| L(s) = 1 | + (0.761 − 1.19i)2-s + (−2.27 − 2.42i)3-s + (−0.839 − 1.81i)4-s + (1.10 + 2.43i)5-s + (−4.61 + 0.859i)6-s + (−2.12 + 1.58i)7-s + (−2.80 − 0.381i)8-s + (−0.525 + 8.01i)9-s + (3.74 + 0.540i)10-s + (−0.263 − 0.423i)11-s + (−2.49 + 6.15i)12-s + (−0.0923 − 0.937i)13-s + (0.270 + 3.73i)14-s + (3.40 − 8.21i)15-s + (−2.58 + 3.04i)16-s + (3.79 + 0.500i)17-s + ⋯ |
| L(s) = 1 | + (0.538 − 0.842i)2-s + (−1.31 − 1.39i)3-s + (−0.419 − 0.907i)4-s + (0.494 + 1.09i)5-s + (−1.88 + 0.350i)6-s + (−0.801 + 0.598i)7-s + (−0.990 − 0.134i)8-s + (−0.175 + 2.67i)9-s + (1.18 + 0.171i)10-s + (−0.0794 − 0.127i)11-s + (−0.719 + 1.77i)12-s + (−0.0256 − 0.260i)13-s + (0.0723 + 0.997i)14-s + (0.878 − 2.12i)15-s + (−0.647 + 0.762i)16-s + (0.921 + 0.121i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.893584 - 0.188182i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.893584 - 0.188182i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.761 + 1.19i)T \) |
| 7 | \( 1 + (2.12 - 1.58i)T \) |
| good | 3 | \( 1 + (2.27 + 2.42i)T + (-0.196 + 2.99i)T^{2} \) |
| 5 | \( 1 + (-1.10 - 2.43i)T + (-3.29 + 3.75i)T^{2} \) |
| 11 | \( 1 + (0.263 + 0.423i)T + (-4.86 + 9.86i)T^{2} \) |
| 13 | \( 1 + (0.0923 + 0.937i)T + (-12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (-3.79 - 0.500i)T + (16.4 + 4.39i)T^{2} \) |
| 19 | \( 1 + (-0.311 + 1.88i)T + (-17.9 - 6.10i)T^{2} \) |
| 23 | \( 1 + (-4.68 - 5.34i)T + (-3.00 + 22.8i)T^{2} \) |
| 29 | \( 1 + (0.586 + 1.09i)T + (-16.1 + 24.1i)T^{2} \) |
| 31 | \( 1 + (2.85 - 0.764i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (3.89 - 10.3i)T + (-27.8 - 24.3i)T^{2} \) |
| 41 | \( 1 + (-1.25 - 6.28i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (5.81 - 1.76i)T + (35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (-1.20 + 1.57i)T + (-12.1 - 45.3i)T^{2} \) |
| 53 | \( 1 + (1.02 + 1.64i)T + (-23.4 + 47.5i)T^{2} \) |
| 59 | \( 1 + (-2.05 + 2.87i)T + (-18.9 - 55.8i)T^{2} \) |
| 61 | \( 1 + (-13.0 + 3.03i)T + (54.7 - 26.9i)T^{2} \) |
| 67 | \( 1 + (-6.51 - 6.95i)T + (-4.38 + 66.8i)T^{2} \) |
| 71 | \( 1 + (-5.14 - 7.69i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (14.2 - 7.03i)T + (44.4 - 57.9i)T^{2} \) |
| 79 | \( 1 + (-1.54 - 11.7i)T + (-76.3 + 20.4i)T^{2} \) |
| 83 | \( 1 + (-3.44 + 4.19i)T + (-16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (-3.75 - 11.0i)T + (-70.6 + 54.1i)T^{2} \) |
| 97 | \( 1 + (6.67 - 6.67i)T - 97iT^{2} \) |
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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
−10.27455029126283808794172818066, −9.738675617095851741218215413628, −8.274328690007758947948038430603, −6.93221620802324941861457914373, −6.58926969862474051384489358552, −5.62941831534410129337910049298, −5.22197887174057282636460897369, −3.25460165586112294831122373281, −2.44131486191003337765235940263, −1.21116427463650464179136360710,
0.49012912167561857003262674936, 3.43648873118759094582006264133, 4.19424102816392735114168444063, 5.07664103692545469319317509498, 5.55940095749412113326075167603, 6.39675523379253825700460826444, 7.29351922195818954343229414540, 8.845532103822485611469448461185, 9.277554891520916467529024005708, 10.11370088587110618886208448917
