| 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 | \( 1 + (7.39 - 1.30i)T + (91.1 - 33.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.75444254577986543582946911651, −12.12053532604212563468919005072, −10.37025421764173889659585917555, −9.654715627262385585513047628787, −9.206881308836855062576847911115, −8.116380671761360686388880360893, −5.94686951288342220573150704820, −5.30564693104672539223613299969, −4.35862928247509445702655408326, −2.38884878189798592950316704726,
1.16496615970913042726666512638, 3.01805418033197642339351454531, 5.18543226776160894802675501363, 5.96533956360794596200441718414, 6.98034922645417734797632387048, 8.302638149423456721018087390275, 9.431974436269578145191131820065, 10.43037153223938641515371769865, 11.10173306295755237287431233188, 12.80140143075419997159395623802
