| L(s) = 1 | − 0.409i·2-s + 1.83·4-s − 4.13i·5-s − 5.18i·7-s − 1.56i·8-s − 1.69·10-s + i·11-s + (2.62 + 2.47i)13-s − 2.12·14-s + 3.02·16-s − 0.488·17-s − 0.446i·19-s − 7.58i·20-s + 0.409·22-s + 5.50·23-s + ⋯ |
| L(s) = 1 | − 0.289i·2-s + 0.916·4-s − 1.85i·5-s − 1.95i·7-s − 0.554i·8-s − 0.535·10-s + 0.301i·11-s + (0.726 + 0.686i)13-s − 0.566·14-s + 0.755·16-s − 0.118·17-s − 0.102i·19-s − 1.69i·20-s + 0.0872·22-s + 1.14·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 + 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.686 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.187141449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.187141449\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-2.62 - 2.47i)T \) |
| good | 2 | \( 1 + 0.409iT - 2T^{2} \) |
| 5 | \( 1 + 4.13iT - 5T^{2} \) |
| 7 | \( 1 + 5.18iT - 7T^{2} \) |
| 17 | \( 1 + 0.488T + 17T^{2} \) |
| 19 | \( 1 + 0.446iT - 19T^{2} \) |
| 23 | \( 1 - 5.50T + 23T^{2} \) |
| 29 | \( 1 - 6.58T + 29T^{2} \) |
| 31 | \( 1 - 1.52iT - 31T^{2} \) |
| 37 | \( 1 - 7.87iT - 37T^{2} \) |
| 41 | \( 1 - 8.78iT - 41T^{2} \) |
| 43 | \( 1 - 4.57T + 43T^{2} \) |
| 47 | \( 1 - 5.42iT - 47T^{2} \) |
| 53 | \( 1 + 8.66T + 53T^{2} \) |
| 59 | \( 1 - 5.60iT - 59T^{2} \) |
| 61 | \( 1 + 0.855T + 61T^{2} \) |
| 67 | \( 1 + 3.87iT - 67T^{2} \) |
| 71 | \( 1 - 8.49iT - 71T^{2} \) |
| 73 | \( 1 + 5.95iT - 73T^{2} \) |
| 79 | \( 1 + 8.91T + 79T^{2} \) |
| 83 | \( 1 + 0.457iT - 83T^{2} \) |
| 89 | \( 1 + 4.68iT - 89T^{2} \) |
| 97 | \( 1 + 2.06iT - 97T^{2} \) |
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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
−9.491070338023510307833979252017, −8.506274115306691391400527277109, −7.75652867675908100585798571114, −6.93325939832301351823207524860, −6.17077092384537102363776790105, −4.72190976920788852979507241744, −4.37285896243308034275840457581, −3.23329658689383957131335173512, −1.43913408478516504430430008265, −1.00854974209107800343645063027,
2.11322879883306743695345118154, 2.78681061224394910953214534455, 3.41031765514103511992517531986, 5.37997230227675458314646501798, 5.99875198943250184686781825560, 6.53848801784987017532632534076, 7.37785084898478134075596594630, 8.246470144326339777538375639290, 9.034327371040932898037938813434, 10.15409899340011214306689553405
