| L(s) = 1 | + (0.998 + 1.00i)2-s + (−0.477 − 0.714i)3-s + (−0.00472 + 1.99i)4-s + (0.0517 + 0.260i)5-s + (0.238 − 1.19i)6-s + (−0.515 − 1.24i)7-s + (−2.00 + 1.99i)8-s + (0.865 − 2.08i)9-s + (−0.208 + 0.311i)10-s + (−4.11 − 2.74i)11-s + (1.43 − 0.951i)12-s + (−0.650 + 3.26i)13-s + (0.730 − 1.75i)14-s + (0.161 − 0.161i)15-s + (−3.99 − 0.0188i)16-s + (1.10 + 1.10i)17-s + ⋯ |
| L(s) = 1 | + (0.706 + 0.707i)2-s + (−0.275 − 0.412i)3-s + (−0.00236 + 0.999i)4-s + (0.0231 + 0.116i)5-s + (0.0973 − 0.486i)6-s + (−0.194 − 0.470i)7-s + (−0.709 + 0.704i)8-s + (0.288 − 0.696i)9-s + (−0.0660 + 0.0985i)10-s + (−1.24 − 0.829i)11-s + (0.413 − 0.274i)12-s + (−0.180 + 0.906i)13-s + (0.195 − 0.469i)14-s + (0.0416 − 0.0416i)15-s + (−0.999 − 0.00472i)16-s + (0.267 + 0.267i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03176 + 0.365055i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03176 + 0.365055i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.998 - 1.00i)T \) |
| good | 3 | \( 1 + (0.477 + 0.714i)T + (-1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (-0.0517 - 0.260i)T + (-4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (0.515 + 1.24i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (4.11 + 2.74i)T + (4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (0.650 - 3.26i)T + (-12.0 - 4.97i)T^{2} \) |
| 17 | \( 1 + (-1.10 - 1.10i)T + 17iT^{2} \) |
| 19 | \( 1 + (-2.56 - 0.510i)T + (17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (-3.70 - 1.53i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (5.46 - 3.65i)T + (11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 - 8.22iT - 31T^{2} \) |
| 37 | \( 1 + (-7.58 + 1.50i)T + (34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (10.4 + 4.33i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-2.31 + 3.46i)T + (-16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (-2.33 - 2.33i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.33 + 2.23i)T + (20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (-1.16 - 5.84i)T + (-54.5 + 22.5i)T^{2} \) |
| 61 | \( 1 + (6.89 + 10.3i)T + (-23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + (-1.94 - 2.91i)T + (-25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (5.39 + 13.0i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (0.375 - 0.906i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-1.20 + 1.20i)T - 79iT^{2} \) |
| 83 | \( 1 + (-15.1 - 3.00i)T + (76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (1.66 - 0.689i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 - 15.4iT - 97T^{2} \) |
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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
−15.01170613862610804823098151561, −13.90171067910332104147234724866, −13.02245323824127859646180274670, −12.06604075388387294782888357044, −10.77563890795248584023248338300, −9.011331578191609137499275122855, −7.52498349893677102016778967967, −6.58549273483988319359752609579, −5.21999981164846299599233953222, −3.41486717933226148283128215035,
2.68585626354920518872783328907, 4.72832501698847284114950017194, 5.60562136602461403907805469203, 7.60709714104221497565711128193, 9.564782163818971895604056695621, 10.40411989913549372731957231414, 11.44779479469800337328206311983, 12.80970303854058268279889431278, 13.30196397107095710327815641368, 15.00342103821547749955456897513
