| L(s) = 1 | + (−2.36 − 1.84i)3-s + (−7.65 + 1.34i)5-s + (−10.4 − 8.78i)7-s + (2.18 + 8.73i)9-s + (−2.55 − 0.450i)11-s + (−8.23 − 2.99i)13-s + (20.5 + 10.9i)15-s + (15.2 − 8.82i)17-s + (−1.46 + 2.54i)19-s + (8.55 + 40.1i)21-s + (11.8 + 14.1i)23-s + (33.2 − 12.1i)25-s + (10.9 − 24.6i)27-s + (1.05 + 2.91i)29-s + (−30.6 + 25.7i)31-s + ⋯ |
| L(s) = 1 | + (−0.788 − 0.615i)3-s + (−1.53 + 0.269i)5-s + (−1.49 − 1.25i)7-s + (0.242 + 0.970i)9-s + (−0.232 − 0.0409i)11-s + (−0.633 − 0.230i)13-s + (1.37 + 0.728i)15-s + (0.899 − 0.519i)17-s + (−0.0773 + 0.133i)19-s + (0.407 + 1.91i)21-s + (0.517 + 0.616i)23-s + (1.33 − 0.484i)25-s + (0.405 − 0.914i)27-s + (0.0365 + 0.100i)29-s + (−0.988 + 0.829i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.293652 + 0.0863075i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.293652 + 0.0863075i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.36 + 1.84i)T \) |
| good | 5 | \( 1 + (7.65 - 1.34i)T + (23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (10.4 + 8.78i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (2.55 + 0.450i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (8.23 + 2.99i)T + (129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-15.2 + 8.82i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (1.46 - 2.54i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-11.8 - 14.1i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-1.05 - 2.91i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (30.6 - 25.7i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (12.8 + 22.2i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-21.2 + 58.3i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (5.71 - 32.4i)T + (-1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-33.4 + 39.8i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 53.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (102. - 18.0i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (4.56 + 3.82i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-44.4 - 16.1i)T + (3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-77.5 + 44.7i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-33.6 + 58.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (34.1 - 12.4i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (6.94 + 19.0i)T + (-5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-83.7 - 48.3i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (11.6 - 66.2i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−10.88662054129238783538667779870, −10.51428604412518980453373152505, −9.337919279349358420620127529534, −7.65660832614011467476662714290, −7.41586862769152723893792013279, −6.63493611355925134647276456837, −5.30812927138631326712167174148, −3.99717093867925339006436684696, −3.11976632493453052260858563452, −0.69853049567773953267844353796,
0.24385423327944028017171382633, 2.98856249067199407456408551654, 3.92137282296975839984325702288, 5.05491022707275486211409134106, 6.04706683174323702147377818217, 7.00878010975256717962196575555, 8.197958393401055588478665199134, 9.217606749793804331223377994474, 9.880043765119571938183586431953, 11.00636614402684231580165992874
