| L(s) = 1 | + (0.940 + 1.05i)2-s + (0.235 + 0.0232i)3-s + (−0.230 + 1.98i)4-s + (0.471 + 0.881i)5-s + (0.197 + 0.270i)6-s + (0.890 + 4.47i)7-s + (−2.31 + 1.62i)8-s + (−2.88 − 0.574i)9-s + (−0.487 + 1.32i)10-s + (−0.120 − 0.0992i)11-s + (−0.100 + 0.462i)12-s + (−1.70 − 0.911i)13-s + (−3.89 + 5.15i)14-s + (0.0906 + 0.218i)15-s + (−3.89 − 0.914i)16-s + (2.27 − 5.49i)17-s + ⋯ |
| L(s) = 1 | + (0.665 + 0.746i)2-s + (0.136 + 0.0134i)3-s + (−0.115 + 0.993i)4-s + (0.210 + 0.394i)5-s + (0.0805 + 0.110i)6-s + (0.336 + 1.69i)7-s + (−0.818 + 0.574i)8-s + (−0.962 − 0.191i)9-s + (−0.154 + 0.419i)10-s + (−0.0364 − 0.0299i)11-s + (−0.0289 + 0.133i)12-s + (−0.473 − 0.252i)13-s + (−1.03 + 1.37i)14-s + (0.0234 + 0.0564i)15-s + (−0.973 − 0.228i)16-s + (0.551 − 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.852 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.529794 + 1.87425i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.529794 + 1.87425i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.940 - 1.05i)T \) |
| 5 | \( 1 + (-0.471 - 0.881i)T \) |
| good | 3 | \( 1 + (-0.235 - 0.0232i)T + (2.94 + 0.585i)T^{2} \) |
| 7 | \( 1 + (-0.890 - 4.47i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (0.120 + 0.0992i)T + (2.14 + 10.7i)T^{2} \) |
| 13 | \( 1 + (1.70 + 0.911i)T + (7.22 + 10.8i)T^{2} \) |
| 17 | \( 1 + (-2.27 + 5.49i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-4.91 + 1.49i)T + (15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (7.35 - 4.91i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-4.20 - 5.12i)T + (-5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (-5.97 + 5.97i)T - 31iT^{2} \) |
| 37 | \( 1 + (2.12 - 6.99i)T + (-30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (-4.21 - 6.30i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-10.1 + 0.995i)T + (42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (-5.19 - 2.15i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (6.34 - 7.72i)T + (-10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (-3.38 + 1.81i)T + (32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.226 + 2.29i)T + (-59.8 - 11.9i)T^{2} \) |
| 67 | \( 1 + (0.984 - 9.99i)T + (-65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (-0.0978 + 0.0194i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-1.74 + 8.76i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-7.06 + 2.92i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (2.39 + 7.89i)T + (-69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (6.22 + 4.15i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-1.97 + 1.97i)T - 97iT^{2} \) |
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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
−11.44623458413089330004562145592, −9.705753577278861991811015088022, −9.111163583281881219289754013208, −8.131467691454179122708212935377, −7.46701614906184348691306274186, −6.10693940438872450177667499587, −5.63499190258325091798034550785, −4.80641887434947974836935346134, −3.06085389353223037115877550493, −2.60991871674159016848163696301,
0.849389705763669336777626121853, 2.25241656442829314350501748181, 3.68756457693693476964993345926, 4.37650688006801364767431438087, 5.47238526779236672723146224774, 6.38578813542099299809256332460, 7.65206862203678719959292326750, 8.472994184859162861214150813222, 9.742263753789331202575352010786, 10.36389827057202025601380641244
