| 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
−11.00636614402684231580165992874, −9.880043765119571938183586431953, −9.217606749793804331223377994474, −8.197958393401055588478665199134, −7.00878010975256717962196575555, −6.04706683174323702147377818217, −5.05491022707275486211409134106, −3.92137282296975839984325702288, −2.98856249067199407456408551654, −0.24385423327944028017171382633,
0.69853049567773953267844353796, 3.11976632493453052260858563452, 3.99717093867925339006436684696, 5.30812927138631326712167174148, 6.63493611355925134647276456837, 7.41586862769152723893792013279, 7.65660832614011467476662714290, 9.337919279349358420620127529534, 10.51428604412518980453373152505, 10.88662054129238783538667779870
