| L(s) = 1 | + (0.972 − 1.74i)2-s + (−1.83 + 2.37i)3-s + (−2.10 − 3.39i)4-s + (−0.431 − 2.44i)5-s + (2.37 + 5.51i)6-s + (9.31 + 7.81i)7-s + (−7.99 + 0.376i)8-s + (−2.29 − 8.70i)9-s + (−4.69 − 1.62i)10-s + (3.17 − 18.0i)11-s + (11.9 + 1.21i)12-s + (7.62 − 20.9i)13-s + (22.7 − 8.67i)14-s + (6.60 + 3.45i)15-s + (−7.11 + 14.3i)16-s + (−4.71 + 2.72i)17-s + ⋯ |
| L(s) = 1 | + (0.486 − 0.873i)2-s + (−0.610 + 0.792i)3-s + (−0.526 − 0.849i)4-s + (−0.0863 − 0.489i)5-s + (0.395 + 0.918i)6-s + (1.33 + 1.11i)7-s + (−0.998 + 0.0470i)8-s + (−0.255 − 0.966i)9-s + (−0.469 − 0.162i)10-s + (0.288 − 1.63i)11-s + (0.994 + 0.101i)12-s + (0.586 − 1.61i)13-s + (1.62 − 0.619i)14-s + (0.440 + 0.230i)15-s + (−0.444 + 0.895i)16-s + (−0.277 + 0.160i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0599 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0599 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.09814 - 1.16612i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09814 - 1.16612i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.972 + 1.74i)T \) |
| 3 | \( 1 + (1.83 - 2.37i)T \) |
| good | 5 | \( 1 + (0.431 + 2.44i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-9.31 - 7.81i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-3.17 + 18.0i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-7.62 + 20.9i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (4.71 - 2.72i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-3.59 - 2.07i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (8.24 + 9.82i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (17.4 - 6.36i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-15.8 + 13.3i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (23.5 - 13.6i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-0.450 + 1.23i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-26.4 - 4.66i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-45.1 + 53.7i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 16.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-9.85 - 55.9i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (44.4 - 52.9i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (28.4 - 78.2i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (50.0 - 28.8i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (35.7 - 61.8i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (37.1 - 13.5i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-7.73 + 2.81i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-46.1 - 26.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-30.7 + 174. i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.69848785008246814210785814224, −11.03666588754384245286644335546, −10.31682195995136458969386272408, −8.700413386970093996991955600272, −8.582995617032882224120631862308, −5.75691896343148664627912184420, −5.57461247466358492541506032703, −4.31638436699244500005474981907, −2.98788593865531788189397910650, −0.916856440533620317992406210547,
1.75881977894630803458927452640, 4.19249803845760011164721618268, 4.90926347278955980144312131305, 6.48842861347467777640368211343, 7.20305895167429597446428153314, 7.73602730789559321174483961043, 9.153179823393579970952809002721, 10.73282081224895746837873461847, 11.57835976616508582608872788451, 12.34182023512286691237157497918
