L(s) = 1 | + 40.2i·3-s − 324. i·5-s + 956.·7-s + 569.·9-s + 5.45e3i·11-s + 6.28e3i·13-s + 1.30e4·15-s + 3.45e4·17-s − 1.45e4i·19-s + 3.84e4i·21-s + 2.46e4·23-s − 2.71e4·25-s + 1.10e5i·27-s − 1.71e5i·29-s − 1.11e5·31-s + ⋯ |
L(s) = 1 | + 0.859i·3-s − 1.16i·5-s + 1.05·7-s + 0.260·9-s + 1.23i·11-s + 0.793i·13-s + 0.998·15-s + 1.70·17-s − 0.488i·19-s + 0.906i·21-s + 0.422·23-s − 0.347·25-s + 1.08i·27-s − 1.30i·29-s − 0.673·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.513i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.94119 + 0.536973i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.94119 + 0.536973i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 40.2iT - 2.18e3T^{2} \) |
| 5 | \( 1 + 324. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 956.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.45e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 6.28e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 3.45e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.45e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 2.46e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.71e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 1.11e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.03e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 7.16e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.28e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 1.19e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.04e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 2.25e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 1.55e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 3.16e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 5.38e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.22e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.89e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 4.37e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.84e6T + 8.07e13T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.40519455865784082939914881480, −14.39203259543494557078906210496, −12.77714150327853309273502533641, −11.67262015276301913742300086757, −10.08137739312159876278443291319, −9.061706111576974386139262456565, −7.57580825931589858403040245539, −5.12002923692211082562771860120, −4.30330922035465681614105333630, −1.47233357333067625863462355272,
1.24965527033562442954500987866, 3.17791790466598936175492726342, 5.67346551674896208644101448274, 7.21701212440006962384602812542, 8.183953889847568431944729384834, 10.35064371880875854341313670993, 11.36647871761704734964409800463, 12.71417190014099770460378063826, 14.11774184109986775246027220360, 14.77216062182119151495401459518