| L(s) = 1 | − 1.73i·3-s + 6·5-s − 6.92i·7-s − 2.99·9-s + 20.7i·11-s − 14·13-s − 10.3i·15-s − 6·17-s + 6.92i·19-s − 11.9·21-s + 11·25-s + 5.19i·27-s + 30·29-s − 20.7i·31-s + 36·33-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.20·5-s − 0.989i·7-s − 0.333·9-s + 1.88i·11-s − 1.07·13-s − 0.692i·15-s − 0.352·17-s + 0.364i·19-s − 0.571·21-s + 0.440·25-s + 0.192i·27-s + 1.03·29-s − 0.670i·31-s + 1.09·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.19474 - 0.320131i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19474 - 0.320131i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| good | 5 | \( 1 - 6T + 25T^{2} \) |
| 7 | \( 1 + 6.92iT - 49T^{2} \) |
| 11 | \( 1 - 20.7iT - 121T^{2} \) |
| 13 | \( 1 + 14T + 169T^{2} \) |
| 17 | \( 1 + 6T + 289T^{2} \) |
| 19 | \( 1 - 6.92iT - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 30T + 841T^{2} \) |
| 31 | \( 1 + 20.7iT - 961T^{2} \) |
| 37 | \( 1 - 26T + 1.36e3T^{2} \) |
| 41 | \( 1 + 54T + 1.68e3T^{2} \) |
| 43 | \( 1 - 20.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 41.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 18T + 2.80e3T^{2} \) |
| 59 | \( 1 + 20.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 70T + 3.72e3T^{2} \) |
| 67 | \( 1 + 117. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 83.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 82T + 5.32e3T^{2} \) |
| 79 | \( 1 + 76.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 20.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 114T + 7.92e3T^{2} \) |
| 97 | \( 1 - 34T + 9.40e3T^{2} \) |
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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
−15.07500885625071748659671228376, −14.05566349905762346618557539727, −13.11797727171950896711337448925, −12.11142725612442185527413440496, −10.30235033541122667450196192061, −9.586859779110616616445593412586, −7.60205019610170951882175267444, −6.57443135448348026528823162544, −4.77526890252897284178461225069, −2.03879976439589166055243716567,
2.76052191293842014090323933196, 5.20828591504842669058027768061, 6.23404957279477630116313023620, 8.538657292628325343749570992944, 9.464515365343886291359577295796, 10.68013238286750354977213958990, 11.97197820720414649674190343116, 13.45785257766367994033987543481, 14.32956604717959136396154646130, 15.55805893655081970887031122188
