L(s) = 1 | + (0.0159 + 0.00280i)2-s + (−0.402 − 1.68i)3-s + (−1.87 − 0.683i)4-s + (3.75 + 1.36i)5-s + (−0.00168 − 0.0279i)6-s + (0.157 − 2.64i)7-s + (−0.0559 − 0.0323i)8-s + (−2.67 + 1.35i)9-s + (0.0558 + 0.0322i)10-s + (−1.76 − 4.84i)11-s + (−0.395 + 3.44i)12-s + (−0.220 + 0.606i)13-s + (0.00992 − 0.0415i)14-s + (0.789 − 6.86i)15-s + (3.06 + 2.57i)16-s + (1.69 − 2.93i)17-s + ⋯ |
L(s) = 1 | + (0.0112 + 0.00198i)2-s + (−0.232 − 0.972i)3-s + (−0.939 − 0.341i)4-s + (1.67 + 0.610i)5-s + (−0.000686 − 0.0114i)6-s + (0.0596 − 0.998i)7-s + (−0.0197 − 0.0114i)8-s + (−0.891 + 0.452i)9-s + (0.0176 + 0.0102i)10-s + (−0.531 − 1.46i)11-s + (−0.114 + 0.993i)12-s + (−0.0612 + 0.168i)13-s + (0.00265 − 0.0111i)14-s + (0.203 − 1.77i)15-s + (0.765 + 0.642i)16-s + (0.410 − 0.711i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.109 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.819776 - 0.734128i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.819776 - 0.734128i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.402 + 1.68i)T \) |
| 7 | \( 1 + (-0.157 + 2.64i)T \) |
good | 2 | \( 1 + (-0.0159 - 0.00280i)T + (1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (-3.75 - 1.36i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (1.76 + 4.84i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (0.220 - 0.606i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.69 + 2.93i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.328 + 0.189i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.993 - 0.175i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.97 - 5.41i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (2.20 - 6.06i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 - 6.16T + 37T^{2} \) |
| 41 | \( 1 + (-0.266 - 0.0969i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.45 - 8.22i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.91 + 2.51i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-1.71 + 0.992i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.62 - 2.20i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (2.75 + 7.57i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.63 - 9.29i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-8.57 + 4.94i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 7.08iT - 73T^{2} \) |
| 79 | \( 1 + (0.222 - 1.26i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (8.07 - 2.93i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (2.67 + 4.63i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.39 + 1.30i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.80140143075419997159395623802, −11.10173306295755237287431233188, −10.43037153223938641515371769865, −9.431974436269578145191131820065, −8.302638149423456721018087390275, −6.98034922645417734797632387048, −5.96533956360794596200441718414, −5.18543226776160894802675501363, −3.01805418033197642339351454531, −1.16496615970913042726666512638,
2.38884878189798592950316704726, 4.35862928247509445702655408326, 5.30564693104672539223613299969, 5.94686951288342220573150704820, 8.116380671761360686388880360893, 9.206881308836855062576847911115, 9.654715627262385585513047628787, 10.37025421764173889659585917555, 12.12053532604212563468919005072, 12.75444254577986543582946911651