L(s) = 1 | + (−1.5 + 0.866i)3-s + (−0.5 − 2.17i)5-s + (1.13 + 2.38i)7-s + (−2.63 − 4.56i)11-s + 2.62i·13-s + (2.63 + 2.83i)15-s + (−0.362 + 0.209i)17-s + (−1.63 + 2.83i)19-s + (−3.77 − 2.59i)21-s + (−6.77 − 3.91i)23-s + (−4.50 + 2.17i)25-s − 5.19i·27-s − 4.27·29-s + (1.63 + 2.83i)31-s + (7.91 + 4.56i)33-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.499i)3-s + (−0.223 − 0.974i)5-s + (0.429 + 0.902i)7-s + (−0.795 − 1.37i)11-s + 0.728i·13-s + (0.680 + 0.732i)15-s + (−0.0879 + 0.0507i)17-s + (−0.375 + 0.650i)19-s + (−0.823 − 0.566i)21-s + (−1.41 − 0.815i)23-s + (−0.900 + 0.435i)25-s − 0.999i·27-s − 0.793·29-s + (0.294 + 0.509i)31-s + (1.37 + 0.795i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + (0.5 + 2.17i)T \) |
| 7 | \( 1 + (-1.13 - 2.38i)T \) |
good | 3 | \( 1 + (1.5 - 0.866i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (2.63 + 4.56i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.62iT - 13T^{2} \) |
| 17 | \( 1 + (0.362 - 0.209i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.63 - 2.83i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.77 + 3.91i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4.27T + 29T^{2} \) |
| 31 | \( 1 + (-1.63 - 2.83i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (8.63 + 4.98i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3.72T + 41T^{2} \) |
| 43 | \( 1 - 2.15iT - 43T^{2} \) |
| 47 | \( 1 + (5.63 + 3.25i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.91 - 2.83i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.63 - 2.83i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.77 + 11.7i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.04 + 1.76i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4.54T + 71T^{2} \) |
| 73 | \( 1 + (-5.63 + 3.25i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.63 - 6.30i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.40iT - 83T^{2} \) |
| 89 | \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−10.55361515005613159974552744882, −9.472057989153108793835802710455, −8.375480160114893620531556279430, −8.160651813752996349875241656597, −6.32384526515233771265880517222, −5.52095851591609928055688148879, −4.96296139480504841661510012882, −3.80761665812797061162829850565, −2.04787434083982636997707824342, 0,
1.94998155691387685553302570066, 3.47601079616546679901494109684, 4.71374788668524788033126068387, 5.76380294453075210618616015452, 6.85507084151694328354603410731, 7.34917811486977815086666317352, 8.168436854909239360464105070761, 9.870588490477249757855820206283, 10.34711401525902996788728072237