| L(s) = 1 | + (6.65 − 9.15i)2-s + (−17.2 + 53.0i)3-s + (−39.5 − 121. i)4-s + (−95.9 + 69.7i)5-s + (371. + 510. i)6-s + (3.85e3 − 1.25e3i)7-s + (−1.37e3 − 447. i)8-s + (2.78e3 + 2.02e3i)9-s + 1.34e3i·10-s + (1.40e4 + 4.11e3i)11-s + 7.14e3·12-s + (8.97e3 − 1.23e4i)13-s + (1.41e4 − 4.36e4i)14-s + (−2.04e3 − 6.29e3i)15-s + (−1.32e4 + 9.63e3i)16-s + (−5.69e4 − 7.83e4i)17-s + ⋯ |
| L(s) = 1 | + (0.415 − 0.572i)2-s + (−0.212 + 0.655i)3-s + (−0.154 − 0.475i)4-s + (−0.153 + 0.111i)5-s + (0.286 + 0.394i)6-s + (1.60 − 0.521i)7-s + (−0.336 − 0.109i)8-s + (0.424 + 0.308i)9-s + 0.134i·10-s + (0.959 + 0.280i)11-s + 0.344·12-s + (0.314 − 0.432i)13-s + (0.369 − 1.13i)14-s + (−0.0404 − 0.124i)15-s + (−0.202 + 0.146i)16-s + (−0.681 − 0.938i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.346i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.937 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.26649 - 0.405739i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.26649 - 0.405739i\) |
| \(L(5)\) |
|
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 | \( 1 + (-1.40e4 - 4.11e3i)T \) |
| good | 3 | \( 1 + (17.2 - 53.0i)T + (-5.30e3 - 3.85e3i)T^{2} \) |
| 5 | \( 1 + (95.9 - 69.7i)T + (1.20e5 - 3.71e5i)T^{2} \) |
| 7 | \( 1 + (-3.85e3 + 1.25e3i)T + (4.66e6 - 3.38e6i)T^{2} \) |
| 13 | \( 1 + (-8.97e3 + 1.23e4i)T + (-2.52e8 - 7.75e8i)T^{2} \) |
| 17 | \( 1 + (5.69e4 + 7.83e4i)T + (-2.15e9 + 6.63e9i)T^{2} \) |
| 19 | \( 1 + (-2.01e5 - 6.55e4i)T + (1.37e10 + 9.98e9i)T^{2} \) |
| 23 | \( 1 + 1.96e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + (-1.50e5 + 4.89e4i)T + (4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (7.81e5 + 5.67e5i)T + (2.63e11 + 8.11e11i)T^{2} \) |
| 37 | \( 1 + (-4.31e5 - 1.32e6i)T + (-2.84e12 + 2.06e12i)T^{2} \) |
| 41 | \( 1 + (-2.24e6 - 7.28e5i)T + (6.45e12 + 4.69e12i)T^{2} \) |
| 43 | \( 1 + 4.86e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (1.07e6 - 3.29e6i)T + (-1.92e13 - 1.39e13i)T^{2} \) |
| 53 | \( 1 + (4.87e6 + 3.54e6i)T + (1.92e13 + 5.92e13i)T^{2} \) |
| 59 | \( 1 + (4.22e6 + 1.30e7i)T + (-1.18e14 + 8.63e13i)T^{2} \) |
| 61 | \( 1 + (3.88e6 + 5.34e6i)T + (-5.92e13 + 1.82e14i)T^{2} \) |
| 67 | \( 1 + 1.28e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + (3.39e7 - 2.46e7i)T + (1.99e14 - 6.14e14i)T^{2} \) |
| 73 | \( 1 + (-1.02e7 + 3.32e6i)T + (6.52e14 - 4.74e14i)T^{2} \) |
| 79 | \( 1 + (1.61e7 - 2.22e7i)T + (-4.68e14 - 1.44e15i)T^{2} \) |
| 83 | \( 1 + (-1.17e7 - 1.61e7i)T + (-6.95e14 + 2.14e15i)T^{2} \) |
| 89 | \( 1 + 1.09e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (-5.99e7 - 4.35e7i)T + (2.42e15 + 7.45e15i)T^{2} \) |
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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
−15.89960565555912417089161481630, −14.64214661811183157234539965222, −13.63441333918398325725732306033, −11.73233814421709036211942093142, −10.96745880430639709772574768960, −9.574974185897256132853087736247, −7.58355569579191875493283858309, −5.15398920802855864057426914958, −3.98886645227530151664827768333, −1.46516776995258991941394813702,
1.50158674009844835911962651201, 4.33127730043621472499879320276, 6.06007223176853502573787794404, 7.56756455091282923715860783429, 8.888392274122235493648348688717, 11.39415667483803752491064790335, 12.25430909008563295838007737599, 13.81986731923659021085685466777, 14.81386522604015417000720237136, 16.10963370353823193285021433341
