L(s) = 1 | + (−2.54 + 0.925i)2-s + (4.07 − 3.41i)4-s + (−0.290 − 1.64i)5-s + (0.383 + 0.321i)7-s + (−4.48 + 7.77i)8-s + (2.26 + 3.91i)10-s + (−0.333 + 1.88i)11-s + (−2.92 − 1.06i)13-s + (−1.27 − 0.463i)14-s + (2.36 − 13.4i)16-s + (−1.33 − 2.30i)17-s + (−2.89 + 5.02i)19-s + (−6.81 − 5.71i)20-s + (−0.900 − 5.10i)22-s + (−3.55 + 2.98i)23-s + ⋯ |
L(s) = 1 | + (−1.79 + 0.654i)2-s + (2.03 − 1.70i)4-s + (−0.129 − 0.736i)5-s + (0.144 + 0.121i)7-s + (−1.58 + 2.74i)8-s + (0.715 + 1.23i)10-s + (−0.100 + 0.569i)11-s + (−0.811 − 0.295i)13-s + (−0.340 − 0.123i)14-s + (0.592 − 3.35i)16-s + (−0.323 − 0.559i)17-s + (−0.664 + 1.15i)19-s + (−1.52 − 1.27i)20-s + (−0.192 − 1.08i)22-s + (−0.742 + 0.622i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.149778 + 0.298233i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.149778 + 0.298233i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.54 - 0.925i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (0.290 + 1.64i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.383 - 0.321i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.333 - 1.88i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (2.92 + 1.06i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.33 + 2.30i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.89 - 5.02i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.55 - 2.98i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.45 + 0.894i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (3.53 - 2.96i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-2.42 - 4.20i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-10.8 - 3.94i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.56 - 8.86i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.23 - 4.39i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 5.43T + 53T^{2} \) |
| 59 | \( 1 + (0.380 + 2.15i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-5.24 - 4.39i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (11.7 + 4.26i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.41 - 2.45i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.96 - 8.60i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.99 - 1.81i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.56 + 0.933i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (5.60 - 9.71i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.19 - 6.78i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.26252707620123999601468295779, −9.687589480422830658298374501554, −8.961242986834450686672855105057, −8.074115538825501914607118537300, −7.61890780958636599485441658513, −6.59160522947024877945325204798, −5.65872864960025546466334631009, −4.60997044032844731067269077002, −2.49824103714018130905329292631, −1.28268480527986092238839871168,
0.32192466494411410996763868704, 2.09502879219841906909182670020, 2.88369969911416657165521413241, 4.18058599152447143514268383789, 6.10896113995363236472094304773, 7.04931734678439284441771131449, 7.60021400757286370796288717745, 8.653638109731386837124267386905, 9.160662495908217116419120554125, 10.22014362096744383578467774710