L(s) = 1 | + (−1.89 + 0.691i)2-s + (1.59 − 1.34i)4-s + (0.766 + 0.642i)5-s + (−1.54 − 2.67i)7-s + (−0.0878 + 0.152i)8-s + (−1.89 − 0.691i)10-s + (0.481 − 0.834i)11-s + (−0.513 − 2.91i)13-s + (4.78 + 4.01i)14-s + (−0.663 + 3.76i)16-s + (0.0366 − 0.0133i)17-s + (−4.31 + 0.596i)19-s + 2.08·20-s + (−0.338 + 1.91i)22-s + (−2.97 + 2.49i)23-s + ⋯ |
L(s) = 1 | + (−1.34 + 0.488i)2-s + (0.799 − 0.670i)4-s + (0.342 + 0.287i)5-s + (−0.584 − 1.01i)7-s + (−0.0310 + 0.0538i)8-s + (−0.600 − 0.218i)10-s + (0.145 − 0.251i)11-s + (−0.142 − 0.808i)13-s + (1.27 + 1.07i)14-s + (−0.165 + 0.940i)16-s + (0.00887 − 0.00323i)17-s + (−0.990 + 0.136i)19-s + 0.466·20-s + (−0.0720 + 0.408i)22-s + (−0.620 + 0.520i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00735523 - 0.0415021i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00735523 - 0.0415021i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 + (4.31 - 0.596i)T \) |
good | 2 | \( 1 + (1.89 - 0.691i)T + (1.53 - 1.28i)T^{2} \) |
| 7 | \( 1 + (1.54 + 2.67i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.481 + 0.834i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.513 + 2.91i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.0366 + 0.0133i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (2.97 - 2.49i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (8.76 + 3.19i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-4.68 - 8.12i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.11T + 37T^{2} \) |
| 41 | \( 1 + (2.09 - 11.8i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (3.79 + 3.18i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-7.53 - 2.74i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (4.64 - 3.89i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-2.60 + 0.948i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (7.68 - 6.44i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (8.78 + 3.19i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (9.64 + 8.09i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.155 + 0.883i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.54 + 8.75i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (4.80 + 8.31i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.51 - 8.61i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (7.31 - 2.66i)T + (74.3 - 62.3i)T^{2} \) |
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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
−9.795065011893696194725037694480, −9.001886895209614848091693629254, −8.054440694607076805478517790726, −7.41145093187799625077850207494, −6.58502077822751553185695614091, −5.86590427599427967893675254334, −4.30189278066795494703747552421, −3.18574364272067173992496979696, −1.50314339569445336468794478788, −0.03156016782010521147885970706,
1.82836967619592702105782387186, 2.52553177419025236291947497826, 4.13780953447665482469329621810, 5.42311025265904269161438782697, 6.36805345279583360945628695797, 7.36464457544442442375591048909, 8.448893681670000334283581731136, 9.016529922683902533227182672115, 9.563511921213207852002139895806, 10.29533004682933068117033167196