L(s) = 1 | + (0.5 + 0.866i)2-s + (0.800 + 1.53i)3-s + (−0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.929 + 1.46i)6-s − 4.66·7-s − 0.999·8-s + (−1.71 + 2.45i)9-s + (−0.866 − 0.499i)10-s − 3.04i·11-s + (−1.73 − 0.0746i)12-s + (3.56 + 2.06i)13-s + (−2.33 − 4.03i)14-s + (−1.46 − 0.929i)15-s + (−0.5 − 0.866i)16-s + (−2.22 + 1.28i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.462 + 0.886i)3-s + (−0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s + (−0.379 + 0.596i)6-s − 1.76·7-s − 0.353·8-s + (−0.572 + 0.819i)9-s + (−0.273 − 0.158i)10-s − 0.918i·11-s + (−0.499 − 0.0215i)12-s + (0.989 + 0.571i)13-s + (−0.623 − 1.07i)14-s + (−0.377 − 0.240i)15-s + (−0.125 − 0.216i)16-s + (−0.540 + 0.312i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.193057 - 0.928388i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.193057 - 0.928388i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.800 - 1.53i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 19 | \( 1 + (3.71 - 2.28i)T \) |
good | 7 | \( 1 + 4.66T + 7T^{2} \) |
| 11 | \( 1 + 3.04iT - 11T^{2} \) |
| 13 | \( 1 + (-3.56 - 2.06i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.22 - 1.28i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (0.586 + 0.338i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.992 + 1.71i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 4.29iT - 31T^{2} \) |
| 37 | \( 1 - 4.49iT - 37T^{2} \) |
| 41 | \( 1 + (-1.65 - 2.86i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.86 - 4.96i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-7.58 - 4.38i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.57 - 7.92i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.02 + 10.4i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.95 - 12.0i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.56 + 1.48i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.25 + 10.8i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.18 - 3.78i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.74 + 3.89i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.61iT - 83T^{2} \) |
| 89 | \( 1 + (-4.81 + 8.33i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.00 - 4.62i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.00505877198079593554645274783, −10.35676650247891507547703800175, −9.221299055817145844089736366875, −8.750289748865433238002910524955, −7.74808061778110176420510291228, −6.31360270403463517914322945449, −6.12229069852028395081582854865, −4.44883361572849332455007064670, −3.64134809944785795703163527776, −2.91511612201068862062538581078,
0.43277348805457912531406931271, 2.23045865770019044534966811794, 3.27117982737242307035934933095, 4.16233761654337585725334158713, 5.79338952311017453834068820565, 6.60712429333047605943216785206, 7.41450210884285968720180635539, 8.718987148392376785523515619970, 9.280188670308504045592237088150, 10.26645570503031647344911383220