| L(s) = 1 | + (1.28 + 0.587i)2-s + (−1.23 + 4.19i)3-s + (1.30 + 1.51i)4-s + (−3.16 − 3.87i)5-s + (−4.05 + 4.67i)6-s + (−0.828 + 5.76i)7-s + (0.796 + 2.71i)8-s + (−8.53 − 5.48i)9-s + (−1.79 − 6.84i)10-s + (−9.75 + 4.45i)11-s + (−7.95 + 3.63i)12-s + (−7.82 + 1.12i)13-s + (−4.44 + 6.92i)14-s + (20.1 − 8.49i)15-s + (−0.569 + 3.95i)16-s + (−6.38 + 7.36i)17-s + ⋯ |
| L(s) = 1 | + (0.643 + 0.293i)2-s + (−0.410 + 1.39i)3-s + (0.327 + 0.377i)4-s + (−0.632 − 0.774i)5-s + (−0.675 + 0.779i)6-s + (−0.118 + 0.822i)7-s + (0.0996 + 0.339i)8-s + (−0.947 − 0.609i)9-s + (−0.179 − 0.684i)10-s + (−0.886 + 0.405i)11-s + (−0.663 + 0.302i)12-s + (−0.602 + 0.0865i)13-s + (−0.317 + 0.494i)14-s + (1.34 − 0.566i)15-s + (−0.0355 + 0.247i)16-s + (−0.375 + 0.433i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0501i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0283448 - 1.12985i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0283448 - 1.12985i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.28 - 0.587i)T \) |
| 5 | \( 1 + (3.16 + 3.87i)T \) |
| 23 | \( 1 + (-7.56 + 21.7i)T \) |
| good | 3 | \( 1 + (1.23 - 4.19i)T + (-7.57 - 4.86i)T^{2} \) |
| 7 | \( 1 + (0.828 - 5.76i)T + (-47.0 - 13.8i)T^{2} \) |
| 11 | \( 1 + (9.75 - 4.45i)T + (79.2 - 91.4i)T^{2} \) |
| 13 | \( 1 + (7.82 - 1.12i)T + (162. - 47.6i)T^{2} \) |
| 17 | \( 1 + (6.38 - 7.36i)T + (-41.1 - 286. i)T^{2} \) |
| 19 | \( 1 + (-10.2 + 8.90i)T + (51.3 - 357. i)T^{2} \) |
| 29 | \( 1 + (21.9 - 25.3i)T + (-119. - 832. i)T^{2} \) |
| 31 | \( 1 + (-27.3 + 8.02i)T + (808. - 519. i)T^{2} \) |
| 37 | \( 1 + (-31.7 - 20.3i)T + (568. + 1.24e3i)T^{2} \) |
| 41 | \( 1 + (26.8 - 17.2i)T + (698. - 1.52e3i)T^{2} \) |
| 43 | \( 1 + (-25.0 - 7.35i)T + (1.55e3 + 9.99e2i)T^{2} \) |
| 47 | \( 1 - 75.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (4.70 - 32.7i)T + (-2.69e3 - 791. i)T^{2} \) |
| 59 | \( 1 + (-10.5 - 73.1i)T + (-3.33e3 + 980. i)T^{2} \) |
| 61 | \( 1 + (5.39 + 18.3i)T + (-3.13e3 + 2.01e3i)T^{2} \) |
| 67 | \( 1 + (36.6 - 80.3i)T + (-2.93e3 - 3.39e3i)T^{2} \) |
| 71 | \( 1 + (-24.0 + 52.7i)T + (-3.30e3 - 3.80e3i)T^{2} \) |
| 73 | \( 1 + (26.6 - 23.1i)T + (758. - 5.27e3i)T^{2} \) |
| 79 | \( 1 + (-154. + 22.2i)T + (5.98e3 - 1.75e3i)T^{2} \) |
| 83 | \( 1 + (77.0 + 49.4i)T + (2.86e3 + 6.26e3i)T^{2} \) |
| 89 | \( 1 + (-38.1 + 129. i)T + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (-25.6 + 16.4i)T + (3.90e3 - 8.55e3i)T^{2} \) |
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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
−12.43244264997365308599134131793, −11.57452348904577842321401709105, −10.66277048632096207984091549600, −9.542523391339532943475446677875, −8.676865515901417190238427795375, −7.48923229563604418630125204038, −5.87917371087651576814308177445, −4.89866030106700641439127814895, −4.41164664000318452580331225495, −2.88790424494140840327257426060,
0.50152432807836366248323192130, 2.36463642194394288050963329284, 3.69360380096835446172290899483, 5.31578919765251185016007850836, 6.51425513975509231691607079396, 7.34817247221515994746320829190, 7.895409294447451895262943831274, 9.913383312704592774119850263600, 10.93517180988589241760638795147, 11.64280875277633880121479275937
