| L(s) = 1 | + (0.371 + 2.10i)2-s + (0.871 − 1.49i)3-s + (−2.41 + 0.877i)4-s + (2.18 − 0.795i)5-s + (3.47 + 1.27i)6-s + (1.27 − 2.32i)7-s + (−0.605 − 1.04i)8-s + (−1.48 − 2.60i)9-s + (2.48 + 4.30i)10-s + (−1.37 − 0.500i)11-s + (−0.787 + 4.37i)12-s + (−5.79 + 2.11i)13-s + (5.35 + 1.81i)14-s + (0.713 − 3.96i)15-s + (−1.94 + 1.63i)16-s + (3.15 + 5.46i)17-s + ⋯ |
| L(s) = 1 | + (0.262 + 1.48i)2-s + (0.503 − 0.864i)3-s + (−1.20 + 0.438i)4-s + (0.977 − 0.355i)5-s + (1.41 + 0.521i)6-s + (0.480 − 0.876i)7-s + (−0.214 − 0.370i)8-s + (−0.493 − 0.869i)9-s + (0.786 + 1.36i)10-s + (−0.414 − 0.150i)11-s + (−0.227 + 1.26i)12-s + (−1.60 + 0.585i)13-s + (1.43 + 0.485i)14-s + (0.184 − 1.02i)15-s + (−0.487 + 0.408i)16-s + (0.765 + 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50169 + 0.706198i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50169 + 0.706198i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.871 + 1.49i)T \) |
| 7 | \( 1 + (-1.27 + 2.32i)T \) |
| good | 2 | \( 1 + (-0.371 - 2.10i)T + (-1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (-2.18 + 0.795i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (1.37 + 0.500i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (5.79 - 2.11i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.15 - 5.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.94 - 5.10i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.628 + 3.56i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.39 + 0.870i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-5.82 + 2.11i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 - 0.375T + 37T^{2} \) |
| 41 | \( 1 + (-6.70 + 2.44i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.123 - 0.699i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.13 - 2.59i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (0.517 - 0.895i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.58 + 2.16i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (5.48 + 1.99i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.04 + 5.94i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-3.69 + 6.40i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 6.77T + 73T^{2} \) |
| 79 | \( 1 + (-0.0335 - 0.190i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-9.88 - 3.59i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (1.46 - 2.53i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.44 + 13.8i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−13.05222155448903217408437074137, −12.26939015007272594253589360218, −10.51524344916013112422599975734, −9.370615770197342219520021227792, −8.115529301738162186694510810096, −7.62605713124735581240759170445, −6.48625155079208820341923167292, −5.65505371772727090533960208160, −4.31627672752550442507215401760, −1.97399951284516209702024646610,
2.42348144249333564029454052305, 2.79897417364096843463214174957, 4.70614738641245896928893804174, 5.40519422258103993172103889644, 7.50253231432315533766677284899, 9.088410287143931737524531614676, 9.727323388538780533210555207146, 10.38295220621776263680543404885, 11.37802034600539515538864403095, 12.26140708932910305187070076526
