| L(s) = 1 | + (0.809 − 0.587i)2-s + (0.913 − 0.406i)3-s + (0.309 − 0.951i)4-s + (−1.64 − 2.85i)5-s + (0.5 − 0.866i)6-s + (2.64 + 2.94i)7-s + (−0.309 − 0.951i)8-s + (0.669 − 0.743i)9-s + (−3.00 − 1.34i)10-s + (−4.22 + 0.897i)11-s + (−0.104 − 0.994i)12-s + (0.129 − 1.22i)13-s + (3.86 + 0.822i)14-s + (−2.66 − 1.93i)15-s + (−0.809 − 0.587i)16-s + (5.03 + 1.07i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.527 − 0.234i)3-s + (0.154 − 0.475i)4-s + (−0.736 − 1.27i)5-s + (0.204 − 0.353i)6-s + (1.00 + 1.11i)7-s + (−0.109 − 0.336i)8-s + (0.223 − 0.247i)9-s + (−0.951 − 0.423i)10-s + (−1.27 + 0.270i)11-s + (−0.0301 − 0.287i)12-s + (0.0358 − 0.340i)13-s + (1.03 + 0.219i)14-s + (−0.688 − 0.500i)15-s + (−0.202 − 0.146i)16-s + (1.22 + 0.259i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42794 - 0.940154i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42794 - 0.940154i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (-0.913 + 0.406i)T \) |
| 31 | \( 1 + (1.50 - 5.36i)T \) |
| good | 5 | \( 1 + (1.64 + 2.85i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.64 - 2.94i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (4.22 - 0.897i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.129 + 1.22i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-5.03 - 1.07i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.129 - 1.22i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.85 - 5.70i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (0.901 - 0.655i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-0.511 + 0.885i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (11.3 + 5.04i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.194 + 1.84i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-1.11 - 0.809i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (4.58 - 5.09i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-7.30 + 3.25i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + (3.99 + 6.91i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-10.3 + 11.4i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (-13.8 + 2.95i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (12.8 + 2.73i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-5.14 - 2.29i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (0.0194 - 0.0598i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.76 + 17.7i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−12.33502758046538924864707757324, −11.91729750489578248402866923046, −10.62017935486438530195983277676, −9.261857471925296163794387729913, −8.271960160147146231666970923944, −7.66894489027552043447345249257, −5.44985043108868720833883150188, −4.96725668000211701759342568051, −3.35255711526274979952868604394, −1.69326271673494629585947511231,
2.80773708483590754187880986360, 3.91646254240753951493073559698, 5.07043253583315232658703546779, 6.80135647085235763840408351627, 7.67114132026161837512171103897, 8.187229959300663309778135666309, 10.11698446045317957988627293208, 10.87913214506653841079917217497, 11.63119139795000705500751385477, 13.10010347930045309697063767837
