| L(s) = 1 | + (−2.55 + 0.554i)3-s + (−0.787 + 2.09i)5-s + (−3.57 + 1.95i)7-s + (3.47 − 1.58i)9-s + (−0.603 − 0.522i)11-s + (−2.63 + 4.82i)13-s + (0.846 − 5.77i)15-s + (−0.466 − 0.349i)17-s + (−1.14 + 7.98i)19-s + (8.03 − 6.96i)21-s + (−4.08 − 2.50i)23-s + (−3.76 − 3.29i)25-s + (−1.70 + 1.27i)27-s + (10.1 − 1.45i)29-s + (−1.92 + 1.23i)31-s + ⋯ |
| L(s) = 1 | + (−1.47 + 0.320i)3-s + (−0.352 + 0.935i)5-s + (−1.35 + 0.737i)7-s + (1.15 − 0.528i)9-s + (−0.181 − 0.157i)11-s + (−0.731 + 1.33i)13-s + (0.218 − 1.49i)15-s + (−0.113 − 0.0846i)17-s + (−0.263 + 1.83i)19-s + (1.75 − 1.51i)21-s + (−0.852 − 0.523i)23-s + (−0.752 − 0.659i)25-s + (−0.328 + 0.246i)27-s + (1.88 − 0.270i)29-s + (−0.344 + 0.221i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0252 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0252 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0602101 - 0.0587115i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0602101 - 0.0587115i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.787 - 2.09i)T \) |
| 23 | \( 1 + (4.08 + 2.50i)T \) |
| good | 3 | \( 1 + (2.55 - 0.554i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (3.57 - 1.95i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (0.603 + 0.522i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.63 - 4.82i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (0.466 + 0.349i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (1.14 - 7.98i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-10.1 + 1.45i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (1.92 - 1.23i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.50 - 0.562i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (1.48 - 3.24i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (2.30 + 10.5i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-2.58 - 2.58i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.41 - 4.42i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (1.90 + 6.47i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (3.02 + 4.71i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-6.52 - 0.466i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-1.93 - 2.23i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (7.53 + 10.0i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-3.62 + 1.06i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (1.38 - 3.70i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (11.7 + 7.52i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-6.24 - 16.7i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.48071916257875852510844618511, −10.21227561178444259632968469602, −9.361885936732847158860414983287, −8.129231721035764424640289817318, −6.85407206406331333930511387498, −6.37620769977193544982611989868, −5.80178847804570357013328033758, −4.57718672221999297938190652919, −3.60224251492591868381474442306, −2.33705829183132462757866965950,
0.06771437928083696149295596262, 0.831024045547292235743044875094, 2.93382069984324103545010015163, 4.33286168443706496184662271335, 5.09469599995041919686684575748, 5.94433744497731543327846806164, 6.82867978466807323127657830763, 7.47810595658931928285216290551, 8.596909630000775169437325587209, 9.766037365755324143592977150192
