L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (−0.500 + 0.866i)4-s − 2i·5-s + (−0.866 − 0.499i)6-s + (3.46 + 2i)7-s − 3i·8-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)10-s + (3.46 − 2i)11-s − 12-s − 3.99·14-s + (1.73 − i)15-s + (0.500 + 0.866i)16-s + (1 − 1.73i)17-s − 0.999i·18-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (−0.250 + 0.433i)4-s − 0.894i·5-s + (−0.353 − 0.204i)6-s + (1.30 + 0.755i)7-s − 1.06i·8-s + (−0.166 + 0.288i)9-s + (0.316 + 0.547i)10-s + (1.04 − 0.603i)11-s − 0.288·12-s − 1.06·14-s + (0.447 − 0.258i)15-s + (0.125 + 0.216i)16-s + (0.242 − 0.420i)17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12210 + 0.591308i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12210 + 0.591308i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.866 - 0.5i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 + (-3.46 - 2i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.46 + 2i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5 - 8.66i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + (1.73 - i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.19 - 3i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6 - 10.3i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (10.3 + 6i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.92 + 4i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + (1.73 - i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.66 - 5i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.07653171725581739287371986267, −9.758777023120490532268167020653, −8.972105478683061547030849688736, −8.523772592159863876121282326895, −7.914854382130781075196214412133, −6.59380417041481276008485035464, −5.17828559136724514226844144433, −4.50907840137758826199335631566, −3.23733454865177329590224391855, −1.32706513678421127496192652455,
1.21258031371111450142897206751, 2.22035890197502677009601545077, 3.90362739056643014903956334910, 5.03343672835818185234682035605, 6.40896089649131685023041641185, 7.26902923345936071442047001961, 8.174592963637014901234646926138, 8.942579938555328393974933150966, 10.12766739455584385987827426668, 10.57716475040532129221519695907