| 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
| \(L(1)\) |
\(\approx\) |
\(1.46690 + 0.153670i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46690 + 0.153670i\) |
| \(L(\frac{3}{2})\) |
|
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
−11.73356242423538875074488227843, −11.51653323872830638845885189680, −10.46404445680996587850171023023, −9.379205151038870486792923260361, −7.911538561249516201825350088777, −7.20038094686424626729559821178, −6.37706604143833495615857724373, −4.48183882656542141336060508462, −3.45513924005130416314990043474, −1.94879556891097156844467792767,
1.49978932016770380955037173260, 3.47568100970377252774720219428, 4.84959335749891059753588645876, 6.05907301532198415726924083886, 7.00510074486454870948101248877, 8.244394137568381913872741687139, 8.823053521871771856248363528396, 10.35149829102117302599837724394, 11.40622945696294274401543425560, 11.77826516770952685124007337799
