L(s) = 1 | + (1.38 + 1.91i)2-s + (−0.951 + 0.309i)3-s + (−1.10 + 3.41i)4-s + (−1.97 + 1.05i)5-s + (−1.91 − 1.38i)6-s + (1.63 + 0.529i)7-s + (−3.56 + 1.15i)8-s + (0.809 − 0.587i)9-s + (−4.75 − 2.31i)10-s + (−0.406 − 3.29i)11-s − 3.58i·12-s + (2.07 + 2.85i)13-s + (1.25 + 3.85i)14-s + (1.55 − 1.60i)15-s + (−1.37 − 0.996i)16-s + (0.219 − 0.302i)17-s + ⋯ |
L(s) = 1 | + (0.982 + 1.35i)2-s + (−0.549 + 0.178i)3-s + (−0.554 + 1.70i)4-s + (−0.882 + 0.469i)5-s + (−0.780 − 0.567i)6-s + (0.616 + 0.200i)7-s + (−1.26 + 0.409i)8-s + (0.269 − 0.195i)9-s + (−1.50 − 0.731i)10-s + (−0.122 − 0.992i)11-s − 1.03i·12-s + (0.576 + 0.792i)13-s + (0.334 + 1.02i)14-s + (0.400 − 0.415i)15-s + (−0.342 − 0.249i)16-s + (0.0532 − 0.0732i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.468714 + 1.35015i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.468714 + 1.35015i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.951 - 0.309i)T \) |
| 5 | \( 1 + (1.97 - 1.05i)T \) |
| 11 | \( 1 + (0.406 + 3.29i)T \) |
good | 2 | \( 1 + (-1.38 - 1.91i)T + (-0.618 + 1.90i)T^{2} \) |
| 7 | \( 1 + (-1.63 - 0.529i)T + (5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (-2.07 - 2.85i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.219 + 0.302i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.78 - 5.48i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 3.80iT - 23T^{2} \) |
| 29 | \( 1 + (-1.08 + 3.34i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-8.16 + 5.93i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (10.9 + 3.56i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.747 + 2.30i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 0.224iT - 43T^{2} \) |
| 47 | \( 1 + (4.44 - 1.44i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.43 - 1.97i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.65 - 5.09i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.75 - 2.00i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 9.08iT - 67T^{2} \) |
| 71 | \( 1 + (-7.09 - 5.15i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (14.2 + 4.62i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.11 + 5.16i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-4.37 + 6.02i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + (-8.61 - 11.8i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−13.61112301535289979099972120503, −12.20929177425922387457901139369, −11.58099559378460273682454592298, −10.47461619016622507576033156015, −8.547113827595787415114608404285, −7.82658703645907851348917297770, −6.63073796898061446624166063987, −5.81450566880712771354196506783, −4.57646638114460284287985882889, −3.59671288796755792180807898432,
1.31313711757966032634140415655, 3.26500644525415542160931094227, 4.64714352045491912050042402627, 5.18680860547769293082393635050, 7.06621489772671121740120191150, 8.360465502998344884325402114357, 9.938920706166474608776319009175, 10.90435938491640088153453030202, 11.59925435530210926159147349843, 12.32855001879961526394278433554