| L(s) = 1 | + (0.0810 − 0.563i)2-s + (1.06 + 2.33i)3-s + (1.60 + 0.472i)4-s + (−0.715 − 0.825i)5-s + (1.40 − 0.412i)6-s + (−0.841 − 0.540i)7-s + (0.869 − 1.90i)8-s + (−2.36 + 2.72i)9-s + (−0.523 + 0.336i)10-s + (−0.0174 − 0.121i)11-s + (0.612 + 4.26i)12-s + (−4.06 + 2.61i)13-s + (−0.373 + 0.430i)14-s + (1.16 − 2.55i)15-s + (1.81 + 1.16i)16-s + (6.98 − 2.05i)17-s + ⋯ |
| L(s) = 1 | + (0.0573 − 0.398i)2-s + (0.616 + 1.35i)3-s + (0.803 + 0.236i)4-s + (−0.320 − 0.369i)5-s + (0.573 − 0.168i)6-s + (−0.317 − 0.204i)7-s + (0.307 − 0.673i)8-s + (−0.787 + 0.909i)9-s + (−0.165 + 0.106i)10-s + (−0.00527 − 0.0366i)11-s + (0.176 + 1.23i)12-s + (−1.12 + 0.724i)13-s + (−0.0997 + 0.115i)14-s + (0.301 − 0.659i)15-s + (0.453 + 0.291i)16-s + (1.69 − 0.497i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44243 + 0.388921i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44243 + 0.388921i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (1.63 + 4.50i)T \) |
| good | 2 | \( 1 + (-0.0810 + 0.563i)T + (-1.91 - 0.563i)T^{2} \) |
| 3 | \( 1 + (-1.06 - 2.33i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (0.715 + 0.825i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (0.0174 + 0.121i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (4.06 - 2.61i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-6.98 + 2.05i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (6.86 + 2.01i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.186 + 0.0547i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (3.13 - 6.86i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-1.15 + 1.33i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (2.22 + 2.56i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.993 - 2.17i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 0.343T + 47T^{2} \) |
| 53 | \( 1 + (4.70 + 3.02i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (8.56 - 5.50i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-0.198 + 0.434i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.597 - 4.15i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.704 - 4.89i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-15.6 - 4.60i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-3.13 + 2.01i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-7.60 + 8.77i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.0462 - 0.101i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-0.804 - 0.928i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−12.62460244241641257108198684861, −12.01997723075783420615612632612, −10.69095746135679690015170277385, −10.10513664329855811914561862075, −9.103633132213790135704814780876, −7.943339959807015632237246544447, −6.67042162187598593176424460954, −4.82087936801209202897632805760, −3.81164911248125991319951657138, −2.60592392954242020646053156683,
1.91147325230707111322170047210, 3.18129833549011379382392854939, 5.62863219168258718191290221444, 6.60838651703441367398724389651, 7.70537353693718913974253024798, 7.943820844816913386072799941769, 9.743881854936077633566951537672, 10.90264196872656849410066270905, 12.19062245089237499888193699074, 12.62513340888972819858160304621
