L(s) = 1 | + (1.58 + 0.575i)2-s + (0.639 + 0.536i)4-s + (0.766 − 0.642i)5-s + (0.274 − 0.474i)7-s + (−0.981 − 1.69i)8-s + (1.58 − 0.575i)10-s + (0.165 + 0.286i)11-s + (0.837 − 4.74i)13-s + (0.707 − 0.593i)14-s + (−0.863 − 4.89i)16-s + (4.96 + 1.80i)17-s + (4.30 + 0.670i)19-s + 0.834·20-s + (0.0968 + 0.549i)22-s + (0.850 + 0.713i)23-s + ⋯ |
L(s) = 1 | + (1.11 + 0.407i)2-s + (0.319 + 0.268i)4-s + (0.342 − 0.287i)5-s + (0.103 − 0.179i)7-s + (−0.346 − 0.600i)8-s + (0.500 − 0.182i)10-s + (0.0499 + 0.0864i)11-s + (0.232 − 1.31i)13-s + (0.189 − 0.158i)14-s + (−0.215 − 1.22i)16-s + (1.20 + 0.438i)17-s + (0.988 + 0.153i)19-s + 0.186·20-s + (0.0206 + 0.117i)22-s + (0.177 + 0.148i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.79317 - 0.519710i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.79317 - 0.519710i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (-4.30 - 0.670i)T \) |
good | 2 | \( 1 + (-1.58 - 0.575i)T + (1.53 + 1.28i)T^{2} \) |
| 7 | \( 1 + (-0.274 + 0.474i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.165 - 0.286i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.837 + 4.74i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-4.96 - 1.80i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-0.850 - 0.713i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (3.01 - 1.09i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.01 + 5.21i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.67T + 37T^{2} \) |
| 41 | \( 1 + (-1.37 - 7.79i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.25 + 1.05i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (4.32 - 1.57i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (5.15 + 4.32i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (6.39 + 2.32i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.520 - 0.436i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-7.30 + 2.65i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.832 + 0.698i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-2.42 - 13.7i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.243 + 1.38i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.427 - 0.740i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.52 - 14.3i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-6.59 - 2.40i)T + (74.3 + 62.3i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.985672458863693385763970163815, −9.506843609918766976858116716008, −8.188891653053125959634014213787, −7.49306687836902865174267324940, −6.33133002062596076816184415455, −5.56454089562266103700629352345, −5.02696842916046767046100237874, −3.81115053822249594175985995512, −3.01022401831637769703626618868, −1.10047973249215099581522493131,
1.73471076375379939021056895179, 2.96127596027046497058845427676, 3.76946826715701928158501240172, 4.89872598172084773399793685759, 5.56835728821320821067359938544, 6.55450491125411308799596081889, 7.51386370511313934791665769697, 8.700424214057694478915540706885, 9.395893034089577983212972900007, 10.40761377366092736229591037983