| L(s) = 1 | + (−0.907 − 0.761i)2-s + (0.509 + 1.65i)3-s + (−0.103 − 0.588i)4-s + (−1.20 − 3.32i)5-s + (0.797 − 1.88i)6-s + (0.600 − 1.04i)7-s + (−1.53 + 2.66i)8-s + (−2.48 + 1.68i)9-s + (−1.43 + 3.93i)10-s − 3.37i·11-s + (0.921 − 0.471i)12-s + (1.06 − 2.92i)13-s + (−1.33 + 0.486i)14-s + (4.88 − 3.69i)15-s + (2.30 − 0.837i)16-s + (−1.91 − 5.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.641 − 0.538i)2-s + (0.294 + 0.955i)3-s + (−0.0518 − 0.294i)4-s + (−0.540 − 1.48i)5-s + (0.325 − 0.771i)6-s + (0.227 − 0.393i)7-s + (−0.543 + 0.941i)8-s + (−0.826 + 0.562i)9-s + (−0.453 + 1.24i)10-s − 1.01i·11-s + (0.265 − 0.136i)12-s + (0.295 − 0.810i)13-s + (−0.357 + 0.130i)14-s + (1.26 − 0.954i)15-s + (0.575 − 0.209i)16-s + (−0.463 − 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.446371 - 0.583021i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.446371 - 0.583021i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.509 - 1.65i)T \) |
| 19 | \( 1 + (-3.51 - 2.57i)T \) |
| good | 2 | \( 1 + (0.907 + 0.761i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (1.20 + 3.32i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.600 + 1.04i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 3.37iT - 11T^{2} \) |
| 13 | \( 1 + (-1.06 + 2.92i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.91 + 5.24i)T + (-13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-2.71 + 0.479i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.75 - 9.93i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 - 0.427iT - 31T^{2} \) |
| 37 | \( 1 - 0.0841iT - 37T^{2} \) |
| 41 | \( 1 + (1.55 + 1.30i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.0600 - 0.340i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.15 + 1.26i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (4.91 - 4.12i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-0.479 + 2.71i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.68 - 1.70i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (4.40 + 5.25i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-10.6 - 8.94i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-2.30 + 13.0i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (5.23 + 14.3i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-9.41 - 5.43i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.172 + 0.976i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (2.27 - 2.71i)T + (-16.8 - 95.5i)T^{2} \) |
| 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
−12.16535799090382865708410121603, −11.19233292896659173920612936224, −10.49341944881524843787152308001, −9.246881787889943961793972993962, −8.798704652896941384625621432548, −7.88423215658436029650622072864, −5.50182540291232594483824618590, −4.81863748914292214632504121745, −3.25009320401758096458039573191, −0.854407424032179166530072191532,
2.43807473500303177502030308082, 3.84744174477944293881614676412, 6.31117554980177475478495396935, 6.99563481853313714030130355287, 7.72234610289217912244281683929, 8.666512081195970839426785436283, 9.813970511008913837184098609407, 11.26910730721708126867770562357, 11.97541570094268394037994782808, 13.03319090747087097381278669515
