| L(s) = 1 | + (−4.85 + 10.0i)2-s + (−1.63 − 1.30i)3-s + (1.80 + 2.26i)4-s + (−133. − 64.5i)5-s + (21.0 − 10.1i)6-s + (−384. + 482. i)7-s + (−1.42e3 + 325. i)8-s + (−485. − 2.12e3i)9-s + (1.30e3 − 1.03e3i)10-s + (533. + 121. i)11-s − 6.04i·12-s + (1.39e3 − 6.11e3i)13-s + (−2.99e3 − 6.22e3i)14-s + (134. + 279. i)15-s + (3.56e3 − 1.56e4i)16-s − 2.68e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.428 + 0.890i)2-s + (−0.0349 − 0.0278i)3-s + (0.0140 + 0.0176i)4-s + (−0.479 − 0.230i)5-s + (0.0397 − 0.0191i)6-s + (−0.424 + 0.531i)7-s + (−0.985 + 0.224i)8-s + (−0.222 − 0.972i)9-s + (0.411 − 0.327i)10-s + (0.120 + 0.0275i)11-s − 0.00100i·12-s + (0.176 − 0.772i)13-s + (−0.291 − 0.605i)14-s + (0.0103 + 0.0214i)15-s + (0.217 − 0.952i)16-s − 1.32i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0685 + 0.997i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0685 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.146886 - 0.157329i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.146886 - 0.157329i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (3.69e4 - 1.26e5i)T \) |
| good | 2 | \( 1 + (4.85 - 10.0i)T + (-79.8 - 100. i)T^{2} \) |
| 3 | \( 1 + (1.63 + 1.30i)T + (486. + 2.13e3i)T^{2} \) |
| 5 | \( 1 + (133. + 64.5i)T + (4.87e4 + 6.10e4i)T^{2} \) |
| 7 | \( 1 + (384. - 482. i)T + (-1.83e5 - 8.02e5i)T^{2} \) |
| 11 | \( 1 + (-533. - 121. i)T + (1.75e7 + 8.45e6i)T^{2} \) |
| 13 | \( 1 + (-1.39e3 + 6.11e3i)T + (-5.65e7 - 2.72e7i)T^{2} \) |
| 17 | \( 1 + 2.68e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (-2.84e3 + 2.26e3i)T + (1.98e8 - 8.71e8i)T^{2} \) |
| 23 | \( 1 + (7.21e4 - 3.47e4i)T + (2.12e9 - 2.66e9i)T^{2} \) |
| 31 | \( 1 + (8.25e4 - 1.71e5i)T + (-1.71e10 - 2.15e10i)T^{2} \) |
| 37 | \( 1 + (5.21e4 - 1.18e4i)T + (8.55e10 - 4.11e10i)T^{2} \) |
| 41 | \( 1 + 4.77e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + (-1.05e5 - 2.18e5i)T + (-1.69e11 + 2.12e11i)T^{2} \) |
| 47 | \( 1 + (9.73e5 + 2.22e5i)T + (4.56e11 + 2.19e11i)T^{2} \) |
| 53 | \( 1 + (1.04e6 + 5.05e5i)T + (7.32e11 + 9.18e11i)T^{2} \) |
| 59 | \( 1 + 9.07e4T + 2.48e12T^{2} \) |
| 61 | \( 1 + (-1.89e6 - 1.51e6i)T + (6.99e11 + 3.06e12i)T^{2} \) |
| 67 | \( 1 + (-5.45e4 - 2.38e5i)T + (-5.46e12 + 2.62e12i)T^{2} \) |
| 71 | \( 1 + (2.15e5 - 9.46e5i)T + (-8.19e12 - 3.94e12i)T^{2} \) |
| 73 | \( 1 + (-3.68e5 - 7.64e5i)T + (-6.88e12 + 8.63e12i)T^{2} \) |
| 79 | \( 1 + (2.15e6 - 4.91e5i)T + (1.73e13 - 8.33e12i)T^{2} \) |
| 83 | \( 1 + (7.18e5 + 9.00e5i)T + (-6.03e12 + 2.64e13i)T^{2} \) |
| 89 | \( 1 + (1.06e5 - 2.21e5i)T + (-2.75e13 - 3.45e13i)T^{2} \) |
| 97 | \( 1 + (-9.57e6 + 7.63e6i)T + (1.79e13 - 7.87e13i)T^{2} \) |
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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
−15.68361077050250533821138853681, −14.40651290023019684846875825416, −12.50899646074131306089847596247, −11.67260398286186142509934081868, −9.526380861732727000912334721460, −8.418385172104932364623011049642, −7.05631331783007264944962801659, −5.70594172724141331306772222769, −3.24436398273406987279114332473, −0.11458037803922843618467580004,
1.94632770969209991370170410291, 3.84137437580365796196763192972, 6.23622207166141614304825011810, 8.032668118080785482330983939332, 9.722769565917779873363420185912, 10.79946706076985288375820257163, 11.70805425251826588870160878170, 13.18002116249571121747183159577, 14.62830977546230257559270978285, 15.97464580401500514885255093805
