L(s) = 1 | + (−1 + i)2-s + (0.866 − 1.5i)3-s − 2i·4-s + (−3 + 1.73i)5-s + (0.633 + 2.36i)6-s + (−2.59 + 0.5i)7-s + (2 + 2i)8-s + (−1.5 − 2.59i)9-s + (1.26 − 4.73i)10-s + (1.73 + i)11-s + (−3 − 1.73i)12-s + (−4.5 − 2.59i)13-s + (2.09 − 3.09i)14-s + 6i·15-s − 4·16-s + (−4.5 + 2.59i)17-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.499 − 0.866i)3-s − i·4-s + (−1.34 + 0.774i)5-s + (0.258 + 0.965i)6-s + (−0.981 + 0.188i)7-s + (0.707 + 0.707i)8-s + (−0.5 − 0.866i)9-s + (0.400 − 1.49i)10-s + (0.522 + 0.301i)11-s + (−0.866 − 0.499i)12-s + (−1.24 − 0.720i)13-s + (0.560 − 0.827i)14-s + 1.54i·15-s − 16-s + (−1.09 + 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - i)T \) |
| 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 7 | \( 1 + (2.59 - 0.5i)T \) |
good | 5 | \( 1 + (3 - 1.73i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.73 - i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.5 + 2.59i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (4.5 - 2.59i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.866 - 1.5i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.46 - 2i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.5 + 4.33i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.19T + 31T^{2} \) |
| 37 | \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 + 0.866i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-9.52 + 5.5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 1.73T + 47T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 1.73T + 59T^{2} \) |
| 61 | \( 1 - 5.19iT - 61T^{2} \) |
| 67 | \( 1 + 9iT - 67T^{2} \) |
| 71 | \( 1 - 2iT - 71T^{2} \) |
| 73 | \( 1 + (10.5 - 6.06i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 3iT - 79T^{2} \) |
| 83 | \( 1 + (2.59 + 4.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.5 - 4.33i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.5 - 0.866i)T + (48.5 - 84.0i)T^{2} \) |
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
−11.70432835884398746422242965906, −10.46484735022776030681777696681, −9.482680718981277138923513597914, −8.392180430447137745710025233043, −7.56185125510803068444613010408, −6.92029228221117370305331795787, −6.04111805822899657810248528764, −4.00911864210596074217135043914, −2.50556179857420631260033446627, 0,
2.71545191740239012378783046545, 3.99029659077505534818667743828, 4.58309559017506750217701112909, 6.92931409694225584808319047439, 7.928608073454293163366949066974, 9.017003666081641634336198264152, 9.341191250229922405363263977740, 10.51325038401309153850925318532, 11.50227286605910580364539247021