| L(s) = 1 | + (−0.313 − 0.0553i)2-s + (1.24 − 1.20i)3-s + (−1.78 − 0.649i)4-s + (−2.68 − 0.977i)5-s + (−0.456 + 0.310i)6-s + (−2.48 + 0.913i)7-s + (1.07 + 0.621i)8-s + (0.0851 − 2.99i)9-s + (0.788 + 0.455i)10-s + (−1.16 − 3.20i)11-s + (−2.99 + 1.34i)12-s + (−1.17 + 3.21i)13-s + (0.830 − 0.149i)14-s + (−4.51 + 2.02i)15-s + (2.60 + 2.18i)16-s + (2.33 − 4.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.221 − 0.0391i)2-s + (0.717 − 0.697i)3-s + (−0.891 − 0.324i)4-s + (−1.20 − 0.437i)5-s + (−0.186 + 0.126i)6-s + (−0.938 + 0.345i)7-s + (0.380 + 0.219i)8-s + (0.0283 − 0.999i)9-s + (0.249 + 0.144i)10-s + (−0.351 − 0.965i)11-s + (−0.865 + 0.388i)12-s + (−0.324 + 0.891i)13-s + (0.221 − 0.0398i)14-s + (−1.16 + 0.523i)15-s + (0.651 + 0.546i)16-s + (0.565 − 0.979i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.163675 - 0.567916i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.163675 - 0.567916i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.24 + 1.20i)T \) |
| 7 | \( 1 + (2.48 - 0.913i)T \) |
| good | 2 | \( 1 + (0.313 + 0.0553i)T + (1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (2.68 + 0.977i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (1.16 + 3.20i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (1.17 - 3.21i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.33 + 4.03i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.78 + 2.76i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.35 - 0.238i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.81 + 7.73i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.756 + 2.07i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + 1.76T + 37T^{2} \) |
| 41 | \( 1 + (-4.96 - 1.80i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.57 - 8.95i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.87 + 1.04i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-1.30 + 0.756i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.25 - 4.41i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-3.36 - 9.24i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.66 + 9.44i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (9.64 - 5.56i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 5.91iT - 73T^{2} \) |
| 79 | \( 1 + (-1.98 + 11.2i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-6.72 + 2.44i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-6.12 - 10.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.47 + 0.436i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.17608563026894617693509788662, −11.50830915511552241017571230315, −9.693074827671973149173118579033, −9.147467875483370150979789843421, −8.153485319501302511560917011345, −7.34806685467150638423153188111, −5.83203884943547788282807327826, −4.27568291182342968777515778690, −3.05098076693775852914602620468, −0.54022383259787437040262735551,
3.30547110079981480068675956805, 3.85600645559787422371959193515, 5.20499416360862987582487733293, 7.41099035357330059126163004417, 7.81677444742250327309269174519, 9.002015913816926875878054492063, 10.05283696775161760087532900442, 10.52733039760389813648884729245, 12.26046493464549781448447467347, 12.87048879350258579115704960661
