| L(s) = 1 | + (0.817 − 0.240i)3-s + (2.57 + 1.65i)5-s + (−0.181 − 1.26i)7-s + (−1.91 + 1.22i)9-s + (0.0891 − 0.195i)11-s + (0.133 − 0.926i)13-s + (2.50 + 0.735i)15-s + (1.65 + 1.91i)17-s + (2.99 − 3.45i)19-s + (−0.452 − 0.990i)21-s + (−3.67 − 3.08i)23-s + (1.81 + 3.98i)25-s + (−2.94 + 3.39i)27-s + (−1.37 − 1.58i)29-s + (−6.97 − 2.04i)31-s + ⋯ |
| L(s) = 1 | + (0.471 − 0.138i)3-s + (1.15 + 0.740i)5-s + (−0.0687 − 0.477i)7-s + (−0.637 + 0.409i)9-s + (0.0268 − 0.0588i)11-s + (0.0369 − 0.257i)13-s + (0.646 + 0.189i)15-s + (0.402 + 0.464i)17-s + (0.687 − 0.793i)19-s + (−0.0986 − 0.216i)21-s + (−0.766 − 0.642i)23-s + (0.363 + 0.797i)25-s + (−0.566 + 0.653i)27-s + (−0.254 − 0.293i)29-s + (−1.25 − 0.368i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48236 + 0.0991809i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48236 + 0.0991809i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (3.67 + 3.08i)T \) |
| good | 3 | \( 1 + (-0.817 + 0.240i)T + (2.52 - 1.62i)T^{2} \) |
| 5 | \( 1 + (-2.57 - 1.65i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (0.181 + 1.26i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-0.0891 + 0.195i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.133 + 0.926i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-1.65 - 1.91i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-2.99 + 3.45i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (1.37 + 1.58i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (6.97 + 2.04i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (8.67 - 5.57i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-4.48 - 2.88i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (9.26 - 2.72i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 5.35T + 47T^{2} \) |
| 53 | \( 1 + (2.04 + 14.2i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (0.584 - 4.06i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (6.83 + 2.00i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-1.66 - 3.64i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-2.94 - 6.43i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-2.53 + 2.93i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (1.92 - 13.3i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-8.81 + 5.66i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-4.62 + 1.35i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-9.77 - 6.28i)T + (40.2 + 88.2i)T^{2} \) |
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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
−12.96004086814453849754121705130, −11.50713797285152977518661164953, −10.52111440043194189985904814826, −9.782903057080590890614365684450, −8.622882596549457712380716201332, −7.48300385979779943668639600783, −6.35424725554119430989964163257, −5.29472971939934674861060434863, −3.38055353496730746764729192911, −2.13419264130997883025339260075,
1.90228671487078440457882589827, 3.49870233159668770648382169131, 5.31504315112052571143868072114, 5.96306276764303344610382613939, 7.59542105784530688634747867650, 9.027524452199123163094834196417, 9.234210428085550807125074221035, 10.39842505801745192076370310503, 11.86863811006206424631881190246, 12.54651527786567509902195479585
