| L(s) = 1 | + (1.73 + i)2-s + (0.588 + 5.16i)3-s + (1.99 + 3.46i)4-s + (4.27 − 7.41i)5-s + (−4.14 + 9.53i)6-s + (−6.41 + 17.3i)7-s + 7.99i·8-s + (−26.3 + 6.07i)9-s + (14.8 − 8.55i)10-s + (53.8 − 31.0i)11-s + (−16.7 + 12.3i)12-s − 61.7i·13-s + (−28.4 + 23.6i)14-s + (40.7 + 17.7i)15-s + (−8 + 13.8i)16-s + (−13.3 − 23.0i)17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.113 + 0.993i)3-s + (0.249 + 0.433i)4-s + (0.382 − 0.662i)5-s + (−0.281 + 0.648i)6-s + (−0.346 + 0.938i)7-s + 0.353i·8-s + (−0.974 + 0.224i)9-s + (0.468 − 0.270i)10-s + (1.47 − 0.852i)11-s + (−0.401 + 0.297i)12-s − 1.31i·13-s + (−0.543 + 0.451i)14-s + (0.701 + 0.305i)15-s + (−0.125 + 0.216i)16-s + (−0.190 − 0.329i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.53436 + 1.05376i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.53436 + 1.05376i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.73 - i)T \) |
| 3 | \( 1 + (-0.588 - 5.16i)T \) |
| 7 | \( 1 + (6.41 - 17.3i)T \) |
| good | 5 | \( 1 + (-4.27 + 7.41i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-53.8 + 31.0i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 61.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (13.3 + 23.0i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (58.6 + 33.8i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-45.8 - 26.4i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 55.1iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (134. - 77.4i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (157. - 273. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 210.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 351.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (115. - 200. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (232. - 134. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (9.14 + 15.8i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (72.3 + 41.7i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (64.7 + 112. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 804. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-370. + 213. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (609. - 1.05e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.37e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-386. + 670. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 848. iT - 9.12e5T^{2} \) |
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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
−15.61341861374427362801064543496, −14.80836036887547422571341155760, −13.56143576976020769509725485992, −12.33176031120480011345436565758, −11.06465504305904643224200884189, −9.345550918434592611817204920722, −8.554359323291460432849977401234, −6.14734207628190542332130500300, −5.05133158937831106508375494036, −3.28149325960991510892352155582,
1.85197233113115655136756724682, 3.96520433960802959689091070994, 6.45508656112513757957908812061, 7.02953830936597440439492626947, 9.225881536598052232066186137860, 10.73080274409045154381984140309, 11.95080204213682770220030274399, 12.99918898408203486185425229878, 14.24515993891904135982493516718, 14.54205381327905705190950309187
