| L(s) = 1 | + (1.59 − 0.617i)2-s + (−1.40 − 1.00i)3-s + (0.683 − 0.622i)4-s + (−0.740 + 2.60i)5-s + (−2.86 − 0.739i)6-s + (2.46 − 3.71i)7-s + (−0.819 + 1.64i)8-s + (0.963 + 2.84i)9-s + (0.426 + 4.60i)10-s + (0.620 + 0.499i)11-s + (−1.59 + 0.187i)12-s + (−3.71 + 5.60i)13-s + (1.63 − 7.44i)14-s + (3.66 − 2.91i)15-s + (−0.460 + 4.97i)16-s + (3.21 − 1.72i)17-s + ⋯ |
| L(s) = 1 | + (1.12 − 0.436i)2-s + (−0.812 − 0.582i)3-s + (0.341 − 0.311i)4-s + (−0.331 + 1.16i)5-s + (−1.17 − 0.301i)6-s + (0.930 − 1.40i)7-s + (−0.289 + 0.581i)8-s + (0.321 + 0.946i)9-s + (0.134 + 1.45i)10-s + (0.187 + 0.150i)11-s + (−0.459 + 0.0541i)12-s + (−1.03 + 1.55i)13-s + (0.435 − 1.99i)14-s + (0.946 − 0.752i)15-s + (−0.115 + 1.24i)16-s + (0.780 − 0.419i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.89430 + 0.437252i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89430 + 0.437252i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.40 + 1.00i)T \) |
| 103 | \( 1 + (9.16 + 4.36i)T \) |
| good | 2 | \( 1 + (-1.59 + 0.617i)T + (1.47 - 1.34i)T^{2} \) |
| 5 | \( 1 + (0.740 - 2.60i)T + (-4.25 - 2.63i)T^{2} \) |
| 7 | \( 1 + (-2.46 + 3.71i)T + (-2.72 - 6.44i)T^{2} \) |
| 11 | \( 1 + (-0.620 - 0.499i)T + (2.35 + 10.7i)T^{2} \) |
| 13 | \( 1 + (3.71 - 5.60i)T + (-5.06 - 11.9i)T^{2} \) |
| 17 | \( 1 + (-3.21 + 1.72i)T + (9.39 - 14.1i)T^{2} \) |
| 19 | \( 1 + (-0.858 - 2.43i)T + (-14.8 + 11.9i)T^{2} \) |
| 23 | \( 1 + (-0.662 - 4.26i)T + (-21.9 + 6.97i)T^{2} \) |
| 29 | \( 1 + (-1.14 + 4.01i)T + (-24.6 - 15.2i)T^{2} \) |
| 31 | \( 1 + (-3.46 + 1.59i)T + (20.1 - 23.5i)T^{2} \) |
| 37 | \( 1 + (3.87 - 5.13i)T + (-10.1 - 35.5i)T^{2} \) |
| 41 | \( 1 + (-9.77 - 2.45i)T + (36.1 + 19.3i)T^{2} \) |
| 43 | \( 1 + (-0.728 - 0.964i)T + (-11.7 + 41.3i)T^{2} \) |
| 47 | \( 1 + (-3.83 - 6.64i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.22 + 6.09i)T + (-8.12 - 52.3i)T^{2} \) |
| 59 | \( 1 + (-4.60 - 6.95i)T + (-22.9 + 54.3i)T^{2} \) |
| 61 | \( 1 + (10.2 - 5.53i)T + (33.6 - 50.8i)T^{2} \) |
| 67 | \( 1 + (-2.17 + 4.37i)T + (-40.3 - 53.4i)T^{2} \) |
| 71 | \( 1 + (8.88 + 2.23i)T + (62.5 + 33.5i)T^{2} \) |
| 73 | \( 1 + (-0.113 - 0.400i)T + (-62.0 + 38.4i)T^{2} \) |
| 79 | \( 1 + (-1.09 - 1.12i)T + (-2.43 + 78.9i)T^{2} \) |
| 83 | \( 1 + (0.612 + 1.22i)T + (-50.0 + 66.2i)T^{2} \) |
| 89 | \( 1 + (7.98 + 7.28i)T + (8.21 + 88.6i)T^{2} \) |
| 97 | \( 1 + (15.8 - 9.80i)T + (43.2 - 86.8i)T^{2} \) |
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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
−10.57680024905007751268718537312, −9.684180650098522558335251815236, −7.956749972121297562651156530563, −7.35405205047425505579733173137, −6.75493296383813338907488341836, −5.65620899019582866547308960648, −4.55443045335960471328483983851, −4.12638826719439448686547079812, −2.78643645018892609229436090575, −1.53216443908108426109198106350,
0.76565541510331834193251153685, 2.87499999423745000729939008873, 4.20133153096757586757352458367, 5.04550729698200018632172100586, 5.32706601721356879505923924676, 5.96516562614087057009642833012, 7.30860277564450700553391353114, 8.438587640419532365345428643012, 9.067843765578907311670953990843, 10.04308377401964814632590484166
