| L(s) = 1 | + (7.92 + 1.39i)2-s + (−179. − 65.4i)4-s + (625. + 745. i)5-s + (845. − 307. i)7-s + (−3.11e3 − 1.79e3i)8-s + (3.91e3 + 6.77e3i)10-s + (1.59e4 − 1.89e4i)11-s + (4.11e3 + 2.33e4i)13-s + (7.13e3 − 1.25e3i)14-s + (1.53e4 + 1.28e4i)16-s + (−9.06e4 + 5.23e4i)17-s + (1.42e4 − 2.47e4i)19-s + (−6.36e4 − 1.74e5i)20-s + (1.52e5 − 1.28e5i)22-s + (−1.10e5 + 3.03e5i)23-s + ⋯ |
| L(s) = 1 | + (0.495 + 0.0873i)2-s + (−0.702 − 0.255i)4-s + (1.00 + 1.19i)5-s + (0.352 − 0.128i)7-s + (−0.760 − 0.439i)8-s + (0.391 + 0.677i)10-s + (1.08 − 1.29i)11-s + (0.144 + 0.817i)13-s + (0.185 − 0.0327i)14-s + (0.233 + 0.196i)16-s + (−1.08 + 0.626i)17-s + (0.109 − 0.189i)19-s + (−0.397 − 1.09i)20-s + (0.651 − 0.546i)22-s + (−0.394 + 1.08i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.18451 + 1.37713i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.18451 + 1.37713i\) |
| \(L(5)\) |
|
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 | \( 1 + (-7.92 - 1.39i)T + (240. + 87.5i)T^{2} \) |
| 5 | \( 1 + (-625. - 745. i)T + (-6.78e4 + 3.84e5i)T^{2} \) |
| 7 | \( 1 + (-845. + 307. i)T + (4.41e6 - 3.70e6i)T^{2} \) |
| 11 | \( 1 + (-1.59e4 + 1.89e4i)T + (-3.72e7 - 2.11e8i)T^{2} \) |
| 13 | \( 1 + (-4.11e3 - 2.33e4i)T + (-7.66e8 + 2.78e8i)T^{2} \) |
| 17 | \( 1 + (9.06e4 - 5.23e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.42e4 + 2.47e4i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.10e5 - 3.03e5i)T + (-5.99e10 - 5.03e10i)T^{2} \) |
| 29 | \( 1 + (-9.09e5 - 1.60e5i)T + (4.70e11 + 1.71e11i)T^{2} \) |
| 31 | \( 1 + (-1.44e6 - 5.24e5i)T + (6.53e11 + 5.48e11i)T^{2} \) |
| 37 | \( 1 + (-4.32e5 - 7.49e5i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (1.02e6 - 1.81e5i)T + (7.50e12 - 2.73e12i)T^{2} \) |
| 43 | \( 1 + (-2.74e6 - 2.30e6i)T + (2.02e12 + 1.15e13i)T^{2} \) |
| 47 | \( 1 + (-4.26e4 - 1.17e5i)T + (-1.82e13 + 1.53e13i)T^{2} \) |
| 53 | \( 1 - 6.71e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (5.75e6 + 6.85e6i)T + (-2.54e13 + 1.44e14i)T^{2} \) |
| 61 | \( 1 + (3.21e6 - 1.17e6i)T + (1.46e14 - 1.23e14i)T^{2} \) |
| 67 | \( 1 + (-4.19e6 - 2.37e7i)T + (-3.81e14 + 1.38e14i)T^{2} \) |
| 71 | \( 1 + (5.60e6 - 3.23e6i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (2.52e6 - 4.37e6i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.07e7 + 6.11e7i)T + (-1.42e15 - 5.18e14i)T^{2} \) |
| 83 | \( 1 + (-2.06e7 - 3.63e6i)T + (2.11e15 + 7.70e14i)T^{2} \) |
| 89 | \( 1 + (5.38e7 + 3.11e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (-1.11e8 - 9.38e7i)T + (1.36e15 + 7.71e15i)T^{2} \) |
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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
−13.45662854476812880387383209984, −11.74575371515487908322903750426, −10.72534889038665698732647875587, −9.583036005236068169876251257583, −8.596899338314855269206751706966, −6.55430945268888075505955556603, −6.05195700194417452882938822798, −4.40661956620556834326879507222, −3.08069945722309576715785428392, −1.33424994608547132069978316449,
0.75923513711880618486399096571, 2.27990140543886823821328868119, 4.34821486151935100116833848128, 4.98495927909328931816492094162, 6.33851177687188859098076062075, 8.269854385008708697026546848864, 9.146955161410007369133702824154, 10.00844497621405339264454970571, 11.93774059155099765965917721375, 12.59801290916997164299143409189
