L(s) = 1 | + (−0.135 + 0.765i)2-s + (1.31 + 0.477i)4-s + (−1.82 + 1.52i)5-s + (2.35 − 0.855i)7-s + (−1.32 + 2.28i)8-s + (−0.924 − 1.60i)10-s + (−2.40 − 2.01i)11-s + (0.232 + 1.31i)13-s + (0.337 + 1.91i)14-s + (0.564 + 0.473i)16-s + (3.13 + 5.43i)17-s + (−4.03 + 6.98i)19-s + (−3.11 + 1.13i)20-s + (1.87 − 1.57i)22-s + (3.81 + 1.38i)23-s + ⋯ |
L(s) = 1 | + (−0.0954 + 0.541i)2-s + (0.655 + 0.238i)4-s + (−0.814 + 0.683i)5-s + (0.888 − 0.323i)7-s + (−0.466 + 0.808i)8-s + (−0.292 − 0.506i)10-s + (−0.725 − 0.608i)11-s + (0.0643 + 0.365i)13-s + (0.0902 + 0.512i)14-s + (0.141 + 0.118i)16-s + (0.760 + 1.31i)17-s + (−0.925 + 1.60i)19-s + (−0.696 + 0.253i)20-s + (0.399 − 0.334i)22-s + (0.794 + 0.289i)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
\(L(1)\) |
\(\approx\) |
\(0.632298 + 1.25900i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.632298 + 1.25900i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.135 - 0.765i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (1.82 - 1.52i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-2.35 + 0.855i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (2.40 + 2.01i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.232 - 1.31i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.13 - 5.43i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.03 - 6.98i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.81 - 1.38i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.61 + 9.14i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.66 + 0.968i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (2.76 + 4.79i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.23 - 6.99i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.79 - 1.50i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (4.33 - 1.57i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 0.135T + 53T^{2} \) |
| 59 | \( 1 + (-3.06 + 2.57i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.321 - 0.116i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.75 - 9.96i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.09 - 7.09i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.15 + 10.6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.708 + 4.01i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (0.158 - 0.899i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (1.86 - 3.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.59 + 3.85i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.96698355773572549157141440921, −10.05259643704533610185017044444, −8.412525216572854827918291203702, −7.994993271353724637877885983946, −7.45514061986920784779674468079, −6.33035030295798733444331824558, −5.62407144568970486630070735507, −4.14733115677932738061804938173, −3.24627131995384667517310158913, −1.85413185377550065953813161562,
0.74309652909208492248116972486, 2.20666869344269542209821326743, 3.24930917477564873574310466927, 4.81827161688150518265058162356, 5.18641764574832030058359066016, 6.82217388554008466840426957736, 7.46472847739533431468441887057, 8.469178349751270241929313556184, 9.198954687134584910606758945917, 10.34414360119954009209318712915