L(s) = 1 | + (2.63 − 1.52i)3-s + (0.866 − 1.5i)5-s + (2.14 + 1.54i)7-s + (3.12 − 5.40i)9-s + (−1.09 − 1.88i)11-s − 5.91·13-s − 5.26i·15-s + (3.62 − 2.09i)17-s + (−3.27 − 1.88i)19-s + (8.00 + 0.792i)21-s + (−1.88 − 1.09i)23-s + (1 + 1.73i)25-s − 9.85i·27-s + 7.41i·29-s + (4.03 + 6.99i)31-s + ⋯ |
L(s) = 1 | + (1.52 − 0.877i)3-s + (0.387 − 0.670i)5-s + (0.812 + 0.582i)7-s + (1.04 − 1.80i)9-s + (−0.328 − 0.569i)11-s − 1.64·13-s − 1.35i·15-s + (0.878 − 0.507i)17-s + (−0.750 − 0.433i)19-s + (1.74 + 0.173i)21-s + (−0.393 − 0.227i)23-s + (0.200 + 0.346i)25-s − 1.89i·27-s + 1.37i·29-s + (0.725 + 1.25i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.293 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.293 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.28179 - 1.68595i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.28179 - 1.68595i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.14 - 1.54i)T \) |
good | 3 | \( 1 + (-2.63 + 1.52i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.866 + 1.5i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.09 + 1.88i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.91T + 13T^{2} \) |
| 17 | \( 1 + (-3.62 + 2.09i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.27 + 1.88i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.88 + 1.09i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 7.41iT - 29T^{2} \) |
| 31 | \( 1 + (-4.03 - 6.99i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.07 - 0.621i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 7.34iT - 41T^{2} \) |
| 43 | \( 1 - 6.16T + 43T^{2} \) |
| 47 | \( 1 + (-4.56 + 7.89i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.37 + 0.792i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.63 + 1.52i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.07 - 1.86i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.81 + 8.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.72iT - 71T^{2} \) |
| 73 | \( 1 + (11.7 - 6.77i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.23 + 4.17i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12.1iT - 83T^{2} \) |
| 89 | \( 1 + (-9.98 - 5.76i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5.91iT - 97T^{2} \) |
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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
−9.568028952662590373282293670871, −8.920005700290769565028563990771, −8.328586376988395596560863370897, −7.61461561188680506394428689997, −6.83040791686934319518098522309, −5.44902205005609021944946998708, −4.66472908765727104513046448558, −3.07694468948993586070366096155, −2.35841707794882064578360371324, −1.27505801620927099975937200672,
2.12147282670581813421301849873, 2.67956500599733170783384400560, 4.08043362222592805996984717419, 4.53147557529532969626388446166, 5.84626079956696315715056329159, 7.44129728752202659605485548404, 7.67295089993096353090396047995, 8.589945618969112381542732352204, 9.717699431538216207635629327553, 10.09664601734747938384298183601