L(s) = 1 | + (1.87 + 0.684i)4-s + (0.939 − 0.342i)7-s + (0.868 + 4.92i)13-s + (3.06 + 2.57i)16-s + (3.5 − 6.06i)19-s + (−0.868 + 4.92i)25-s + 2·28-s + (3.75 + 1.36i)31-s + (−5.5 − 9.52i)37-s + (6.12 + 5.14i)43-s + (−4.59 + 3.85i)49-s + (−1.73 + 9.84i)52-s + (0.939 − 0.342i)61-s + (4.00 + 6.92i)64-s + (0.868 + 4.92i)67-s + ⋯ |
L(s) = 1 | + (0.939 + 0.342i)4-s + (0.355 − 0.129i)7-s + (0.240 + 1.36i)13-s + (0.766 + 0.642i)16-s + (0.802 − 1.39i)19-s + (−0.173 + 0.984i)25-s + 0.377·28-s + (0.675 + 0.245i)31-s + (−0.904 − 1.56i)37-s + (0.934 + 0.784i)43-s + (−0.656 + 0.550i)49-s + (−0.240 + 1.36i)52-s + (0.120 − 0.0437i)61-s + (0.500 + 0.866i)64-s + (0.106 + 0.601i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.95479 + 0.463295i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.95479 + 0.463295i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.939 + 0.342i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.868 - 4.92i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.75 - 1.36i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.12 - 5.14i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.939 + 0.342i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.868 - 4.92i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.95 + 16.7i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (14.5 + 12.2i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.74184171131108221650325611893, −9.495088441770129377999257989015, −8.804880235518497228374776897340, −7.62991530431921058117347079900, −7.06161528353037875473470237816, −6.20666735076104675126060596197, −5.02359505910417287553351515004, −3.89140524968830114346084977688, −2.72184952383379176943349457570, −1.53827353629654300169931148094,
1.21879467525710883604211949684, 2.55910998280373179008796778097, 3.62090610091713417808477378267, 5.15651556945074164642761987560, 5.85837831947177063687722787420, 6.75858138093147949467261678963, 7.87305415028100094866877835358, 8.296594641017158319484057693855, 9.818383213622276193531768044828, 10.29771174607494653914169656129