L(s) = 1 | + (1 + 1.73i)2-s + (−0.999 + 1.73i)4-s + (0.5 − 0.866i)5-s + 1.99·10-s + (2.5 + 4.33i)11-s + (−2 + 3.46i)13-s + (1.99 + 3.46i)16-s + 4·17-s − 5·19-s + (1 + 1.73i)20-s + (−5 + 8.66i)22-s + (3 − 5.19i)23-s + (−0.499 − 0.866i)25-s − 7.99·26-s + (−2.5 − 4.33i)29-s + ⋯ |
L(s) = 1 | + (0.707 + 1.22i)2-s + (−0.499 + 0.866i)4-s + (0.223 − 0.387i)5-s + 0.632·10-s + (0.753 + 1.30i)11-s + (−0.554 + 0.960i)13-s + (0.499 + 0.866i)16-s + 0.970·17-s − 1.14·19-s + (0.223 + 0.387i)20-s + (−1.06 + 1.84i)22-s + (0.625 − 1.08i)23-s + (−0.0999 − 0.173i)25-s − 1.56·26-s + (−0.464 − 0.804i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.35834 + 1.61881i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.35834 + 1.61881i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.5 + 4.33i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.5 + 7.79i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + (-3.5 + 6.06i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3 - 5.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 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
−11.88657537476453472470380889455, −10.45515097635752729043078404077, −9.543883624788080384083895375648, −8.567252287017957271268606885159, −7.44701784852275684292512964663, −6.75452567382348551595363908591, −5.86997298511438985141290507324, −4.67611950493511795609400749324, −4.15047423315279489157591034152, −1.98367875033956582307624119637,
1.35269131782032640114823389683, 2.97748964713865780379152016304, 3.54160068514967504957584078213, 4.98202575537214199687654671054, 5.91207776707295930885432489899, 7.19786071499130443615413172839, 8.380892804581133615615982842053, 9.512393993195809474810266301505, 10.54949728709666003795157553614, 10.93935445073101914610824133097