| L(s) = 1 | − 1.72·2-s − 3-s + 0.973·4-s + 1.98·5-s + 1.72·6-s + 2.67·7-s + 1.77·8-s + 9-s − 3.41·10-s + 3.40·11-s − 0.973·12-s + 0.577·13-s − 4.60·14-s − 1.98·15-s − 4.99·16-s + 0.893·17-s − 1.72·18-s − 2.02·19-s + 1.92·20-s − 2.67·21-s − 5.86·22-s − 0.0407·23-s − 1.77·24-s − 1.07·25-s − 0.996·26-s − 27-s + 2.59·28-s + ⋯ |
| L(s) = 1 | − 1.21·2-s − 0.577·3-s + 0.486·4-s + 0.885·5-s + 0.703·6-s + 1.00·7-s + 0.626·8-s + 0.333·9-s − 1.08·10-s + 1.02·11-s − 0.280·12-s + 0.160·13-s − 1.23·14-s − 0.511·15-s − 1.24·16-s + 0.216·17-s − 0.406·18-s − 0.464·19-s + 0.430·20-s − 0.583·21-s − 1.25·22-s − 0.00850·23-s − 0.361·24-s − 0.215·25-s − 0.195·26-s − 0.192·27-s + 0.491·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8974995581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8974995581\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 223 | \( 1 - T \) |
| good | 2 | \( 1 + 1.72T + 2T^{2} \) |
| 5 | \( 1 - 1.98T + 5T^{2} \) |
| 7 | \( 1 - 2.67T + 7T^{2} \) |
| 11 | \( 1 - 3.40T + 11T^{2} \) |
| 13 | \( 1 - 0.577T + 13T^{2} \) |
| 17 | \( 1 - 0.893T + 17T^{2} \) |
| 19 | \( 1 + 2.02T + 19T^{2} \) |
| 23 | \( 1 + 0.0407T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 + 0.750T + 31T^{2} \) |
| 37 | \( 1 + 2.32T + 37T^{2} \) |
| 41 | \( 1 + 0.161T + 41T^{2} \) |
| 43 | \( 1 - 1.74T + 43T^{2} \) |
| 47 | \( 1 - 8.72T + 47T^{2} \) |
| 53 | \( 1 + 6.59T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 3.01T + 61T^{2} \) |
| 67 | \( 1 - 6.47T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 2.20T + 73T^{2} \) |
| 79 | \( 1 + 0.611T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 9.02T + 89T^{2} \) |
| 97 | \( 1 + 0.169T + 97T^{2} \) |
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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
−10.35664914261751766222184418718, −9.667579119590231599283080065224, −8.855370640676598071652684570824, −8.126392077226111386525008851741, −7.08767959784046320315076953604, −6.20543726535842507938511587220, −5.13522314737895467469935100923, −4.14419956558645784743425456176, −2.04111552109992997515798115945, −1.10326869546823870611010537859,
1.10326869546823870611010537859, 2.04111552109992997515798115945, 4.14419956558645784743425456176, 5.13522314737895467469935100923, 6.20543726535842507938511587220, 7.08767959784046320315076953604, 8.126392077226111386525008851741, 8.855370640676598071652684570824, 9.667579119590231599283080065224, 10.35664914261751766222184418718
