L(s) = 1 | + (−1.22 − 0.707i)2-s + (−0.866 − 1.5i)3-s + (0.999 + 1.73i)4-s + 2.44i·6-s + (−6.98 − 0.440i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (2.76 + 4.79i)11-s + (1.73 − 2.99i)12-s + 3.50·13-s + (8.24 + 5.47i)14-s + (−2.00 + 3.46i)16-s + (−4.06 − 7.04i)17-s + (3.67 − 2.12i)18-s + (15.3 + 8.86i)19-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.288 − 0.5i)3-s + (0.249 + 0.433i)4-s + 0.408i·6-s + (−0.998 − 0.0628i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.251 + 0.435i)11-s + (0.144 − 0.249i)12-s + 0.269·13-s + (0.588 + 0.391i)14-s + (−0.125 + 0.216i)16-s + (−0.239 − 0.414i)17-s + (0.204 − 0.117i)18-s + (0.808 + 0.466i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7202812882\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7202812882\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 + 0.707i)T \) |
| 3 | \( 1 + (0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (6.98 + 0.440i)T \) |
good | 11 | \( 1 + (-2.76 - 4.79i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 3.50T + 169T^{2} \) |
| 17 | \( 1 + (4.06 + 7.04i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-15.3 - 8.86i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-10.3 - 5.96i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 8.52T + 841T^{2} \) |
| 31 | \( 1 + (6.68 - 3.85i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (24.2 + 14.0i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 3.14iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 43.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-9.35 + 16.2i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-2.44 + 1.40i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-26.1 + 15.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (41.0 + 23.6i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (5.70 - 3.29i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 97.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + (30.5 + 52.9i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-61.5 + 106. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 89.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-102. - 59.0i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 65.1T + 9.40e3T^{2} \) |
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
−9.412548231002377432200252986694, −8.797971300068463812460239392254, −7.64860479683143668879407288827, −7.02676392654388169850210081343, −6.25431359196958996437814940555, −5.21004417167934061652018367965, −3.82778170373435261819769643900, −2.88174942263948410495413994135, −1.62933922293769788452726382371, −0.35202897984745051059788810677,
1.00384436264255268218561504514, 2.76424762494258445813258372860, 3.75355964794678948001179326887, 4.97055053319433064686539396400, 5.96581600254203888502423950201, 6.56717195679841542997795309132, 7.47249558205569347590185538765, 8.611587500241515830775210991136, 9.172896367861092372317828359307, 9.918549855744535914069491861669