| L(s) = 1 | + (−1.73 + i)2-s + (−4.17 + 3.09i)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 + (7.89 − 25.8i)9-s + (14.8 + 8.55i)10-s + (−53.8 − 31.0i)11-s + (2.35 + 20.6i)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 + (−0.803 + 0.594i)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 + (0.292 − 0.956i)9-s + (0.468 + 0.270i)10-s + (−1.47 − 0.852i)11-s + (0.0565 + 0.496i)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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.143466 - 0.229439i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.143466 - 0.229439i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (4.17 - 3.09i)T \) |
| 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
−15.85589822027428514693837949345, −14.13551625778436389183945278631, −12.66049936898668967646713124962, −11.21335762921504670409582259321, −10.31440841251495776586887841955, −9.003941420664072757730604550869, −7.47877831339954383621927553114, −5.88152120473588487766555649746, −4.28256869631074807568177319796, −0.28038133432348573849515631207,
2.58873764215124798777500597205, 5.47196291967181265122752230585, 7.09185261419439840659339362748, 8.220344551994546573576270942905, 10.19492343611679360655162139043, 10.97023405673379165755044096152, 12.40092457713519809602630188608, 12.94602446582281319872810523253, 15.14143744650377864297092181054, 15.90280181791300280533735760474
