L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2.63 − 0.189i)7-s + 0.999·8-s + (3.44 − 1.99i)11-s − 0.0681·13-s + (1.48 − 2.19i)14-s + (−0.5 + 0.866i)16-s + (−6.34 + 3.66i)17-s + (1.76 + 1.01i)19-s + 3.98i·22-s + (1.86 − 3.23i)23-s + (0.0340 − 0.0590i)26-s + (1.15 + 2.38i)28-s + 0.898i·29-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.997 − 0.0716i)7-s + 0.353·8-s + (1.03 − 0.600i)11-s − 0.0189·13-s + (0.396 − 0.585i)14-s + (−0.125 + 0.216i)16-s + (−1.53 + 0.888i)17-s + (0.404 + 0.233i)19-s + 0.848i·22-s + (0.389 − 0.673i)23-s + (0.00668 − 0.0115i)26-s + (0.218 + 0.449i)28-s + 0.166i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.141902639\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.141902639\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.63 + 0.189i)T \) |
good | 11 | \( 1 + (-3.44 + 1.99i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 0.0681T + 13T^{2} \) |
| 17 | \( 1 + (6.34 - 3.66i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.76 - 1.01i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.86 + 3.23i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.898iT - 29T^{2} \) |
| 31 | \( 1 + (4.18 - 2.41i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.52 - 2.03i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 - 0.964iT - 43T^{2} \) |
| 47 | \( 1 + (1.43 + 0.830i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.61 + 11.4i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.32 - 9.21i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.51 - 3.76i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.23 + 5.33i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9.93iT - 71T^{2} \) |
| 73 | \( 1 + (-5.82 - 10.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.77 + 15.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 14.3iT - 83T^{2} \) |
| 89 | \( 1 + (-0.913 + 1.58i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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
−8.832014736892397204655362801437, −8.181430577757245021958272759843, −7.11432182472998625061646251681, −6.50243215157981628721201492554, −6.14641596895531882880517182036, −5.08259399693302777394785640872, −4.07769704204470253285730781911, −3.39154013557628853924873472540, −2.10445356163746052383419909438, −0.76869102993845650164557105187,
0.58995651033237408201130671946, 1.94049754630946054162080584897, 2.81239791582229750266283543143, 3.75334296532567013095148958588, 4.45080251655195405481370754003, 5.46458263483850679140014246968, 6.58598866684375140466699664409, 6.96851505930503298196854602919, 7.84263231461225991946265248244, 8.953498081557671321202298235009