L(s) = 1 | + (0.866 + 1.5i)2-s + (−0.5 + 0.866i)4-s + (−1.73 + 3i)5-s + (−0.5 − 0.866i)7-s + 1.73·8-s − 6·10-s + (1.73 + 3i)11-s + (−1 + 1.73i)13-s + (0.866 − 1.5i)14-s + (2.49 + 4.33i)16-s − 3.46·17-s − 4·19-s + (−1.73 − 3i)20-s + (−3 + 5.19i)22-s + (−1.73 + 3i)23-s + ⋯ |
L(s) = 1 | + (0.612 + 1.06i)2-s + (−0.250 + 0.433i)4-s + (−0.774 + 1.34i)5-s + (−0.188 − 0.327i)7-s + 0.612·8-s − 1.89·10-s + (0.522 + 0.904i)11-s + (−0.277 + 0.480i)13-s + (0.231 − 0.400i)14-s + (0.624 + 1.08i)16-s − 0.840·17-s − 0.917·19-s + (−0.387 − 0.670i)20-s + (−0.639 + 1.10i)22-s + (−0.361 + 0.625i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.284418 + 1.61301i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.284418 + 1.61301i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.866 - 1.5i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1.73 - 3i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.73 - 3i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (1.73 - 3i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-5.19 + 9i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.46 - 6i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 + (3.46 - 6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.05423883244977917188550343849, −10.43953060216553406060245286484, −9.360876963356583217147566993449, −7.997396278611694818498002768736, −7.18035384163905091696878695690, −6.79623694807527420097752725568, −5.93280252992874800060660115368, −4.40664850805581626324290538819, −3.93664318576452541472925165667, −2.28847172101989537014774372157,
0.77910926802181957947642901557, 2.35527024725192331897776410508, 3.67056451189458619865424150178, 4.41333633923385585149827293939, 5.29091289216703518474563171252, 6.59529482658329391243123707675, 8.046791761818013617381776348305, 8.561654210230042977964727218984, 9.541958645927193377953188071678, 10.72476188011815244473904433107