| L(s) = 1 | + (0.939 + 0.342i)2-s + (−1.60 − 0.638i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)5-s + (−1.29 − 1.15i)6-s + (−3.84 + 3.22i)7-s + (0.500 + 0.866i)8-s + (2.18 + 2.05i)9-s + (−0.5 + 0.866i)10-s + (0.698 + 3.96i)11-s + (−0.822 − 1.52i)12-s + (3.16 − 1.15i)13-s + (−4.71 + 1.71i)14-s + (0.908 − 1.47i)15-s + (0.173 + 0.984i)16-s + (0.383 − 0.663i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (−0.929 − 0.368i)3-s + (0.383 + 0.321i)4-s + (−0.0776 + 0.440i)5-s + (−0.528 − 0.469i)6-s + (−1.45 + 1.21i)7-s + (0.176 + 0.306i)8-s + (0.727 + 0.685i)9-s + (−0.158 + 0.273i)10-s + (0.210 + 1.19i)11-s + (−0.237 − 0.440i)12-s + (0.877 − 0.319i)13-s + (−1.25 + 0.458i)14-s + (0.234 − 0.380i)15-s + (0.0434 + 0.246i)16-s + (0.0929 − 0.161i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.718291 + 0.832144i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.718291 + 0.832144i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 3 | \( 1 + (1.60 + 0.638i)T \) |
| 5 | \( 1 + (0.173 - 0.984i)T \) |
| good | 7 | \( 1 + (3.84 - 3.22i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.698 - 3.96i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-3.16 + 1.15i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.383 + 0.663i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.22 - 2.12i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.93 + 4.14i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (7.15 + 2.60i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-6.55 - 5.50i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (1.42 - 2.47i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.89 + 1.41i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.42 - 8.06i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.79 + 5.70i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 0.439T + 53T^{2} \) |
| 59 | \( 1 + (-0.211 + 1.20i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-7.11 + 5.97i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-6.96 + 2.53i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.75 + 4.76i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.36 - 11.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.46 - 0.898i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.10 + 0.401i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-6.11 - 10.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.21 - 12.5i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−12.35787999360892731803976523818, −11.60391912393810465342395219037, −10.37426350323177650267035014456, −9.544965160951732618719426299254, −8.036772632425421580207821935439, −6.75674361073808865884453764068, −6.24581583761531762066627572992, −5.33945407961490238434692702529, −3.84065914253877126812326834410, −2.36062113315012603184328613214,
0.78020389730342078013953256866, 3.57266808803914635991240198683, 4.08101812152485490848444513102, 5.69949842422799650350429911656, 6.29622552356539134558753795494, 7.39211896143231967050409216820, 9.135186659777962739407122505899, 10.00375088472483856393130928193, 10.90723160584261952316767425347, 11.58048469238720460589849495232
