| L(s) = 1 | − 1.73i·2-s + 1.00·4-s − 3.46i·5-s + 2·7-s − 8.66i·8-s − 5.99·10-s − 1.73i·11-s − 4·13-s − 3.46i·14-s − 10.9·16-s + 15.5i·17-s + 11·19-s − 3.46i·20-s − 2.99·22-s + 27.7i·23-s + ⋯ |
| L(s) = 1 | − 0.866i·2-s + 0.250·4-s − 0.692i·5-s + 0.285·7-s − 1.08i·8-s − 0.599·10-s − 0.157i·11-s − 0.307·13-s − 0.247i·14-s − 0.687·16-s + 0.916i·17-s + 0.578·19-s − 0.173i·20-s − 0.136·22-s + 1.20i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.01152 - 1.01152i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01152 - 1.01152i\) |
| \(L(2)\) |
|
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 | \( 1 + 1.73iT - 4T^{2} \) |
| 5 | \( 1 + 3.46iT - 25T^{2} \) |
| 7 | \( 1 - 2T + 49T^{2} \) |
| 11 | \( 1 + 1.73iT - 121T^{2} \) |
| 13 | \( 1 + 4T + 169T^{2} \) |
| 17 | \( 1 - 15.5iT - 289T^{2} \) |
| 19 | \( 1 - 11T + 361T^{2} \) |
| 23 | \( 1 - 27.7iT - 529T^{2} \) |
| 29 | \( 1 - 45.0iT - 841T^{2} \) |
| 31 | \( 1 - 32T + 961T^{2} \) |
| 37 | \( 1 + 34T + 1.36e3T^{2} \) |
| 41 | \( 1 - 12.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 61T + 1.84e3T^{2} \) |
| 47 | \( 1 + 48.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 + 50.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 56T + 3.72e3T^{2} \) |
| 67 | \( 1 + 31T + 4.48e3T^{2} \) |
| 71 | \( 1 + 31.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 65T + 5.32e3T^{2} \) |
| 79 | \( 1 - 38T + 6.24e3T^{2} \) |
| 83 | \( 1 + 48.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 124. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 115T + 9.40e3T^{2} \) |
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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
−13.51448435198373463008825810544, −12.54067656517139737040194692719, −11.72877688759363686535417870674, −10.67495667156671470596313301077, −9.600538665297463033232587871654, −8.301424638954119487778952888415, −6.82388651889356636195690988481, −5.12602344708600068734670111311, −3.42511858789640607936777590708, −1.50473706047180571496209978742,
2.65536699747951785015437441508, 4.91767619016850799903901619266, 6.37807969982557596340368666874, 7.28003725026865185845865828635, 8.377248914493064909821380960478, 9.968739564659839038337420143615, 11.16580948122811582814913551340, 12.06995674983017770374194116785, 13.74604169606526625164836059246, 14.55966552129171568996663950980
