L(s) = 1 | + (−0.984 + 1.70i)2-s + (−0.939 − 1.62i)4-s + (1.85 + 3.20i)5-s + (1.17 − 2.03i)7-s − 0.237·8-s − 7.29·10-s + (−1.08 + 1.88i)11-s + (2.35 + 4.08i)13-s + (2.31 + 4.00i)14-s + (2.11 − 3.66i)16-s + 2.93·17-s − 6.22·19-s + (3.47 − 6.02i)20-s + (−2.14 − 3.71i)22-s + (0.259 + 0.449i)23-s + ⋯ |
L(s) = 1 | + (−0.696 + 1.20i)2-s + (−0.469 − 0.813i)4-s + (0.827 + 1.43i)5-s + (0.443 − 0.768i)7-s − 0.0839·8-s − 2.30·10-s + (−0.328 + 0.568i)11-s + (0.654 + 1.13i)13-s + (0.617 + 1.07i)14-s + (0.528 − 0.915i)16-s + 0.711·17-s − 1.42·19-s + (0.777 − 1.34i)20-s + (−0.457 − 0.792i)22-s + (0.0541 + 0.0937i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(-1.09970i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-1.09970i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (0.984 - 1.70i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.85 - 3.20i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.17 + 2.03i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.08 - 1.88i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.35 - 4.08i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 2.93T + 17T^{2} \) |
| 19 | \( 1 + 6.22T + 19T^{2} \) |
| 23 | \( 1 + (-0.259 - 0.449i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.74 - 3.02i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.15 - 3.72i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2.41T + 37T^{2} \) |
| 41 | \( 1 + (-1.24 - 2.16i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.532 - 0.921i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.118 - 0.205i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 4.66T + 53T^{2} \) |
| 59 | \( 1 + (6.65 + 11.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.83 + 3.18i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.14 + 12.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.20T + 71T^{2} \) |
| 73 | \( 1 + 4.68T + 73T^{2} \) |
| 79 | \( 1 + (-6.40 + 11.0i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.65 - 9.78i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 0.699T + 89T^{2} \) |
| 97 | \( 1 + (-3.54 + 6.13i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.60683512157953762822604188156, −9.860869380236037278176401275307, −9.056670494994224602220298480244, −7.991951455583657035487670697358, −7.25297311019405750819793495838, −6.56033652583126607303429966780, −6.05395578117833403698671848018, −4.68783395964814139339494249893, −3.27315380605514617259578191652, −1.83876358888238986358815443859,
0.72657846915495905354737958519, 1.82545267961753784815488932767, 2.84274357397700214846368122985, 4.30042438692821367574023599096, 5.66270878049593023214333996607, 5.88799114522623059189722411254, 8.061097215311714611062430662636, 8.539292526321715990636444872386, 9.121994724563930969890079568869, 10.02946403472072155405919370059