L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)7-s + 8-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)14-s + (0.5 − 0.866i)16-s + (−0.499 − 0.866i)22-s + (−1 − 1.73i)23-s + (−0.5 + 0.866i)25-s + (−1 + 1.73i)29-s − 37-s + (0.5 − 0.866i)43-s − 1.99·46-s + (−0.499 − 0.866i)49-s + (0.499 + 0.866i)50-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)7-s + 8-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)14-s + (0.5 − 0.866i)16-s + (−0.499 − 0.866i)22-s + (−1 − 1.73i)23-s + (−0.5 + 0.866i)25-s + (−1 + 1.73i)29-s − 37-s + (0.5 − 0.866i)43-s − 1.99·46-s + (−0.499 − 0.866i)49-s + (0.499 + 0.866i)50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.212361801\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.212361801\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−11.01213510776796008077466787828, −10.26492597290797364311265623501, −9.126884492599423864605666810890, −8.425168318342960032448786383955, −7.21691880131830335540995874591, −6.17336019709093762902976537575, −5.17074570175580096443816158640, −3.86900875947489756213189227700, −3.07743310728371945458667310491, −1.89819266723662397843145634222,
1.80881972249632096017421423116, 3.76844469612622508811687578974, 4.49668122754269432802427301427, 5.74549199459339738671089215640, 6.47272779143865155221813273215, 7.39511579303658622609112329814, 7.922766317425265524038110269641, 9.578563332375894758161409830910, 9.965614092200162167904342640292, 11.03894343978437635354470649680