| L(s) = 1 | + (9.88 − 30.4i)2-s + (598. − 435. i)3-s + (−828. − 601. i)4-s + (2.26e3 + 6.97e3i)5-s + (−7.32e3 − 2.25e4i)6-s + (−5.87e4 − 4.27e4i)7-s + (−2.65e4 + 1.92e4i)8-s + (1.14e5 − 3.52e5i)9-s + 2.34e5·10-s + (−4.45e5 − 2.94e5i)11-s − 7.58e5·12-s + (2.92e5 − 9.00e5i)13-s + (−1.88e6 + 1.36e6i)14-s + (4.39e6 + 3.19e6i)15-s + (3.24e5 + 9.97e5i)16-s + (−2.43e5 − 7.48e5i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (1.42 − 1.03i)3-s + (−0.404 − 0.293i)4-s + (0.324 + 0.998i)5-s + (−0.384 − 1.18i)6-s + (−1.32 − 0.960i)7-s + (−0.286 + 0.207i)8-s + (0.647 − 1.99i)9-s + 0.742·10-s + (−0.834 − 0.551i)11-s − 0.879·12-s + (0.218 − 0.672i)13-s + (−0.934 + 0.679i)14-s + (1.49 + 1.08i)15-s + (0.0772 + 0.237i)16-s + (−0.0415 − 0.127i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.393i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.518909 - 2.53085i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.518909 - 2.53085i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-9.88 + 30.4i)T \) |
| 11 | \( 1 + (4.45e5 + 2.94e5i)T \) |
| good | 3 | \( 1 + (-598. + 435. i)T + (5.47e4 - 1.68e5i)T^{2} \) |
| 5 | \( 1 + (-2.26e3 - 6.97e3i)T + (-3.95e7 + 2.87e7i)T^{2} \) |
| 7 | \( 1 + (5.87e4 + 4.27e4i)T + (6.11e8 + 1.88e9i)T^{2} \) |
| 13 | \( 1 + (-2.92e5 + 9.00e5i)T + (-1.44e12 - 1.05e12i)T^{2} \) |
| 17 | \( 1 + (2.43e5 + 7.48e5i)T + (-2.77e13 + 2.01e13i)T^{2} \) |
| 19 | \( 1 + (-1.25e7 + 9.12e6i)T + (3.59e13 - 1.10e14i)T^{2} \) |
| 23 | \( 1 + 1.84e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (-5.14e7 - 3.74e7i)T + (3.77e15 + 1.16e16i)T^{2} \) |
| 31 | \( 1 + (5.44e7 - 1.67e8i)T + (-2.05e16 - 1.49e16i)T^{2} \) |
| 37 | \( 1 + (-6.79e8 - 4.93e8i)T + (5.49e16 + 1.69e17i)T^{2} \) |
| 41 | \( 1 + (-5.37e8 + 3.90e8i)T + (1.70e17 - 5.23e17i)T^{2} \) |
| 43 | \( 1 - 7.01e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (1.25e8 - 9.08e7i)T + (7.63e17 - 2.35e18i)T^{2} \) |
| 53 | \( 1 + (6.85e8 - 2.11e9i)T + (-7.49e18 - 5.44e18i)T^{2} \) |
| 59 | \( 1 + (3.20e9 + 2.33e9i)T + (9.31e18 + 2.86e19i)T^{2} \) |
| 61 | \( 1 + (2.84e9 + 8.75e9i)T + (-3.52e19 + 2.55e19i)T^{2} \) |
| 67 | \( 1 - 6.35e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (7.48e9 + 2.30e10i)T + (-1.86e20 + 1.35e20i)T^{2} \) |
| 73 | \( 1 + (-2.61e10 - 1.90e10i)T + (9.69e19 + 2.98e20i)T^{2} \) |
| 79 | \( 1 + (-4.84e9 + 1.49e10i)T + (-6.05e20 - 4.39e20i)T^{2} \) |
| 83 | \( 1 + (8.91e9 + 2.74e10i)T + (-1.04e21 + 7.56e20i)T^{2} \) |
| 89 | \( 1 - 3.93e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-2.82e10 + 8.69e10i)T + (-5.78e21 - 4.20e21i)T^{2} \) |
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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
−14.16282605555513324793191636198, −13.56594208642352522224050823929, −12.71876302616004831949721836023, −10.63631567372529066901929517699, −9.466349874871530520925038836797, −7.72028161986637430241192617564, −6.50745025198164561919593138583, −3.26607893178014982983086666019, −2.77370215167841117609762167161, −0.78224080610941228287884776886,
2.57293700183699021829878541316, 4.11141594815645636718509869296, 5.63670029214893945510265013051, 7.955559112752588101330440484245, 9.257152306763194262683609621260, 9.692758203670033910381568926630, 12.60189350227698944743490267044, 13.56236661328592333046853295290, 14.86401411824382841208652316009, 16.00466198042856250498049888953
