| L(s) = 1 | + (−19.4 − 11.2i)2-s + (124 + 214. i)4-s + (−194. + 112. i)5-s + (875 − 1.51e3i)7-s + 179. i·8-s + 5.04e3·10-s + (−6.02e3 − 3.47e3i)11-s + (−1.28e4 − 2.22e4i)13-s + (−3.40e4 + 1.96e4i)14-s + (3.37e4 − 5.84e4i)16-s + 7.48e4i·17-s + 1.89e4·19-s + (−4.82e4 − 2.78e4i)20-s + (7.81e4 + 1.35e5i)22-s + (−4.07e5 + 2.35e5i)23-s + ⋯ |
| L(s) = 1 | + (−1.21 − 0.701i)2-s + (0.484 + 0.838i)4-s + (−0.311 + 0.179i)5-s + (0.364 − 0.631i)7-s + 0.0438i·8-s + 0.504·10-s + (−0.411 − 0.237i)11-s + (−0.450 − 0.780i)13-s + (−0.885 + 0.511i)14-s + (0.515 − 0.892i)16-s + 0.896i·17-s + 0.145·19-s + (−0.301 − 0.173i)20-s + (0.333 + 0.577i)22-s + (−1.45 + 0.840i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.659880 - 0.0577321i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.659880 - 0.0577321i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (19.4 + 11.2i)T + (128 + 221. i)T^{2} \) |
| 5 | \( 1 + (194. - 112. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + (-875 + 1.51e3i)T + (-2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (6.02e3 + 3.47e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + (1.28e4 + 2.22e4i)T + (-4.07e8 + 7.06e8i)T^{2} \) |
| 17 | \( 1 - 7.48e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.89e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + (4.07e5 - 2.35e5i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (3.99e5 + 2.30e5i)T + (2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-1.75e5 - 3.04e5i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 - 1.33e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (-1.62e6 + 9.37e5i)T + (3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-1.76e6 + 3.05e6i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-3.53e6 - 2.04e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + 6.60e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-1.18e7 + 6.85e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (3.76e5 - 6.52e5i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.13e6 + 1.96e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 1.70e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.76e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-1.14e7 + 1.99e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-4.01e7 - 2.31e7i)T + (1.12e15 + 1.95e15i)T^{2} \) |
| 89 | \( 1 - 7.26e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (7.36e7 - 1.27e8i)T + (-3.91e15 - 6.78e15i)T^{2} \) |
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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
−12.30796861312619917384193903064, −11.20203587345832914102932461398, −10.46452377159925688304837085530, −9.542770807702947580491264480089, −8.124201190581286465892162237412, −7.55306154539202547676694903213, −5.56741581034135348180565652314, −3.72642963574617506583762016779, −2.14907356405108403945777559391, −0.76263154303218937539059003787,
0.45842670445318595436787465684, 2.21092675827630173278218333157, 4.39399955133605347117491956910, 5.99810159198207773931778732918, 7.32155137525356874015971309843, 8.189665942224407446366116575572, 9.200875424654462829543551360597, 10.11195185361960647541908871515, 11.55590808686884334884346506621, 12.53911059800234375216215022990
