| L(s) = 1 | + (1.83 − 26.9i)3-s + (86.3 + 49.8i)5-s + (−207. − 359. i)7-s + (−722. − 98.5i)9-s + (1.64e3 − 947. i)11-s + (−335. + 580. i)13-s + (1.50e3 − 2.23e3i)15-s − 594. i·17-s − 9.51e3·19-s + (−1.00e4 + 4.93e3i)21-s + (−1.37e4 − 7.94e3i)23-s + (−2.83e3 − 4.91e3i)25-s + (−3.97e3 + 1.92e4i)27-s + (−1.43e4 + 8.27e3i)29-s + (2.29e4 − 3.97e4i)31-s + ⋯ |
| L(s) = 1 | + (0.0677 − 0.997i)3-s + (0.691 + 0.398i)5-s + (−0.605 − 1.04i)7-s + (−0.990 − 0.135i)9-s + (1.23 − 0.711i)11-s + (−0.152 + 0.264i)13-s + (0.444 − 0.662i)15-s − 0.121i·17-s − 1.38·19-s + (−1.08 + 0.532i)21-s + (−1.13 − 0.653i)23-s + (−0.181 − 0.314i)25-s + (−0.202 + 0.979i)27-s + (−0.587 + 0.339i)29-s + (0.771 − 1.33i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.392648 - 1.35915i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.392648 - 1.35915i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.83 + 26.9i)T \) |
| good | 5 | \( 1 + (-86.3 - 49.8i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (207. + 359. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.64e3 + 947. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (335. - 580. i)T + (-2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 + 594. iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 9.51e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (1.37e4 + 7.94e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (1.43e4 - 8.27e3i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-2.29e4 + 3.97e4i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + 2.07e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (4.53e4 + 2.61e4i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-5.15e4 - 8.92e4i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (4.23e4 - 2.44e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + 1.52e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-2.97e5 - 1.71e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-3.30e4 - 5.72e4i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.73e5 + 3.01e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 5.64e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 6.75e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-2.39e5 - 4.15e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-5.20e5 + 3.00e5i)T + (1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 - 3.33e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-2.72e5 - 4.71e5i)T + (-4.16e11 + 7.21e11i)T^{2} \) |
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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
−13.18586099161627217678489585142, −11.97023337633961245809470735264, −10.76851092722281894683546535864, −9.544015053772810272233991535172, −8.180782361226772939853951664374, −6.68538197331150649799801140742, −6.22315873999529911111417872746, −3.84924841124910120248390443524, −2.12157108939659940146420700987, −0.51678446146989536623146729820,
2.11537060191571756546948660682, 3.85467435995742102073227016491, 5.32905053290048299304125170972, 6.39905481108052887552784186839, 8.578810038492844791330731989971, 9.396878706952240288004889778021, 10.17700977389731930834542946028, 11.72728142120180618644747071677, 12.66064524633774119203703658668, 14.04124210984285390709179498737
