L(s) = 1 | + (−1.85 + 1.07i)2-s + (−1.70 − 0.297i)3-s + (1.29 − 2.23i)4-s + (0.866 + 0.5i)5-s + (3.48 − 1.27i)6-s + (−4.18 + 2.41i)7-s + 1.24i·8-s + (2.82 + 1.01i)9-s − 2.14·10-s + (−0.777 + 0.448i)11-s + (−2.87 + 3.43i)12-s + (3.48 + 0.925i)13-s + (5.17 − 8.96i)14-s + (−1.32 − 1.11i)15-s + (1.24 + 2.15i)16-s − 5.40·17-s + ⋯ |
L(s) = 1 | + (−1.31 + 0.756i)2-s + (−0.985 − 0.171i)3-s + (0.645 − 1.11i)4-s + (0.387 + 0.223i)5-s + (1.42 − 0.520i)6-s + (−1.58 + 0.913i)7-s + 0.441i·8-s + (0.940 + 0.338i)9-s − 0.677·10-s + (−0.234 + 0.135i)11-s + (−0.828 + 0.991i)12-s + (0.966 + 0.256i)13-s + (1.38 − 2.39i)14-s + (−0.343 − 0.286i)15-s + (0.311 + 0.539i)16-s − 1.31·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0928 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0928 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0265727 - 0.0242104i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0265727 - 0.0242104i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.70 + 0.297i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 + (-3.48 - 0.925i)T \) |
good | 2 | \( 1 + (1.85 - 1.07i)T + (1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (4.18 - 2.41i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.777 - 0.448i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 5.40T + 17T^{2} \) |
| 19 | \( 1 - 6.78iT - 19T^{2} \) |
| 23 | \( 1 + (-0.544 + 0.943i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.28 + 2.22i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.42 + 2.55i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.86iT - 37T^{2} \) |
| 41 | \( 1 + (3.50 + 2.02i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.08 - 3.61i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.64 + 3.83i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6.85T + 53T^{2} \) |
| 59 | \( 1 + (13.0 + 7.55i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.04 + 5.27i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.26 - 3.04i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 13.3iT - 71T^{2} \) |
| 73 | \( 1 + 11.8iT - 73T^{2} \) |
| 79 | \( 1 + (1.01 + 1.76i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.21 - 0.700i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 3.44iT - 89T^{2} \) |
| 97 | \( 1 + (15.9 - 9.23i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.32203018210955691751481311559, −9.472764310882774908419145019007, −8.946196469867433210449466577049, −7.77752757064352785036866533345, −6.69413570241673537446549031646, −6.24359778760225735708640026750, −5.63862816092594537361575940933, −3.79042148859993916539568311767, −1.94673837807131833663351946752, −0.04140698656280195061128951307,
1.09263683166803443507752575617, 2.85130636392665584449219174714, 4.12568968112760080445862103926, 5.53914960058629902116508039010, 6.65661549463330571175126584791, 7.22299874797572061112865638426, 8.791354119535471238001354969592, 9.314563120288592986015163080734, 10.14975795281705901043925473160, 10.80694288283752669291657046659