L(s) = 1 | + (0.5 + 0.240i)2-s + (−1.05 − 1.32i)4-s + (−0.321 − 1.40i)5-s + (2.57 + 0.588i)7-s + (−0.455 − 1.99i)8-s + (0.178 − 0.781i)10-s + (−3.32 − 1.60i)11-s + (−5.25 − 2.52i)13-s + (1.14 + 0.915i)14-s + (−0.500 + 2.19i)16-s + (1.81 − 2.27i)17-s + 1.93·19-s + (−1.52 + 1.91i)20-s + (−1.27 − 1.60i)22-s + (−0.815 − 1.02i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.170i)2-s + (−0.527 − 0.661i)4-s + (−0.143 − 0.630i)5-s + (0.974 + 0.222i)7-s + (−0.161 − 0.706i)8-s + (0.0564 − 0.247i)10-s + (−1.00 − 0.482i)11-s + (−1.45 − 0.701i)13-s + (0.306 + 0.244i)14-s + (−0.125 + 0.547i)16-s + (0.440 − 0.552i)17-s + 0.444·19-s + (−0.340 + 0.427i)20-s + (−0.272 − 0.341i)22-s + (−0.170 − 0.213i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.798851 - 0.938376i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.798851 - 0.938376i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.57 - 0.588i)T \) |
good | 2 | \( 1 + (-0.5 - 0.240i)T + (1.24 + 1.56i)T^{2} \) |
| 5 | \( 1 + (0.321 + 1.40i)T + (-4.50 + 2.16i)T^{2} \) |
| 11 | \( 1 + (3.32 + 1.60i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (5.25 + 2.52i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-1.81 + 2.27i)T + (-3.78 - 16.5i)T^{2} \) |
| 19 | \( 1 - 1.93T + 19T^{2} \) |
| 23 | \( 1 + (0.815 + 1.02i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (-4.92 + 6.17i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + 3.24T + 31T^{2} \) |
| 37 | \( 1 + (1.13 - 1.42i)T + (-8.23 - 36.0i)T^{2} \) |
| 41 | \( 1 + (1.69 + 7.41i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.475 + 2.08i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-4.02 - 1.93i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (-8.85 - 11.0i)T + (-11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (1.43 - 6.28i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (1.95 - 2.45i)T + (-13.5 - 59.4i)T^{2} \) |
| 67 | \( 1 - 1.04T + 67T^{2} \) |
| 71 | \( 1 + (-3.79 - 4.75i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-1.23 + 0.593i)T + (45.5 - 57.0i)T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + (4.33 - 2.08i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (8.48 - 4.08i)T + (55.4 - 69.5i)T^{2} \) |
| 97 | \( 1 + 1.35T + 97T^{2} \) |
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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
−10.70719816259838418324658048807, −10.05687956425522629131531581670, −9.023444733562731936347987831999, −8.143970579701107935848249941438, −7.29330737142840852910520420697, −5.63822204577503443275232497041, −5.18513358002557348896410734431, −4.37000963961824166690264547728, −2.60363442661833080602422101610, −0.70661006873585351337156361840,
2.21956291785853385926931319298, 3.38889224175851233801621439773, 4.69080974514985826374556831385, 5.21702598209138893820224473424, 7.03583972210185579560548769002, 7.64677223024131153715308361251, 8.485440067493797681829761670769, 9.667145688750456281378856491918, 10.57352951471469824652638914510, 11.48535975814640161937398486343