| L(s) = 1 | + (1.59 + 1.25i)2-s + (0.924 − 1.29i)3-s + (0.494 + 2.03i)4-s + (−1.87 + 0.362i)5-s + (3.09 − 0.908i)6-s + (2.37 − 1.16i)7-s + (−0.0831 + 0.182i)8-s + (0.150 + 0.436i)9-s + (−3.44 − 1.77i)10-s + (−4.40 + 3.46i)11-s + (3.10 + 1.24i)12-s + (−2.49 + 1.60i)13-s + (5.23 + 1.11i)14-s + (−1.26 + 2.77i)15-s + (3.37 − 1.73i)16-s + (−4.95 − 4.72i)17-s + ⋯ |
| L(s) = 1 | + (1.12 + 0.884i)2-s + (0.533 − 0.749i)3-s + (0.247 + 1.01i)4-s + (−0.840 + 0.161i)5-s + (1.26 − 0.371i)6-s + (0.897 − 0.440i)7-s + (−0.0294 + 0.0644i)8-s + (0.0503 + 0.145i)9-s + (−1.08 − 0.561i)10-s + (−1.32 + 1.04i)11-s + (0.896 + 0.358i)12-s + (−0.691 + 0.444i)13-s + (1.40 + 0.299i)14-s + (−0.326 + 0.716i)15-s + (0.842 − 0.434i)16-s + (−1.20 − 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.94995 + 0.536090i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.94995 + 0.536090i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.37 + 1.16i)T \) |
| 23 | \( 1 + (4.56 - 1.47i)T \) |
| good | 2 | \( 1 + (-1.59 - 1.25i)T + (0.471 + 1.94i)T^{2} \) |
| 3 | \( 1 + (-0.924 + 1.29i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (1.87 - 0.362i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (4.40 - 3.46i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (2.49 - 1.60i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (4.95 + 4.72i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.93 + 2.79i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-6.07 + 1.78i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (0.110 - 0.0105i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-0.464 - 1.34i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (0.915 + 1.05i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.80 - 8.32i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-0.0620 + 0.107i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.316 + 6.64i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (-0.641 - 0.330i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-4.51 - 6.33i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (-1.84 + 0.739i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-1.07 + 7.44i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-1.87 - 7.72i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (0.517 + 10.8i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (10.3 - 11.9i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-3.18 - 0.303i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (4.16 + 4.81i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−13.36217039940706139605596564422, −12.30956562818343898749739330476, −11.36437646680255529180530810354, −9.950606132505928570870046196863, −8.105036982362929723587821986753, −7.44868282183981776037505896089, −6.97113396253733115357077217904, −5.01195614344369094950482890749, −4.43659046737149218927403547762, −2.50126438756050865431159127170,
2.52750436069861749993254323725, 3.73354439404052245951806766069, 4.63053505082394743923602153535, 5.71003388705221244121209602270, 7.997042401861925610008219413969, 8.517734353987754550201107612473, 10.24706189801582450910205600308, 10.92022930600958853352221687712, 11.95610094405362788229132336160, 12.61250984425087712758732263057
