i)3-s + (−39.5 − 121. i)4-s + (−394. + 286. i)5-s + (339. + 467. i)6-s + (1.39e3 − 452. i)7-s + (1.37e3 + 447. i)8-s + (3.19e3 + 2.32e3i)9-s − 5.52e3i·10-s + (2.03e3 + 1.44e4i)11-s − 6.54e3·12-s + (−3.02e4 + 4.16e4i)13-s + (−5.12e3 + 1.57e4i)14-s + (7.70e3 + 2.37e4i)15-s + (−1.32e4 + 9.63e3i)16-s + (2.76e4 + 3.80e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 + 0.572i)2-s + (0.194 − 0.600i)3-s + (−0.154 − 0.475i)4-s + (−0.631 + 0.459i)5-s + (0.262 + 0.360i)6-s + (0.580 − 0.188i)7-s + (0.336 + 0.109i)8-s + (0.486 + 0.353i)9-s − 0.552i·10-s + (0.138 + 0.990i)11-s − 0.315·12-s + (−1.05 + 1.45i)13-s + (−0.133 + 0.410i)14-s + (0.152 + 0.468i)15-s + (−0.202 + 0.146i)16-s + (0.330 + 0.455i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0696 - 0.997i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0696 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.806875 + 0.865174i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.806875 + 0.865174i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (6.65 - 9.15i)T \) |
| 11 | \( 1 + (-2.03e3 - 1.44e4i)T \) |
| good | 3 | \( 1 + (-15.7 + 48.6i)T + (-5.30e3 - 3.85e3i)T^{2} \) |
| 5 | \( 1 + (394. - 286. i)T + (1.20e5 - 3.71e5i)T^{2} \) |
| 7 | \( 1 + (-1.39e3 + 452. i)T + (4.66e6 - 3.38e6i)T^{2} \) |
| 13 | \( 1 + (3.02e4 - 4.16e4i)T + (-2.52e8 - 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-2.76e4 - 3.80e4i)T + (-2.15e9 + 6.63e9i)T^{2} \) |
| 19 | \( 1 + (-1.62e4 - 5.27e3i)T + (1.37e10 + 9.98e9i)T^{2} \) |
| 23 | \( 1 - 3.76e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + (4.38e5 - 1.42e5i)T + (4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (2.66e5 + 1.93e5i)T + (2.63e11 + 8.11e11i)T^{2} \) |
| 37 | \( 1 + (5.69e5 + 1.75e6i)T + (-2.84e12 + 2.06e12i)T^{2} \) |
| 41 | \( 1 + (-2.78e6 - 9.05e5i)T + (6.45e12 + 4.69e12i)T^{2} \) |
| 43 | \( 1 + 5.68e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (1.51e6 - 4.66e6i)T + (-1.92e13 - 1.39e13i)T^{2} \) |
| 53 | \( 1 + (4.84e6 + 3.51e6i)T + (1.92e13 + 5.92e13i)T^{2} \) |
| 59 | \( 1 + (1.36e6 + 4.18e6i)T + (-1.18e14 + 8.63e13i)T^{2} \) |
| 61 | \( 1 + (-3.39e6 - 4.67e6i)T + (-5.92e13 + 1.82e14i)T^{2} \) |
| 67 | \( 1 + 3.85e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + (-1.99e7 + 1.45e7i)T + (1.99e14 - 6.14e14i)T^{2} \) |
| 73 | \( 1 + (-2.06e7 + 6.69e6i)T + (6.52e14 - 4.74e14i)T^{2} \) |
| 79 | \( 1 + (2.57e7 - 3.54e7i)T + (-4.68e14 - 1.44e15i)T^{2} \) |
| 83 | \( 1 + (-3.50e7 - 4.82e7i)T + (-6.95e14 + 2.14e15i)T^{2} \) |
| 89 | \( 1 + 1.44e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (-1.32e8 - 9.63e7i)T + (2.42e15 + 7.45e15i)T^{2} \) |
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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
−16.58033840794641140296711143536, −15.07940665693243019259907708773, −14.26078239964696660282556843227, −12.59971307815634392433277202056, −11.11101922758277758252688790280, −9.454702585011662146226541168254, −7.62611295264714127807153388784, −7.03057425055353283112590704161, −4.58404192607967377759132795807, −1.75457356397508692755089633338,
0.69589761586777241515985396478, 3.23792788248846622649248861439, 4.93183743861804544157349169335, 7.72079183396115828014458510022, 8.988374767533425780583211244073, 10.35960712960498846291256496062, 11.70201203996913081604584510417, 12.89457995919927687820238635372, 14.74539285601831793849569449184, 15.84602602454981709200986249981
