L(s) = 1 | + (−0.939 + 0.342i)3-s + (−0.613 + 3.47i)5-s + (1.85 + 3.21i)7-s + (0.766 − 0.642i)9-s + (−2.64 + 4.58i)11-s + (0.213 + 0.0775i)13-s + (−0.613 − 3.47i)15-s + (−1.26 − 1.06i)17-s + (4.17 + 1.24i)19-s + (−2.84 − 2.38i)21-s + (−1.50 − 8.54i)23-s + (−7.02 − 2.55i)25-s + (−0.500 + 0.866i)27-s + (0.0923 − 0.0775i)29-s + (1.56 + 2.70i)31-s + ⋯ |
L(s) = 1 | + (−0.542 + 0.197i)3-s + (−0.274 + 1.55i)5-s + (0.702 + 1.21i)7-s + (0.255 − 0.214i)9-s + (−0.797 + 1.38i)11-s + (0.0590 + 0.0215i)13-s + (−0.158 − 0.898i)15-s + (−0.307 − 0.257i)17-s + (0.958 + 0.285i)19-s + (−0.621 − 0.521i)21-s + (−0.314 − 1.78i)23-s + (−1.40 − 0.511i)25-s + (−0.0962 + 0.166i)27-s + (0.0171 − 0.0143i)29-s + (0.280 + 0.485i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.140763 + 1.01975i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.140763 + 1.01975i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 + (-4.17 - 1.24i)T \) |
good | 5 | \( 1 + (0.613 - 3.47i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.85 - 3.21i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.64 - 4.58i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.213 - 0.0775i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.26 + 1.06i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (1.50 + 8.54i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.0923 + 0.0775i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.56 - 2.70i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 + (6.67 - 2.43i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.929 + 5.27i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-1.92 + 1.61i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.03 - 5.84i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (0.167 + 0.140i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.273 + 1.55i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (11.8 - 9.95i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.235 + 1.33i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (2.27 - 0.829i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (2.69 - 0.979i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.960 - 1.66i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-11.4 - 4.15i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-13.4 - 11.2i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.43212057727927763624647511763, −9.979488893733830774039269085750, −8.798467409544410252634334668383, −7.77609546790106980615300664417, −7.05754969685970155997017654478, −6.21302080102935703998721174192, −5.23443945525094831282179516213, −4.38866201462903379860327832906, −2.90602185090911719357834615165, −2.15532066579000429664137820055,
0.55231471161788936492440829642, 1.43980836266742677590371950763, 3.48860813624675571914527641771, 4.52941278077148615100382150434, 5.23448674314607879384026985247, 6.00825999312891052057371409912, 7.50237669027401671442885132544, 7.893974529549544641320788230953, 8.748540458396870150856975339690, 9.716064658931575350992546520528