L(s) = 1 | + (−0.363 − 0.5i)2-s + (−0.951 − 0.309i)3-s + (0.5 − 1.53i)4-s + (0.190 + 0.587i)6-s − 1.61i·7-s + (−2.12 + 0.690i)8-s + (−1.61 − 1.17i)9-s + (0.618 − 0.449i)11-s + (−0.951 + 1.30i)12-s + (2.85 − 3.92i)13-s + (−0.809 + 0.587i)14-s + (−1.49 − 1.08i)16-s + (0.726 − 0.236i)17-s + 1.23i·18-s + (1.80 + 5.56i)19-s + ⋯ |
L(s) = 1 | + (−0.256 − 0.353i)2-s + (−0.549 − 0.178i)3-s + (0.250 − 0.769i)4-s + (0.0779 + 0.239i)6-s − 0.611i·7-s + (−0.751 + 0.244i)8-s + (−0.539 − 0.391i)9-s + (0.186 − 0.135i)11-s + (−0.274 + 0.377i)12-s + (0.791 − 1.08i)13-s + (−0.216 + 0.157i)14-s + (−0.374 − 0.272i)16-s + (0.176 − 0.0572i)17-s + 0.291i·18-s + (0.415 + 1.27i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.479492 - 0.625819i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.479492 - 0.625819i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (0.363 + 0.5i)T + (-0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.951 + 0.309i)T + (2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + 1.61iT - 7T^{2} \) |
| 11 | \( 1 + (-0.618 + 0.449i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.85 + 3.92i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.726 + 0.236i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.80 - 5.56i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-4.84 - 6.66i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.427 + 1.31i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.927 + 2.85i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.48 + 3.42i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.23 - 3.07i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 1.85iT - 43T^{2} \) |
| 47 | \( 1 + (1.53 + 0.5i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (5.20 + 1.69i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (3.35 + 2.43i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.80 + 2.76i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (8.78 - 2.85i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.35 - 4.16i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (5.29 + 7.28i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (0.954 - 2.93i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (1.67 - 0.545i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-7.23 + 5.25i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.71 - 0.881i)T + (78.4 + 57.0i)T^{2} \) |
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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
−12.99476048181578427342208010324, −11.75636211985799801969392413784, −11.05541988196755737550548724818, −10.17982921280987079962729677283, −9.088041240447273545525438254320, −7.64122396641957617062150311220, −6.18292146690344596959202432049, −5.46949551689546964019650255098, −3.37012427512599979918941162197, −1.06139388462430419569641658532,
2.80177642705823829414931113637, 4.61827464846276781627458285704, 6.10696930285051884067001288940, 7.06781254693130290727211041631, 8.525745837847073596760729010942, 9.133398696760145106091004822391, 10.88813043030972794353522054889, 11.59132628020091659413299466820, 12.48067015661411841525621784400, 13.63974810285739358097188022684