| L(s) = 1 | + (0.245 + 0.0432i)2-s + (1.36 + 1.06i)3-s + (−1.82 − 0.662i)4-s + (1.99 + 0.727i)5-s + (0.289 + 0.319i)6-s + (0.302 + 2.62i)7-s + (−0.848 − 0.489i)8-s + (0.749 + 2.90i)9-s + (0.458 + 0.264i)10-s + (−0.489 − 1.34i)11-s + (−1.79 − 2.83i)12-s + (1.30 − 3.59i)13-s + (−0.0394 + 0.657i)14-s + (1.96 + 3.11i)15-s + (2.78 + 2.33i)16-s + (0.109 − 0.188i)17-s + ⋯ |
| L(s) = 1 | + (0.173 + 0.0305i)2-s + (0.790 + 0.612i)3-s + (−0.910 − 0.331i)4-s + (0.893 + 0.325i)5-s + (0.118 + 0.130i)6-s + (0.114 + 0.993i)7-s + (−0.300 − 0.173i)8-s + (0.249 + 0.968i)9-s + (0.144 + 0.0836i)10-s + (−0.147 − 0.405i)11-s + (−0.516 − 0.819i)12-s + (0.362 − 0.996i)13-s + (−0.0105 + 0.175i)14-s + (0.507 + 0.804i)15-s + (0.695 + 0.583i)16-s + (0.0264 − 0.0458i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40749 + 0.549573i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40749 + 0.549573i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.36 - 1.06i)T \) |
| 7 | \( 1 + (-0.302 - 2.62i)T \) |
| good | 2 | \( 1 + (-0.245 - 0.0432i)T + (1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (-1.99 - 0.727i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (0.489 + 1.34i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.30 + 3.59i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.109 + 0.188i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.56 + 1.48i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (7.61 - 1.34i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (1.78 + 4.90i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.32 + 3.64i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 - 6.01T + 37T^{2} \) |
| 41 | \( 1 + (10.0 + 3.66i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.29 + 7.34i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (5.51 - 2.00i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-12.1 + 6.98i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.21 - 1.01i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.45 - 3.99i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.59 - 14.6i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (4.29 - 2.47i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 2.46iT - 73T^{2} \) |
| 79 | \( 1 + (0.675 - 3.83i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (0.141 - 0.0515i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-3.40 - 5.89i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.2 - 2.16i)T + (91.1 + 33.1i)T^{2} \) |
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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
−13.19908782774961127423815339036, −11.70708814618209135432980899228, −10.21233083714849181611118976713, −9.807326654448461225736125303175, −8.769286505945915401104043930285, −8.009131364638998357272695507365, −5.92438991914060670908998532288, −5.31835106714656232667183434996, −3.78718810579946057681844467658, −2.38863988067774986687580120689,
1.62776631223814233113410741590, 3.53297375748105547014833633163, 4.66282966517338537264633727757, 6.22341254016753698268513304621, 7.48361777242436134087385162679, 8.428235615188390865712630505532, 9.441031091446585425181385525523, 10.07676679895715191109712385604, 11.81185134629915073540043542174, 12.78735239170443473090049787891
