| 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
| \(L(5)\) |
\(\approx\) |
\(0.750431 + 2.06179i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.750431 + 2.06179i\) |
| \(L(\frac{11}{2})\) |
|
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
−12.74084668394972208858099633062, −11.94455658028056788370071809872, −10.73360085967932654420842019437, −9.817500759472264178940154834807, −8.256457029845607035966496807344, −7.12276300719097014938815695413, −6.17335988599156740565729289299, −4.32172249335086621089573416215, −2.86905019722908484823216701433, −1.94704111393420044280746946576,
0.57203284448338009982210593920, 1.58001663749406013424656422459, 3.36116264280635447140444799933, 5.46469723066502006115109606732, 5.68610545153747466597590874781, 7.45498788479380187386835743765, 8.724496213045390764293056360773, 10.00627865727318261124789545314, 10.78400858340712467442581289820, 12.09927381047030647577757198521
