L(s) = 1 | + (0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (2.38 − 1.37i)5-s − 0.999i·6-s + (0.588 + 2.57i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (1.37 − 2.38i)10-s + (0.600 + 0.346i)11-s + (−0.499 − 0.866i)12-s + (0.924 − 3.48i)13-s + (1.79 + 1.93i)14-s − 2.74i·15-s + (−0.5 − 0.866i)16-s + (−3.18 + 5.51i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + (1.06 − 0.614i)5-s − 0.408i·6-s + (0.222 + 0.974i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.434 − 0.753i)10-s + (0.180 + 0.104i)11-s + (−0.144 − 0.249i)12-s + (0.256 − 0.966i)13-s + (0.480 + 0.518i)14-s − 0.709i·15-s + (−0.125 − 0.216i)16-s + (−0.772 + 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.21318 - 1.42739i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.21318 - 1.42739i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.588 - 2.57i)T \) |
| 13 | \( 1 + (-0.924 + 3.48i)T \) |
good | 5 | \( 1 + (-2.38 + 1.37i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.600 - 0.346i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (3.18 - 5.51i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.78 + 1.02i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.903 - 1.56i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 8.85T + 29T^{2} \) |
| 31 | \( 1 + (-0.817 - 0.471i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.833 + 0.481i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 11.7iT - 41T^{2} \) |
| 43 | \( 1 + 4.97T + 43T^{2} \) |
| 47 | \( 1 + (7.81 - 4.51i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.98 + 3.43i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-10.5 - 6.06i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.03 - 8.71i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.40 - 4.85i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 7.68iT - 71T^{2} \) |
| 73 | \( 1 + (-4.89 - 2.82i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.339 + 0.587i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.32iT - 83T^{2} \) |
| 89 | \( 1 + (11.5 - 6.69i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 9.89iT - 97T^{2} \) |
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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
−10.75497612897089363562224711992, −9.685954157811659200026184288187, −8.941515029236832276496380578866, −8.138955546865258678942001474179, −6.76865215128790349776029648168, −5.68520814451952238724988645115, −5.35786586843913858743453657187, −3.78037975690768414891117106677, −2.40572655532119693815331961337, −1.54738700758617357011416827078,
2.00849937581288247519591392346, 3.28729475812641582605040265472, 4.36168369849859305921477979112, 5.28619369285800784926131543831, 6.51909105331515971065495887178, 7.03724532304687935866157747881, 8.222800465614419365020648053510, 9.452917711165474552298461945309, 9.915715531578660296913396041793, 11.16243896397238542717937050201