| L(s) = 1 | + (1.31 − 2.69i)3-s + (−5.13 + 0.905i)5-s + (8.93 + 7.49i)7-s + (−5.51 − 7.10i)9-s + (−16.7 − 2.95i)11-s + (−12.2 − 4.44i)13-s + (−4.33 + 15.0i)15-s + (−24.5 + 14.1i)17-s + (13.9 − 24.1i)19-s + (31.9 − 14.1i)21-s + (−6.09 − 7.25i)23-s + (2.08 − 0.757i)25-s + (−26.4 + 5.48i)27-s + (4.43 + 12.1i)29-s + (−11.4 + 9.62i)31-s + ⋯ |
| L(s) = 1 | + (0.439 − 0.898i)3-s + (−1.02 + 0.181i)5-s + (1.27 + 1.07i)7-s + (−0.613 − 0.789i)9-s + (−1.52 − 0.268i)11-s + (−0.939 − 0.341i)13-s + (−0.289 + 1.00i)15-s + (−1.44 + 0.834i)17-s + (0.733 − 1.27i)19-s + (1.52 − 0.675i)21-s + (−0.264 − 0.315i)23-s + (0.0832 − 0.0303i)25-s + (−0.979 + 0.203i)27-s + (0.153 + 0.420i)29-s + (−0.370 + 0.310i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0219225 + 0.114901i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0219225 + 0.114901i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.31 + 2.69i)T \) |
| good | 5 | \( 1 + (5.13 - 0.905i)T + (23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-8.93 - 7.49i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (16.7 + 2.95i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (12.2 + 4.44i)T + (129. + 108. i)T^{2} \) |
| 17 | \( 1 + (24.5 - 14.1i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-13.9 + 24.1i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (6.09 + 7.25i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-4.43 - 12.1i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (11.4 - 9.62i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (19.3 + 33.4i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (0.663 - 1.82i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (9.15 - 51.8i)T + (-1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (1.66 - 1.98i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 14.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-16.7 + 2.96i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (17.6 + 14.7i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (50.8 + 18.5i)T + (3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-73.9 + 42.6i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (1.73 - 3.00i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-15.2 + 5.54i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-31.4 - 86.4i)T + (-5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (55.7 + 32.1i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-7.70 + 43.6i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−10.83867734495613924506922608797, −9.170049110350438078665148647412, −8.302331928067697852864390636951, −7.85217105380110350637568148775, −6.99259977950150283224204149831, −5.56659584472966604744200432484, −4.65725897848838740347480962835, −2.94800145143433025198904655436, −2.09923838015563756510473958581, −0.04301406618947109217664241961,
2.24484955280745626530399250128, 3.76338329941237473905663431986, 4.62539698524235631993615899376, 5.18648608694705254614796504499, 7.35008159381325013449718785906, 7.77342317832918352642941039817, 8.533871537745115736102013456790, 9.836772107557030704709163032629, 10.51020400452188645818825873867, 11.34588890753037028623147056435
