| L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.78 − 1.02i)5-s + (−0.289 − 2.62i)7-s + 0.999·8-s + (−1.78 − 1.02i)10-s + (−3.29 + 0.406i)11-s + 6.97i·13-s + (−2.13 + 1.56i)14-s + (−0.5 − 0.866i)16-s + (−2.66 + 4.61i)17-s + (−7.36 + 4.25i)19-s + 2.05i·20-s + (1.99 + 2.64i)22-s + (1.35 − 0.782i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.797 − 0.460i)5-s + (−0.109 − 0.993i)7-s + 0.353·8-s + (−0.563 − 0.325i)10-s + (−0.992 + 0.122i)11-s + 1.93i·13-s + (−0.569 + 0.418i)14-s + (−0.125 − 0.216i)16-s + (−0.646 + 1.11i)17-s + (−1.69 + 0.975i)19-s + 0.460i·20-s + (0.425 + 0.564i)22-s + (0.282 − 0.163i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6630169150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6630169150\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (0.289 + 2.62i)T \) |
| 11 | \( 1 + (3.29 - 0.406i)T \) |
| good | 5 | \( 1 + (-1.78 + 1.02i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 6.97iT - 13T^{2} \) |
| 17 | \( 1 + (2.66 - 4.61i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (7.36 - 4.25i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.35 + 0.782i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.87T + 29T^{2} \) |
| 31 | \( 1 + (4.39 - 7.60i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.82 + 3.15i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2.42T + 41T^{2} \) |
| 43 | \( 1 + 7.55iT - 43T^{2} \) |
| 47 | \( 1 + (-10.2 + 5.91i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.93 - 5.73i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.49 + 2.01i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.39 + 2.53i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.75 - 8.23i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.30iT - 71T^{2} \) |
| 73 | \( 1 + (-0.828 - 0.478i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.75 - 4.47i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7.50T + 83T^{2} \) |
| 89 | \( 1 + (2.99 - 1.72i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5.95T + 97T^{2} \) |
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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
−9.907687877795823435430284931071, −8.810802008355306475631688356492, −8.588024252782134954856190837086, −7.26421043388769735871293080774, −6.62227705199378035450060803665, −5.53631215229131402359259044388, −4.37153181106695402647255482254, −3.86519194938091777661429774558, −2.20323592946772645542370722658, −1.58731698151219985990120535645,
0.28237645536503717670614222059, 2.38221038268867720384691342040, 2.82405045002716253084987122773, 4.65209402058216871205448280197, 5.52936685509802983002300463703, 6.00872030375361280313232911419, 6.92385599261376954582477460458, 7.86872517407382578110309622399, 8.579767132607564149686463593539, 9.310515212810339015284529019537
