L(s) = 1 | + (0.5 − 0.866i)2-s + 1.73i·3-s + (0.500 + 0.866i)4-s + (0.5 + 0.866i)5-s + (1.49 + 0.866i)6-s + (−1 + 1.73i)7-s + 3·8-s − 2.99·9-s + 0.999·10-s + (0.5 − 0.866i)11-s + (−1.49 + 0.866i)12-s + (0.5 + 0.866i)13-s + (0.999 + 1.73i)14-s + (−1.49 + 0.866i)15-s + (0.500 − 0.866i)16-s + 2·17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + 0.999i·3-s + (0.250 + 0.433i)4-s + (0.223 + 0.387i)5-s + (0.612 + 0.353i)6-s + (−0.377 + 0.654i)7-s + 1.06·8-s − 0.999·9-s + 0.316·10-s + (0.150 − 0.261i)11-s + (−0.433 + 0.250i)12-s + (0.138 + 0.240i)13-s + (0.267 + 0.462i)14-s + (−0.387 + 0.223i)15-s + (0.125 − 0.216i)16-s + 0.485·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42564 + 1.19626i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42564 + 1.19626i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.5 - 4.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 14T + 53T^{2} \) |
| 59 | \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7 + 12.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + (-6 + 10.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5 + 8.66i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.73630668251011404097820302396, −10.39112812815007298568815751325, −9.204035616814526049861858735042, −8.556570091677377216414156840985, −7.30949724160604065071813470644, −6.17574306498307210053639030406, −5.19545965994405052571672526068, −3.98285346276636788456939161534, −3.22852679528122311446459928508, −2.21732948289237067147445627025,
0.969386349372659091894649488479, 2.26531065152294420070403497531, 3.97057079452193291999780406948, 5.26612833496443986194770641292, 6.06711973469745507013148682859, 6.84334040578923709037252472116, 7.56466385163742694478419648572, 8.460876341539892122819138262147, 9.677481595227111620669874779139, 10.51811195697517297734044584970