| L(s) = 1 | + (−5.32 − 9.98i)2-s + (−29.4 + 71.0i)3-s + (−71.2 + 106. i)4-s + (373. − 154. i)5-s + (865. − 84.5i)6-s + (−760. + 760. i)7-s + (1.44e3 + 145. i)8-s + (−2.63e3 − 2.63e3i)9-s + (−3.53e3 − 2.90e3i)10-s + (688. + 1.66e3i)11-s + (−5.45e3 − 8.19e3i)12-s + (−1.15e4 − 4.77e3i)13-s + (1.16e4 + 3.54e3i)14-s + 3.10e4i·15-s + (−6.21e3 − 1.51e4i)16-s + 1.13e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.470 − 0.882i)2-s + (−0.629 + 1.51i)3-s + (−0.556 + 0.830i)4-s + (1.33 − 0.553i)5-s + (1.63 − 0.159i)6-s + (−0.837 + 0.837i)7-s + (0.994 + 0.100i)8-s + (−1.20 − 1.20i)9-s + (−1.11 − 0.918i)10-s + (0.156 + 0.376i)11-s + (−0.911 − 1.36i)12-s + (−1.45 − 0.602i)13-s + (1.13 + 0.344i)14-s + 2.37i·15-s + (−0.379 − 0.925i)16-s + 0.559i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0127879 + 0.249635i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0127879 + 0.249635i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (5.32 + 9.98i)T \) |
| good | 3 | \( 1 + (29.4 - 71.0i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (-373. + 154. i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (760. - 760. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (-688. - 1.66e3i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (1.15e4 + 4.77e3i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 - 1.13e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (1.75e4 + 7.26e3i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (5.11e4 + 5.11e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (-6.62e4 + 1.59e5i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 1.27e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (3.44e5 - 1.42e5i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (-4.15e5 - 4.15e5i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (-1.64e5 - 3.97e5i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 + 6.27e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-1.00e5 - 2.41e5i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (-1.21e6 + 5.02e5i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (1.12e6 - 2.70e6i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (1.48e6 - 3.58e6i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (1.93e6 - 1.93e6i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (1.99e6 + 1.99e6i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 + 1.29e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (2.54e6 + 1.05e6i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (-3.12e6 + 3.12e6i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 + 3.06e6T + 8.07e13T^{2} \) |
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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
−16.21417033020091085515209231466, −14.79175447292815664433021488906, −12.96912063503099408916551797832, −12.02434240225316997287482879976, −10.25019473772728938346923799985, −9.855356253224251729980064575806, −8.895804459233571171599426646329, −5.82081682742771368645209861810, −4.52847560348802538286825771445, −2.50246427086968513172479520742,
0.13929423910224118681796734749, 1.84988886134542810325503082562, 5.63082807056546005060640369645, 6.70420098559397906176211977264, 7.32456609204349805013866276061, 9.436977239940528996445428087398, 10.60869213095336425422306677246, 12.52715799877700499237882400928, 13.76322140049350313267719077882, 14.20093804455313797921811068082
