| L(s) = 1 | + (23.8 + 41.2i)2-s + (−121.5 − 70.1i)3-s + (−621. + 1.07e3i)4-s + (4.63e3 − 2.67e3i)5-s − 6.67e3i·6-s + (−5.75e3 − 1.57e4i)7-s − 1.04e4·8-s + (9.84e3 + 1.70e4i)9-s + (2.20e5 + 1.27e5i)10-s + (1.16e5 − 2.02e5i)11-s + (1.51e5 − 8.71e4i)12-s + 3.20e5i·13-s + (5.14e5 − 6.13e5i)14-s − 7.51e5·15-s + (3.88e5 + 6.72e5i)16-s + (5.90e5 + 3.40e5i)17-s + ⋯ |
| L(s) = 1 | + (0.743 + 1.28i)2-s + (−0.5 − 0.288i)3-s + (−0.606 + 1.05i)4-s + (1.48 − 0.856i)5-s − 0.859i·6-s + (−0.342 − 0.939i)7-s − 0.317·8-s + (0.166 + 0.288i)9-s + (2.20 + 1.27i)10-s + (0.725 − 1.25i)11-s + (0.606 − 0.350i)12-s + 0.863i·13-s + (0.955 − 1.14i)14-s − 0.989·15-s + (0.370 + 0.641i)16-s + (0.415 + 0.240i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(2.80378 + 0.680585i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.80378 + 0.680585i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (121.5 + 70.1i)T \) |
| 7 | \( 1 + (5.75e3 + 1.57e4i)T \) |
| good | 2 | \( 1 + (-23.8 - 41.2i)T + (-512 + 886. i)T^{2} \) |
| 5 | \( 1 + (-4.63e3 + 2.67e3i)T + (4.88e6 - 8.45e6i)T^{2} \) |
| 11 | \( 1 + (-1.16e5 + 2.02e5i)T + (-1.29e10 - 2.24e10i)T^{2} \) |
| 13 | \( 1 - 3.20e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 + (-5.90e5 - 3.40e5i)T + (1.00e12 + 1.74e12i)T^{2} \) |
| 19 | \( 1 + (1.45e6 - 8.38e5i)T + (3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (3.23e6 + 5.60e6i)T + (-2.07e13 + 3.58e13i)T^{2} \) |
| 29 | \( 1 - 3.89e7T + 4.20e14T^{2} \) |
| 31 | \( 1 + (-8.51e6 - 4.91e6i)T + (4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + (3.50e7 + 6.06e7i)T + (-2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 - 1.03e6iT - 1.34e16T^{2} \) |
| 43 | \( 1 + 7.79e7T + 2.16e16T^{2} \) |
| 47 | \( 1 + (9.76e6 - 5.63e6i)T + (2.62e16 - 4.55e16i)T^{2} \) |
| 53 | \( 1 + (2.05e8 - 3.56e8i)T + (-8.74e16 - 1.51e17i)T^{2} \) |
| 59 | \( 1 + (-7.52e8 - 4.34e8i)T + (2.55e17 + 4.42e17i)T^{2} \) |
| 61 | \( 1 + (1.35e9 - 7.81e8i)T + (3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (2.66e8 - 4.61e8i)T + (-9.11e17 - 1.57e18i)T^{2} \) |
| 71 | \( 1 + 5.77e8T + 3.25e18T^{2} \) |
| 73 | \( 1 + (-6.26e8 - 3.61e8i)T + (2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (-5.84e8 - 1.01e9i)T + (-4.73e18 + 8.19e18i)T^{2} \) |
| 83 | \( 1 + 6.01e8iT - 1.55e19T^{2} \) |
| 89 | \( 1 + (7.28e9 - 4.20e9i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 - 4.45e9iT - 7.37e19T^{2} \) |
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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
−16.43786439990412967840397166356, −14.07705586490267730903368381572, −13.78432825471805295109951778334, −12.51545487501600472788989834065, −10.34758245328076121014403228355, −8.589331991911839682406208928800, −6.60648799193054094394889206311, −5.91610517948790998629632692457, −4.41379154652120435670625716433, −1.19507804368480801283459737775,
1.77514828887661369522662285912, 2.99611571567939714972433355402, 5.06232996189546522117518739771, 6.41479551818110238178204463542, 9.688154642476043381681109305631, 10.28914845205267651028059301709, 11.79045251912445900671531727165, 12.80992660012507468463398082716, 14.08121204259494731624988632695, 15.26035067735906065181744289176
