| L(s) = 1 | + 0.432i·2-s + 3.81·4-s + 3.81i·5-s + 8.57·7-s + 3.37i·8-s − 1.65·10-s − 13.2i·11-s + 1.78·13-s + 3.70i·14-s + 13.7·16-s + 12.1i·17-s + 23.0·19-s + 14.5i·20-s + 5.72·22-s − 8.79i·23-s + ⋯ |
| L(s) = 1 | + 0.216i·2-s + 0.953·4-s + 0.763i·5-s + 1.22·7-s + 0.422i·8-s − 0.165·10-s − 1.20i·11-s + 0.136·13-s + 0.264i·14-s + 0.861·16-s + 0.713i·17-s + 1.21·19-s + 0.727i·20-s + 0.260·22-s − 0.382i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.758420278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.758420278\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 - 8.42iT \) |
| good | 2 | \( 1 - 0.432iT - 4T^{2} \) |
| 5 | \( 1 - 3.81iT - 25T^{2} \) |
| 7 | \( 1 - 8.57T + 49T^{2} \) |
| 11 | \( 1 + 13.2iT - 121T^{2} \) |
| 13 | \( 1 - 1.78T + 169T^{2} \) |
| 17 | \( 1 - 12.1iT - 289T^{2} \) |
| 19 | \( 1 - 23.0T + 361T^{2} \) |
| 23 | \( 1 + 8.79iT - 529T^{2} \) |
| 29 | \( 1 + 40.9iT - 841T^{2} \) |
| 31 | \( 1 + 53.4T + 961T^{2} \) |
| 37 | \( 1 - 9.58T + 1.36e3T^{2} \) |
| 41 | \( 1 - 62.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 47.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 80.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 77.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 2.47iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 108.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 37.5T + 4.48e3T^{2} \) |
| 73 | \( 1 + 121.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 139.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 145. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 27.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 16.4T + 9.40e3T^{2} \) |
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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
−10.80798547585170519336910501983, −9.722670946626766333860546769638, −8.311674968125769461554439604130, −7.905886698065477397698154562644, −6.88008198609960426239224399755, −6.03741178225994486827219635490, −5.18765365114919050571239932267, −3.65277867352468594340382433480, −2.62558440873589269273504575803, −1.35946888400381330654338019532,
1.24865231561883152100467817909, 2.08800106581076535450470569475, 3.56923659750971991193814601429, 4.94514456510022389432130538633, 5.43014212702246246907852754695, 7.11647447105577592442453886020, 7.39160417326101464871267180502, 8.568919273309348118778004687706, 9.437479390712617860062152801426, 10.42472812848431025932735323508
