L(s) = 1 | + (40.5 + 23.3i)3-s + (632. − 365. i)5-s + (984. − 2.18e3i)7-s + (1.09e3 + 1.89e3i)9-s + (6.35e3 − 1.10e4i)11-s − 2.11e4i·13-s + 3.41e4·15-s + (−8.25e4 − 4.76e4i)17-s + (−1.33e5 + 7.69e4i)19-s + (9.10e4 − 6.56e4i)21-s + (−4.71e4 − 8.15e4i)23-s + (7.14e4 − 1.23e5i)25-s + 1.02e5i·27-s + 5.41e5·29-s + (−1.85e5 − 1.07e5i)31-s + ⋯ |
L(s) = 1 | + (0.5 + 0.288i)3-s + (1.01 − 0.584i)5-s + (0.410 − 0.912i)7-s + (0.166 + 0.288i)9-s + (0.433 − 0.751i)11-s − 0.741i·13-s + 0.674·15-s + (−0.988 − 0.570i)17-s + (−1.02 + 0.590i)19-s + (0.468 − 0.337i)21-s + (−0.168 − 0.291i)23-s + (0.182 − 0.316i)25-s + 0.192i·27-s + 0.765·29-s + (−0.201 − 0.116i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(2.16422 - 1.71975i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.16422 - 1.71975i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-40.5 - 23.3i)T \) |
| 7 | \( 1 + (-984. + 2.18e3i)T \) |
good | 5 | \( 1 + (-632. + 365. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-6.35e3 + 1.10e4i)T + (-1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + 2.11e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (8.25e4 + 4.76e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (1.33e5 - 7.69e4i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (4.71e4 + 8.15e4i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 - 5.41e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (1.85e5 + 1.07e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (3.70e5 + 6.41e5i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 7.82e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 2.21e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-7.13e6 + 4.11e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-6.59e5 + 1.14e6i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (7.52e6 + 4.34e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-1.31e7 + 7.57e6i)T + (9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.99e7 + 3.44e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 9.34e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-2.94e7 - 1.70e7i)T + (4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-2.26e7 - 3.92e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 - 6.58e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (1.09e7 - 6.34e6i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 - 6.15e6iT - 7.83e15T^{2} \) |
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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
−12.68981140912274944075768501892, −11.05430352076333646901557109937, −10.15479631191783430705214276289, −9.032589466229227319450692548917, −8.092507215434615528732534910382, −6.52310573182968975494889717313, −5.14166988964085291077443936527, −3.86567961113634071092691135845, −2.15759967470576661804300021652, −0.75434588099818830649234733394,
1.79697508301208897305381915796, 2.47245478673347349570379229669, 4.41907988757164802498274735896, 6.06741407199783422745234723761, 6.95067881729296740608667443335, 8.587335151113134437735436021386, 9.374940809925590389540347509284, 10.60049067273313692160854315428, 11.87302316422434469940346295753, 12.96715853477216538397288296419