| L(s) = 1 | + (6.65 + 9.15i)2-s + (10.6 + 32.9i)3-s + (−39.5 + 121. i)4-s + (−864. − 627. i)5-s + (−230. + 316. i)6-s + (−1.44e3 − 469. i)7-s + (−1.37e3 + 447. i)8-s + (4.33e3 − 3.15e3i)9-s − 1.20e4i·10-s + (−9.25e3 + 1.13e4i)11-s − 4.42e3·12-s + (−1.75e4 − 2.40e4i)13-s + (−5.30e3 − 1.63e4i)14-s + (1.14e4 − 3.51e4i)15-s + (−1.32e4 − 9.63e3i)16-s + (1.14e4 − 1.57e4i)17-s + ⋯ |
| L(s) = 1 | + (0.415 + 0.572i)2-s + (0.131 + 0.406i)3-s + (−0.154 + 0.475i)4-s + (−1.38 − 1.00i)5-s + (−0.177 + 0.244i)6-s + (−0.601 − 0.195i)7-s + (−0.336 + 0.109i)8-s + (0.661 − 0.480i)9-s − 1.20i·10-s + (−0.631 + 0.775i)11-s − 0.213·12-s + (−0.612 − 0.843i)13-s + (−0.138 − 0.425i)14-s + (0.225 − 0.694i)15-s + (−0.202 − 0.146i)16-s + (0.136 − 0.188i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.576 + 0.817i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.576 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.103671 - 0.200022i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.103671 - 0.200022i\) |
| \(L(5)\) |
|
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 | \( 1 + (9.25e3 - 1.13e4i)T \) |
| good | 3 | \( 1 + (-10.6 - 32.9i)T + (-5.30e3 + 3.85e3i)T^{2} \) |
| 5 | \( 1 + (864. + 627. i)T + (1.20e5 + 3.71e5i)T^{2} \) |
| 7 | \( 1 + (1.44e3 + 469. i)T + (4.66e6 + 3.38e6i)T^{2} \) |
| 13 | \( 1 + (1.75e4 + 2.40e4i)T + (-2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-1.14e4 + 1.57e4i)T + (-2.15e9 - 6.63e9i)T^{2} \) |
| 19 | \( 1 + (1.22e5 - 3.97e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + 3.30e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + (-1.13e6 - 3.68e5i)T + (4.04e11 + 2.94e11i)T^{2} \) |
| 31 | \( 1 + (1.21e6 - 8.83e5i)T + (2.63e11 - 8.11e11i)T^{2} \) |
| 37 | \( 1 + (-5.93e5 + 1.82e6i)T + (-2.84e12 - 2.06e12i)T^{2} \) |
| 41 | \( 1 + (2.24e6 - 7.30e5i)T + (6.45e12 - 4.69e12i)T^{2} \) |
| 43 | \( 1 + 8.39e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (8.84e5 + 2.72e6i)T + (-1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-6.77e6 + 4.92e6i)T + (1.92e13 - 5.92e13i)T^{2} \) |
| 59 | \( 1 + (2.13e6 - 6.56e6i)T + (-1.18e14 - 8.63e13i)T^{2} \) |
| 61 | \( 1 + (-7.62e6 + 1.04e7i)T + (-5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 - 6.19e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + (-2.06e7 - 1.50e7i)T + (1.99e14 + 6.14e14i)T^{2} \) |
| 73 | \( 1 + (1.34e7 + 4.37e6i)T + (6.52e14 + 4.74e14i)T^{2} \) |
| 79 | \( 1 + (3.00e7 + 4.13e7i)T + (-4.68e14 + 1.44e15i)T^{2} \) |
| 83 | \( 1 + (-1.82e7 + 2.50e7i)T + (-6.95e14 - 2.14e15i)T^{2} \) |
| 89 | \( 1 + 4.52e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (9.87e7 - 7.17e7i)T + (2.42e15 - 7.45e15i)T^{2} \) |
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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
−15.77118689232932981530859953605, −14.84650476051719307170543021921, −12.77553124308853786022104359812, −12.30867470607198975258396979849, −10.10540357978977505275537089288, −8.441523107167775712080223151657, −7.18725792126005338162230618318, −4.89831461476543266925724950518, −3.72647229432273213452477315231, −0.091812353205390920710502658992,
2.59700190129221843846387289030, 4.15089849532996524371564602079, 6.62225877945145766418348999408, 8.029321810326934531002804603124, 10.22428099977100124983593580410, 11.41864369543033089810012120321, 12.55662514083652557022892560188, 13.90150996850669732645432473970, 15.22194794874024437453003343370, 16.22478982044016630231158563597
