| L(s) = 1 | + (−1.73 − 2.99i)2-s + (250. − 433. i)4-s + (−702. + 1.21e3i)5-s + (−832. − 1.44e3i)7-s − 3.50e3·8-s + 4.86e3·10-s + (3.18e4 + 5.50e4i)11-s + (5.56e4 − 9.63e4i)13-s + (−2.88e3 + 4.99e3i)14-s + (−1.21e5 − 2.11e5i)16-s − 3.83e5·17-s + 7.44e3·19-s + (3.51e5 + 6.07e5i)20-s + (1.10e5 − 1.90e5i)22-s + (1.30e6 − 2.26e6i)23-s + ⋯ |
| L(s) = 1 | + (−0.0765 − 0.132i)2-s + (0.488 − 0.845i)4-s + (−0.502 + 0.870i)5-s + (−0.131 − 0.227i)7-s − 0.302·8-s + 0.153·10-s + (0.655 + 1.13i)11-s + (0.539 − 0.935i)13-s + (−0.0200 + 0.0347i)14-s + (−0.465 − 0.805i)16-s − 1.11·17-s + 0.0131·19-s + (0.490 + 0.849i)20-s + (0.100 − 0.173i)22-s + (0.975 − 1.68i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.343616 - 0.944079i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.343616 - 0.944079i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (1.73 + 2.99i)T + (-256 + 443. i)T^{2} \) |
| 5 | \( 1 + (702. - 1.21e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 + (832. + 1.44e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-3.18e4 - 5.50e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-5.56e4 + 9.63e4i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + 3.83e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.44e3T + 3.22e11T^{2} \) |
| 23 | \( 1 + (-1.30e6 + 2.26e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (2.89e5 + 5.01e5i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + (1.75e6 - 3.04e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + 9.82e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (8.88e6 - 1.53e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (1.95e7 + 3.38e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (1.33e7 + 2.31e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + 6.04e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (-6.74e7 + 1.16e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (7.03e7 + 1.21e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (3.65e7 - 6.33e7i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 2.72e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.64e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (2.72e8 + 4.71e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (8.69e7 + 1.50e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 - 5.69e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (-2.93e8 - 5.07e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−11.84316272258132824163921677939, −10.80772581182343820965146872010, −10.21856868488829065870789328571, −8.794824681999727405930455593365, −7.08622846106051077563343294768, −6.53273875739643232968345004176, −4.84925938339287974744429319475, −3.24647926693094391407946224102, −1.85708025344547781822012065312, −0.28340959996593971010613981332,
1.43799915980811816146453190573, 3.22863747540908703351470044268, 4.34016126493834644187600174220, 6.07004438705031926803230221775, 7.25867282425757277282920541811, 8.620666163281335125704491723888, 9.049707486322382020356043130478, 11.24235733117605808132921921980, 11.66853864751828369994372641017, 12.87265359900695146773569964444
