| L(s) = 1 | + (−0.154 − 0.121i)2-s + (−1.95 + 2.74i)3-s + (−0.462 − 1.90i)4-s + (−0.731 + 0.140i)5-s + (0.632 − 0.185i)6-s + (−2.43 − 1.02i)7-s + (−0.322 + 0.706i)8-s + (−2.72 − 7.86i)9-s + (0.129 + 0.0668i)10-s + (2.06 − 1.62i)11-s + (6.12 + 2.45i)12-s + (−2.05 + 1.32i)13-s + (0.251 + 0.453i)14-s + (1.04 − 2.27i)15-s + (−3.35 + 1.72i)16-s + (−3.79 − 3.62i)17-s + ⋯ |
| L(s) = 1 | + (−0.108 − 0.0856i)2-s + (−1.12 + 1.58i)3-s + (−0.231 − 0.953i)4-s + (−0.327 + 0.0630i)5-s + (0.258 − 0.0758i)6-s + (−0.922 − 0.386i)7-s + (−0.114 + 0.249i)8-s + (−0.906 − 2.62i)9-s + (0.0410 + 0.0211i)10-s + (0.622 − 0.489i)11-s + (1.76 + 0.708i)12-s + (−0.570 + 0.366i)13-s + (0.0672 + 0.121i)14-s + (0.268 − 0.588i)15-s + (−0.837 + 0.432i)16-s + (−0.921 − 0.878i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0244191 - 0.0672113i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0244191 - 0.0672113i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.43 + 1.02i)T \) |
| 23 | \( 1 + (1.15 - 4.65i)T \) |
| good | 2 | \( 1 + (0.154 + 0.121i)T + (0.471 + 1.94i)T^{2} \) |
| 3 | \( 1 + (1.95 - 2.74i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (0.731 - 0.140i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-2.06 + 1.62i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (2.05 - 1.32i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (3.79 + 3.62i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (3.00 - 2.86i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-2.27 + 0.667i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.74 + 0.166i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-0.00470 - 0.0135i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (5.80 + 6.69i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.01 - 4.41i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-0.583 + 1.00i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.363 + 7.62i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (2.14 + 1.10i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (5.34 + 7.50i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (7.52 - 3.01i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (0.761 - 5.29i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.0725 - 0.299i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (0.199 + 4.19i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-6.42 + 7.41i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-7.15 - 0.682i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-4.25 - 4.91i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.98847049357098558170815970374, −11.30027664868111772369186925059, −10.39425397972792016502572723152, −9.669548672200470631000209411241, −9.037138852595335559441691684709, −6.67969681198778586898051329899, −5.85761879128541621333618741088, −4.68309095512855639920900840556, −3.70527684987744800012808683891, −0.07655386633773235419482694541,
2.44534767560918987254370790914, 4.51617334597143378459106083589, 6.23614510636128464355039237324, 6.82904426646085853025664152281, 7.86666728530975077330725252182, 8.843483774897807349202017969358, 10.52079377792471386629574098353, 11.81138719502462104393487334563, 12.30164406167139850955277473114, 12.91719960509343601133696912708
