| L(s) = 1 | + (21.2 + 78.1i)3-s + (609. + 351. i)5-s + (162. + 2.39e3i)7-s + (−5.65e3 + 3.32e3i)9-s + (−1.59e4 + 9.23e3i)11-s − 5.80e3·13-s + (−1.45e4 + 5.51e4i)15-s + (9.89e4 − 5.71e4i)17-s + (−3.49e4 + 6.06e4i)19-s + (−1.83e5 + 6.36e4i)21-s + (2.55e5 + 1.47e5i)23-s + (5.21e4 + 9.02e4i)25-s + (−3.80e5 − 3.71e5i)27-s + 6.35e5i·29-s + (−4.10e5 − 7.10e5i)31-s + ⋯ |
| L(s) = 1 | + (0.262 + 0.964i)3-s + (0.974 + 0.562i)5-s + (0.0676 + 0.997i)7-s + (−0.861 + 0.507i)9-s + (−1.09 + 0.630i)11-s − 0.203·13-s + (−0.286 + 1.08i)15-s + (1.18 − 0.684i)17-s + (−0.268 + 0.465i)19-s + (−0.944 + 0.327i)21-s + (0.911 + 0.526i)23-s + (0.133 + 0.231i)25-s + (−0.715 − 0.698i)27-s + 0.898i·29-s + (−0.444 − 0.769i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.120043 + 1.89398i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.120043 + 1.89398i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-21.2 - 78.1i)T \) |
| 7 | \( 1 + (-162. - 2.39e3i)T \) |
| good | 5 | \( 1 + (-609. - 351. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (1.59e4 - 9.23e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + 5.80e3T + 8.15e8T^{2} \) |
| 17 | \( 1 + (-9.89e4 + 5.71e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (3.49e4 - 6.06e4i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-2.55e5 - 1.47e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 - 6.35e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (4.10e5 + 7.10e5i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-3.32e5 + 5.75e5i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 2.64e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 5.38e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-3.95e6 - 2.28e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-3.92e6 + 2.26e6i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (1.48e7 - 8.58e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (6.38e6 - 1.10e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (3.15e6 + 5.47e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 3.32e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (3.18e6 + 5.50e6i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (3.46e4 - 5.99e4i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + 8.04e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-8.74e7 - 5.05e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 7.56e7T + 7.83e15T^{2} \) |
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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
−13.29015189587408002279264885310, −12.00148022270201307197797938925, −10.64001786911017519270559904868, −9.886297496332536915664586009788, −9.000046284321580743116544238787, −7.57140656930661499714234758055, −5.78315013867826838990425988007, −5.05358520935073955480523984310, −3.11266786250857558312567614371, −2.16057507834431773927885260231,
0.52122339560101204766995706368, 1.62853608956786554472956068884, 3.11119891599593967322231049852, 5.11480282315937480923020711051, 6.27918483082871990012630036235, 7.58126263294042036415115870993, 8.539496449020214143440689614552, 9.889496784767584014630793090370, 10.97375940785777070259249918773, 12.49120814736262341218357951263
