| L(s) = 1 | + (−9.12 + 2.44i)2-s + (15.7 − 58.6i)3-s + (−33.5 + 19.3i)4-s + (0.602 − 279. i)5-s + 573. i·6-s + (−549. − 722. i)7-s + (1.11e3 − 1.11e3i)8-s + (−1.29e3 − 750. i)9-s + (677. + 2.55e3i)10-s + (2.81e3 + 4.87e3i)11-s + (609. + 2.27e3i)12-s + (−4.26e3 − 4.26e3i)13-s + (6.77e3 + 5.24e3i)14-s + (−1.63e4 − 4.42e3i)15-s + (−4.96e3 + 8.59e3i)16-s + (−2.67e4 − 7.16e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.806 + 0.216i)2-s + (0.336 − 1.25i)3-s + (−0.262 + 0.151i)4-s + (0.00215 − 0.999i)5-s + 1.08i·6-s + (−0.605 − 0.796i)7-s + (0.769 − 0.769i)8-s + (−0.593 − 0.342i)9-s + (0.214 + 0.807i)10-s + (0.637 + 1.10i)11-s + (0.101 + 0.379i)12-s + (−0.538 − 0.538i)13-s + (0.660 + 0.511i)14-s + (−1.25 − 0.338i)15-s + (−0.302 + 0.524i)16-s + (−1.32 − 0.353i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.101016 + 0.398891i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.101016 + 0.398891i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.602 + 279. i)T \) |
| 7 | \( 1 + (549. + 722. i)T \) |
| good | 2 | \( 1 + (9.12 - 2.44i)T + (110. - 64i)T^{2} \) |
| 3 | \( 1 + (-15.7 + 58.6i)T + (-1.89e3 - 1.09e3i)T^{2} \) |
| 11 | \( 1 + (-2.81e3 - 4.87e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (4.26e3 + 4.26e3i)T + 6.27e7iT^{2} \) |
| 17 | \( 1 + (2.67e4 + 7.16e3i)T + (3.55e8 + 2.05e8i)T^{2} \) |
| 19 | \( 1 + (1.84e4 - 3.20e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-6.64e3 - 2.47e4i)T + (-2.94e9 + 1.70e9i)T^{2} \) |
| 29 | \( 1 - 1.24e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + (-1.56e5 + 9.04e4i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (1.41e5 - 3.78e4i)T + (8.22e10 - 4.74e10i)T^{2} \) |
| 41 | \( 1 + 5.76e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + (-2.46e5 + 2.46e5i)T - 2.71e11iT^{2} \) |
| 47 | \( 1 + (-5.34e4 - 1.99e5i)T + (-4.38e11 + 2.53e11i)T^{2} \) |
| 53 | \( 1 + (2.02e5 + 5.41e4i)T + (1.01e12 + 5.87e11i)T^{2} \) |
| 59 | \( 1 + (1.55e6 + 2.68e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.03e5 + 5.98e4i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (3.39e5 - 1.26e6i)T + (-5.24e12 - 3.03e12i)T^{2} \) |
| 71 | \( 1 + 2.18e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (7.12e5 - 2.66e6i)T + (-9.56e12 - 5.52e12i)T^{2} \) |
| 79 | \( 1 + (-7.59e5 - 4.38e5i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (7.08e6 + 7.08e6i)T + 2.71e13iT^{2} \) |
| 89 | \( 1 + (1.09e6 - 1.88e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-8.95e6 + 8.95e6i)T - 8.07e13iT^{2} \) |
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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
−13.93666320508511132095403867682, −12.94566825693435545475193095096, −12.39089647653869163371670562726, −10.06839494459286492789852760834, −8.927146622753574811333964485447, −7.73042350500914775273928056327, −6.83316335380030439966726201663, −4.32061928187759374547413697439, −1.59118683693232905002517350530, −0.23969549088192787524017484308,
2.68813715156497502707478377749, 4.40326337234909530389390047888, 6.40675003808826434349593815593, 8.678081743929749066196904235025, 9.371958960353067447695967013200, 10.46440398691536053929677837847, 11.40253475171748026660302760379, 13.62168273591952144603605030628, 14.77030330117132512679242091585, 15.60727315496904981769336352256
