| L(s) = 1 | + (−24.2 + 13.9i)2-s + (9.01 + 80.4i)3-s + (262. − 455. i)4-s + (−807. + 466. i)5-s + (−1.34e3 − 1.82e3i)6-s + (−2.05e3 − 1.24e3i)7-s + 7.54e3i·8-s + (−6.39e3 + 1.45e3i)9-s + (1.30e4 − 2.25e4i)10-s + (1.11e4 + 6.43e3i)11-s + (3.90e4 + 1.70e4i)12-s + 1.79e4·13-s + (6.71e4 + 1.33e3i)14-s + (−4.48e4 − 6.08e4i)15-s + (−3.82e4 − 6.61e4i)16-s + (−7.67e4 − 4.43e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.51 + 0.873i)2-s + (0.111 + 0.993i)3-s + (1.02 − 1.77i)4-s + (−1.29 + 0.746i)5-s + (−1.03 − 1.40i)6-s + (−0.855 − 0.517i)7-s + 1.84i·8-s + (−0.975 + 0.221i)9-s + (1.30 − 2.25i)10-s + (0.761 + 0.439i)11-s + (1.88 + 0.822i)12-s + 0.628·13-s + (1.74 + 0.0348i)14-s + (−0.885 − 1.20i)15-s + (−0.583 − 1.00i)16-s + (−0.918 − 0.530i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.105009 - 0.0265589i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.105009 - 0.0265589i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-9.01 - 80.4i)T \) |
| 7 | \( 1 + (2.05e3 + 1.24e3i)T \) |
| good | 2 | \( 1 + (24.2 - 13.9i)T + (128 - 221. i)T^{2} \) |
| 5 | \( 1 + (807. - 466. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-1.11e4 - 6.43e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 1.79e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (7.67e4 + 4.43e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-9.33e4 - 1.61e5i)T + (-8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-7.55e4 + 4.36e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + 5.85e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (-1.49e5 + 2.58e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (1.00e6 + 1.73e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + 9.41e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 1.77e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (1.20e6 - 6.93e5i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (6.50e6 + 3.75e6i)T + (3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-4.03e6 - 2.33e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (7.84e6 + 1.35e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.58e7 + 2.73e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 6.90e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (2.54e7 - 4.40e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (1.34e7 + 2.32e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 3.97e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (5.79e6 - 3.34e6i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.38e7T + 7.83e15T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.05968085824816516093568315524, −15.58223805676658156380813342481, −14.39373182681852718262877076437, −11.45148602220156879133273729459, −10.32820130851130593097761108727, −9.205826405742450151061276737841, −7.77667714119415292686029058888, −6.50835892112197048286524544860, −3.74673126885329166493369193515, −0.10477029948960008532336148243,
1.13478606756700712064003184334, 3.19133200362978959759824652577, 6.90812420698674477217804348207, 8.436849246094224228761764601488, 9.033704096942823499102237235551, 11.26018899000118609279275900787, 12.00936229801184918423820750077, 13.10944706889097750524104769385, 15.66972727982406843991876453141, 16.75624804126521327311636379095
