| L(s) = 1 | + (0.151 + 0.119i)2-s + (0.849 − 1.19i)3-s + (−0.462 − 1.90i)4-s + (−2.42 + 0.467i)5-s + (0.270 − 0.0795i)6-s + (−0.112 − 2.64i)7-s + (0.317 − 0.694i)8-s + (0.279 + 0.807i)9-s + (−0.422 − 0.218i)10-s + (3.72 − 2.92i)11-s + (−2.66 − 1.06i)12-s + (−2.71 + 1.74i)13-s + (0.297 − 0.413i)14-s + (−1.50 + 3.28i)15-s + (−3.35 + 1.73i)16-s + (4.22 + 4.03i)17-s + ⋯ |
| L(s) = 1 | + (0.107 + 0.0842i)2-s + (0.490 − 0.688i)3-s + (−0.231 − 0.953i)4-s + (−1.08 + 0.208i)5-s + (0.110 − 0.0324i)6-s + (−0.0423 − 0.999i)7-s + (0.112 − 0.245i)8-s + (0.0932 + 0.269i)9-s + (−0.133 − 0.0689i)10-s + (1.12 − 0.882i)11-s + (−0.770 − 0.308i)12-s + (−0.751 + 0.483i)13-s + (0.0796 − 0.110i)14-s + (−0.387 + 0.849i)15-s + (−0.839 + 0.432i)16-s + (1.02 + 0.978i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.853883 - 0.770358i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.853883 - 0.770358i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (0.112 + 2.64i)T \) |
| 23 | \( 1 + (-4.59 + 1.38i)T \) |
| good | 2 | \( 1 + (-0.151 - 0.119i)T + (0.471 + 1.94i)T^{2} \) |
| 3 | \( 1 + (-0.849 + 1.19i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (2.42 - 0.467i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-3.72 + 2.92i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (2.71 - 1.74i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-4.22 - 4.03i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.35 + 2.24i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-1.35 + 0.397i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (3.67 - 0.350i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-2.92 - 8.45i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-1.98 - 2.28i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (4.17 + 9.14i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (4.76 - 8.25i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0633 + 1.32i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (8.42 + 4.34i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-1.10 - 1.55i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (-11.7 + 4.68i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-0.367 + 2.55i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.326 - 1.34i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-0.0510 - 1.07i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (2.91 - 3.36i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-2.71 - 0.259i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (7.46 + 8.61i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−12.81759150586406444805258229492, −11.56638776373682272052895101354, −10.75269486136604930515808218643, −9.601583566627388381888688098154, −8.328366646382893246924668514854, −7.33433536129473348783188346579, −6.50204513645970584341322911893, −4.76518531345881772824212785539, −3.51756853197452380576449566304, −1.20493635840761307594879012463,
3.01901060427755260735337722702, 3.94943063015503355160217437320, 5.06059390321708692353387427515, 7.13037443222611937264441824779, 8.013181759993135025146194885515, 9.149090746805245172507031351725, 9.659468765580305606887607623788, 11.56551515464035575867401117293, 12.12817069671186855784245950174, 12.69943369235231790260984198951
