L(s) = 1 | + (1.20 + 0.439i)2-s + (−0.266 − 0.223i)4-s + (−0.0775 + 0.439i)5-s + (−2.70 + 2.27i)7-s + (−1.50 − 2.61i)8-s + (−0.286 + 0.497i)10-s + (0.482 + 2.73i)11-s + (−3.09 + 1.12i)13-s + (−4.26 + 1.55i)14-s + (−0.553 − 3.13i)16-s + (−3.51 + 6.09i)17-s + (2.59 + 4.49i)19-s + (0.118 − 0.0996i)20-s + (−0.620 + 3.51i)22-s + (−5.57 − 4.67i)23-s + ⋯ |
L(s) = 1 | + (0.854 + 0.310i)2-s + (−0.133 − 0.111i)4-s + (−0.0346 + 0.196i)5-s + (−1.02 + 0.858i)7-s + (−0.533 − 0.923i)8-s + (−0.0907 + 0.157i)10-s + (0.145 + 0.825i)11-s + (−0.857 + 0.312i)13-s + (−1.14 + 0.415i)14-s + (−0.138 − 0.784i)16-s + (−0.853 + 1.47i)17-s + (0.594 + 1.03i)19-s + (0.0265 − 0.0222i)20-s + (−0.132 + 0.750i)22-s + (−1.16 − 0.975i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.315503 + 0.952136i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.315503 + 0.952136i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.20 - 0.439i)T + (1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (0.0775 - 0.439i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (2.70 - 2.27i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.482 - 2.73i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (3.09 - 1.12i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (3.51 - 6.09i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.59 - 4.49i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.57 + 4.67i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.40 - 1.23i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (1.48 + 1.24i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-1.61 + 2.79i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.56 - 1.66i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1 + 5.67i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (2.31 - 1.93i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 8.77T + 53T^{2} \) |
| 59 | \( 1 + (0.514 - 2.91i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-6.04 + 5.06i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-8.86 + 3.22i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.65 + 4.59i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.777 - 1.34i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-11.1 - 4.07i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (15.2 + 5.56i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-9.21 - 15.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.75 - 9.96i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.55066936860221749298059191787, −9.817566081770541904511853977463, −9.198584820658755066475238062747, −8.132033519872739837977551703474, −6.74690576977672653645835770306, −6.35387416015684554220245374929, −5.37520810952901325181723020949, −4.38337237573744805421804549104, −3.46100967787485729397540049255, −2.15171588119276474888451464654,
0.37255505142840396154959470318, 2.72334442719390605343825038448, 3.40230894992880525534479445315, 4.52669079272444730487601915003, 5.24353859182529025845937833119, 6.46155909254811295058295188937, 7.28552560941149363438600460486, 8.375379842182959369635934655679, 9.344073657720785028035298091939, 9.980716441815785406811748238297