L(s) = 1 | + (5.65 + 9.79i)2-s + (−63.9 + 110. i)4-s + (268. − 155. i)5-s + (−1.57e3 − 1.81e3i)7-s − 1.44e3·8-s + (3.04e3 + 1.75e3i)10-s + (3.66e3 − 6.34e3i)11-s + 1.06e4i·13-s + (8.82e3 − 2.56e4i)14-s + (−8.19e3 − 1.41e4i)16-s + (2.45e4 + 1.41e4i)17-s + (−1.06e5 + 6.16e4i)19-s + 3.97e4i·20-s + 8.28e4·22-s + (1.61e5 + 2.80e5i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.430 − 0.248i)5-s + (−0.656 − 0.754i)7-s − 0.353·8-s + (0.304 + 0.175i)10-s + (0.250 − 0.433i)11-s + 0.372i·13-s + (0.229 − 0.668i)14-s + (−0.125 − 0.216i)16-s + (0.293 + 0.169i)17-s + (−0.819 + 0.473i)19-s + 0.248i·20-s + 0.353·22-s + (0.578 + 1.00i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 - 0.665i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(1.485851618\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.485851618\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-5.65 - 9.79i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.57e3 + 1.81e3i)T \) |
good | 5 | \( 1 + (-268. + 155. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-3.66e3 + 6.34e3i)T + (-1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 - 1.06e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-2.45e4 - 1.41e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (1.06e5 - 6.16e4i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-1.61e5 - 2.80e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + 6.93e4T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-5.86e5 - 3.38e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-6.69e5 - 1.15e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + 1.00e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.01e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (3.43e6 - 1.98e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (6.75e6 - 1.16e7i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-6.99e4 - 4.03e4i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-3.34e6 + 1.93e6i)T + (9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (6.54e5 - 1.13e6i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 3.90e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.19e7 - 6.87e6i)T + (4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (2.54e7 + 4.41e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 - 6.90e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (9.62e7 - 5.55e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 - 3.18e7iT - 7.83e15T^{2} \) |
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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
−12.51281382943977864124438885550, −11.22340592995755564533541481805, −9.970854837749345143659856557878, −9.032091902787607445182493135777, −7.77061669539425863728516083429, −6.65033003970338114954131365829, −5.73682923575496203948484001870, −4.34485662099257166525133480662, −3.20880827987505772310184250102, −1.29854301114577017363233160375,
0.35034238092255017129011595333, 2.07681331599009112548243076086, 3.00732913885826270926808378469, 4.49821739543531424693740410420, 5.81110533235583017430551039872, 6.74641239678478663902253454626, 8.475972020745347380959500016080, 9.559752311081013752267863761867, 10.34773598982332703582613191412, 11.51643850523171751634965520040