| L(s) = 1 | + (−0.0721 − 0.409i)2-s + (1.71 − 0.625i)4-s + (1.69 + 1.42i)5-s + (1.24 + 0.451i)7-s + (−0.795 − 1.37i)8-s + (0.460 − 0.797i)10-s + (−3.99 + 3.35i)11-s + (−0.00313 + 0.0177i)13-s + (0.0952 − 0.539i)14-s + (2.29 − 1.92i)16-s + (1.56 − 2.71i)17-s + (−0.208 − 0.361i)19-s + (3.80 + 1.38i)20-s + (1.65 + 1.39i)22-s + (0.972 − 0.353i)23-s + ⋯ |
| L(s) = 1 | + (−0.0510 − 0.289i)2-s + (0.858 − 0.312i)4-s + (0.758 + 0.636i)5-s + (0.468 + 0.170i)7-s + (−0.281 − 0.486i)8-s + (0.145 − 0.252i)10-s + (−1.20 + 1.01i)11-s + (−0.000869 + 0.00493i)13-s + (0.0254 − 0.144i)14-s + (0.573 − 0.481i)16-s + (0.379 − 0.658i)17-s + (−0.0478 − 0.0829i)19-s + (0.850 + 0.309i)20-s + (0.353 + 0.296i)22-s + (0.202 − 0.0737i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57375 - 0.164864i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57375 - 0.164864i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (0.0721 + 0.409i)T + (-1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (-1.69 - 1.42i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.24 - 0.451i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (3.99 - 3.35i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.00313 - 0.0177i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.56 + 2.71i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.208 + 0.361i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.972 + 0.353i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.35 + 7.68i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (3.50 - 1.27i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (2.21 - 3.83i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.638 - 3.61i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (6.36 - 5.34i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-6.66 - 2.42i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 1.30T + 53T^{2} \) |
| 59 | \( 1 + (2.83 + 2.37i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (6.49 + 2.36i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.91 - 10.8i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-3.04 + 5.26i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.273 - 0.473i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.0849 - 0.481i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.801 - 4.54i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-1.68 - 2.92i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.61 + 6.39i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.91861731091374152413235578565, −11.03882312088696563738672828618, −10.17668127736507280942924757655, −9.639118961252779626702895114572, −7.948225830523985184087278566107, −7.05442492212553151321865045780, −6.00691629560190741522052037127, −4.92718968029792367314396753378, −2.88138239861605174587165751004, −1.96559335472103053662194687295,
1.80226328653260689950495554847, 3.31938531815546667277092779730, 5.24404054047464530077155546490, 5.87790044578573527147396171079, 7.25175054062979100950269671762, 8.175901958662447453885875449981, 9.011875593047442609782388299489, 10.50117215753426721455264522375, 10.98692033739861607334057548029, 12.24210792119864349831145307147
