| L(s) = 1 | + (28.5 − 75.7i)3-s + (−918. + 530. i)5-s + (−1.25e3 + 2.04e3i)7-s + (−4.92e3 − 4.33e3i)9-s + (−2.20e4 − 1.27e4i)11-s + 4.82e4·13-s + (1.39e4 + 8.48e4i)15-s + (−1.51e4 − 8.72e3i)17-s + (6.76e4 + 1.17e5i)19-s + (1.19e5 + 1.53e5i)21-s + (4.28e5 − 2.47e5i)23-s + (3.67e5 − 6.36e5i)25-s + (−4.69e5 + 2.49e5i)27-s + 4.48e5i·29-s + (1.92e5 − 3.32e5i)31-s + ⋯ |
| L(s) = 1 | + (0.352 − 0.935i)3-s + (−1.47 + 0.848i)5-s + (−0.523 + 0.852i)7-s + (−0.750 − 0.660i)9-s + (−1.50 − 0.870i)11-s + 1.69·13-s + (0.275 + 1.67i)15-s + (−0.180 − 0.104i)17-s + (0.519 + 0.899i)19-s + (0.612 + 0.790i)21-s + (1.52 − 0.883i)23-s + (0.941 − 1.63i)25-s + (−0.883 + 0.469i)27-s + 0.633i·29-s + (0.207 − 0.360i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.23030 - 0.282211i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23030 - 0.282211i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-28.5 + 75.7i)T \) |
| 7 | \( 1 + (1.25e3 - 2.04e3i)T \) |
| good | 5 | \( 1 + (918. - 530. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (2.20e4 + 1.27e4i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 4.82e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (1.51e4 + 8.72e3i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-6.76e4 - 1.17e5i)T + (-8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-4.28e5 + 2.47e5i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 4.48e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (-1.92e5 + 3.32e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (4.05e4 + 7.02e4i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 9.81e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.23e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-2.72e6 + 1.57e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (6.10e5 + 3.52e5i)T + (3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.15e7 - 6.67e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-5.42e5 - 9.39e5i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.26e7 + 2.19e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 - 3.64e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (4.68e6 - 8.10e6i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-8.25e5 - 1.42e6i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 3.97e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-2.86e7 + 1.65e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + 1.05e8T + 7.83e15T^{2} \) |
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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
−12.60456813041257772557139094355, −11.48121423961499239539756867972, −10.74858091491543652137034692132, −8.681028518861518357985185783887, −8.076246715487782840019132910074, −6.90670811385225525541514844825, −5.76684218715376679676021061355, −3.46361335397377244113152113812, −2.77385518949527890425348998777, −0.63639990517512890600583905842,
0.68752065318616094904082399717, 3.18727247895415241509560976598, 4.17197048234147477478995748974, 5.14462065828360136351959978998, 7.31527094466522616245205124730, 8.250361607619207250804384264900, 9.290790794113300251850149934070, 10.63721012684633760302977447220, 11.35372932264257816697437308007, 12.90433153689454403723161501069
