| L(s) = 1 | − 1.41·2-s + 2.00·4-s + (−0.174 − 4.99i)5-s − 9.47i·7-s − 2.82·8-s + (0.246 + 7.06i)10-s + 20.3i·11-s − 6.10i·13-s + 13.4i·14-s + 4.00·16-s + 24.0·17-s + 16.0·19-s + (−0.348 − 9.99i)20-s − 28.8i·22-s + 43.2·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s + (−0.0348 − 0.999i)5-s − 1.35i·7-s − 0.353·8-s + (0.0246 + 0.706i)10-s + 1.85i·11-s − 0.469i·13-s + 0.957i·14-s + 0.250·16-s + 1.41·17-s + 0.846·19-s + (−0.0174 − 0.499i)20-s − 1.31i·22-s + 1.88·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0348 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0348 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.340246869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.340246869\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.174 + 4.99i)T \) |
| good | 7 | \( 1 + 9.47iT - 49T^{2} \) |
| 11 | \( 1 - 20.3iT - 121T^{2} \) |
| 13 | \( 1 + 6.10iT - 169T^{2} \) |
| 17 | \( 1 - 24.0T + 289T^{2} \) |
| 19 | \( 1 - 16.0T + 361T^{2} \) |
| 23 | \( 1 - 43.2T + 529T^{2} \) |
| 29 | \( 1 + 19.6iT - 841T^{2} \) |
| 31 | \( 1 - 30.5T + 961T^{2} \) |
| 37 | \( 1 + 44.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 48.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 48.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 68.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 61.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 50.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 48.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 55.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 23.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 40.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 33.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 22.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 79.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 119. iT - 9.40e3T^{2} \) |
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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
−9.798784897406320040541058528113, −9.232583736532435921840663707064, −7.83562432345624506660971919130, −7.62795158685152952798348149570, −6.66431554834524564262839779505, −5.22291132990101407970392217150, −4.52307093444991513798380191116, −3.27684741781832948033444227576, −1.57758519409869178746269204408, −0.68708480865205314799314050672,
1.21412385905775676297975760910, 3.00102306708339986940112458926, 3.10856160620128226801741527308, 5.25732675200615646753559078270, 6.00456973415864844021150914757, 6.79365436416916918517524044331, 7.87304948747086035982598884639, 8.604972434752901028060902421423, 9.342010545202553184013096533254, 10.19447584560023460857026131028
