| 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
−15.00342103821547749955456897513, −13.30196397107095710327815641368, −12.80970303854058268279889431278, −11.44779479469800337328206311983, −10.40411989913549372731957231414, −9.564782163818971895604056695621, −7.60709714104221497565711128193, −5.60562136602461403907805469203, −4.72832501698847284114950017194, −2.68585626354920518872783328907,
3.41486717933226148283128215035, 5.21999981164846299599233953222, 6.58549273483988319359752609579, 7.52498349893677102016778967967, 9.011331578191609137499275122855, 10.77563890795248584023248338300, 12.06604075388387294782888357044, 13.02245323824127859646180274670, 13.90171067910332104147234724866, 15.01170613862610804823098151561
