| L(s) = 1 | + (4.89 + 2.82i)2-s + (15.9 + 27.7i)4-s + (21.4 − 12.4i)5-s + (3.09 − 5.36i)7-s + 181. i·8-s + 140.·10-s + (113. + 65.4i)11-s + (461. + 798. i)13-s + (30.3 − 17.5i)14-s + (−512. + 886. i)16-s + 3.38e3i·17-s + 5.40e3·19-s + (687. + 397. i)20-s + (370. + 641. i)22-s + (−2.04e3 + 1.18e3i)23-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.171 − 0.0992i)5-s + (0.00903 − 0.0156i)7-s + 0.353i·8-s + 0.140·10-s + (0.0851 + 0.0491i)11-s + (0.209 + 0.363i)13-s + (0.0110 − 0.00638i)14-s + (−0.125 + 0.216i)16-s + 0.688i·17-s + 0.787·19-s + (0.0859 + 0.0496i)20-s + (0.0347 + 0.0602i)22-s + (−0.168 + 0.0970i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 - 0.996i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.777272627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.777272627\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4.89 - 2.82i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-21.4 + 12.4i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-3.09 + 5.36i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-113. - 65.4i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-461. - 798. i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 - 3.38e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 5.40e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (2.04e3 - 1.18e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-3.02e4 - 1.74e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (2.33e3 + 4.03e3i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 6.91e3T + 2.56e9T^{2} \) |
| 41 | \( 1 + (4.66e4 - 2.69e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (6.15e4 - 1.06e5i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-8.31e4 - 4.80e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + 1.32e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (2.51e4 - 1.45e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.60e5 - 2.77e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-2.64e5 - 4.58e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 6.28e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.87e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + (2.35e5 - 4.08e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-2.90e4 - 1.67e4i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + 1.01e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (2.09e5 - 3.62e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−12.12628568530816983563610871967, −11.22417109113233148089199204438, −10.00717852578957121291482500260, −8.837854082770574025698349788815, −7.71973660203452497801246413119, −6.59302064842503320212352614015, −5.55246467984070017710544276323, −4.36212220304960894116031636029, −3.09359272372174818933377952781, −1.45877427017401829768352932781,
0.67716411088750949645448610336, 2.27057662456083825419372236583, 3.51229886318853810924615294046, 4.84613263066478634688569853068, 5.93981266219915084449713402763, 7.08897515640434676701653885923, 8.405935909361931091511213521824, 9.716868235526161498573887970795, 10.53304759754674894293455595805, 11.69877763219302206542862469519
