| L(s) = 1 | + (0.180 + 0.0172i)2-s + (−0.690 + 0.723i)3-s + (−1.93 − 0.372i)4-s + (−3.29 + 1.69i)5-s + (−0.136 + 0.118i)6-s + (0.635 − 2.56i)7-s + (−0.689 − 0.202i)8-s + (−0.0475 − 0.998i)9-s + (−0.623 + 0.249i)10-s + (0.277 + 2.90i)11-s + (1.60 − 1.14i)12-s + (6.10 + 0.878i)13-s + (0.158 − 0.452i)14-s + (1.04 − 3.55i)15-s + (3.53 + 1.41i)16-s + (1.72 − 4.97i)17-s + ⋯ |
| L(s) = 1 | + (0.127 + 0.0121i)2-s + (−0.398 + 0.417i)3-s + (−0.965 − 0.186i)4-s + (−1.47 + 0.759i)5-s + (−0.0559 + 0.0484i)6-s + (0.240 − 0.970i)7-s + (−0.243 − 0.0716i)8-s + (−0.0158 − 0.332i)9-s + (−0.197 + 0.0789i)10-s + (0.0835 + 0.874i)11-s + (0.462 − 0.329i)12-s + (1.69 + 0.243i)13-s + (0.0424 − 0.120i)14-s + (0.269 − 0.917i)15-s + (0.882 + 0.353i)16-s + (0.417 − 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.678034 - 0.222543i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.678034 - 0.222543i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.690 - 0.723i)T \) |
| 7 | \( 1 + (-0.635 + 2.56i)T \) |
| 23 | \( 1 + (4.54 + 1.53i)T \) |
| good | 2 | \( 1 + (-0.180 - 0.0172i)T + (1.96 + 0.378i)T^{2} \) |
| 5 | \( 1 + (3.29 - 1.69i)T + (2.90 - 4.07i)T^{2} \) |
| 11 | \( 1 + (-0.277 - 2.90i)T + (-10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (-6.10 - 0.878i)T + (12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-1.72 + 4.97i)T + (-13.3 - 10.5i)T^{2} \) |
| 19 | \( 1 + (1.79 + 5.17i)T + (-14.9 + 11.7i)T^{2} \) |
| 29 | \( 1 + (-0.942 - 1.08i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-7.09 + 1.72i)T + (27.5 - 14.2i)T^{2} \) |
| 37 | \( 1 + (7.24 - 0.345i)T + (36.8 - 3.51i)T^{2} \) |
| 41 | \( 1 + (-4.33 + 6.73i)T + (-17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-2.08 - 7.08i)T + (-36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (-5.92 - 3.42i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.09 + 2.65i)T + (-12.4 + 51.5i)T^{2} \) |
| 59 | \( 1 + (-1.05 - 2.62i)T + (-42.7 + 40.7i)T^{2} \) |
| 61 | \( 1 + (-6.10 + 5.81i)T + (2.90 - 60.9i)T^{2} \) |
| 67 | \( 1 + (4.99 + 3.55i)T + (21.9 + 63.3i)T^{2} \) |
| 71 | \( 1 + (3.13 + 6.86i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-0.943 + 4.89i)T + (-67.7 - 27.1i)T^{2} \) |
| 79 | \( 1 + (-5.57 + 7.08i)T + (-18.6 - 76.7i)T^{2} \) |
| 83 | \( 1 + (-5.40 + 3.47i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (0.473 - 1.95i)T + (-79.1 - 40.7i)T^{2} \) |
| 97 | \( 1 + (4.63 + 2.97i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.88497117080898359195713851630, −10.22192585017426480750844758777, −9.125759687466752014327968086619, −8.141639964333816431925860178988, −7.25897494459805543068281218540, −6.32743274452149243584507527394, −4.72874439914810269009545135422, −4.21146842408634687855133018769, −3.37029941453372824355612824697, −0.59009768362628687103425369780,
1.12006589324773962615421695172, 3.54991012377640480010858190208, 4.15191177928636344077003596544, 5.51025415954020745092211521955, 6.09391076165041603033086936005, 7.960316405962304655576826479394, 8.381222445845528310427318791796, 8.732050702834620112447921984802, 10.31726625985116009816903326038, 11.36041208087714368684582753015
