| L(s) = 1 | − 0.801·2-s + 3-s − 1.35·4-s + 3.04·5-s − 0.801·6-s − 7-s + 2.69·8-s + 9-s − 2.44·10-s − 2.15·11-s − 1.35·12-s + 0.801·14-s + 3.04·15-s + 0.554·16-s + 0.801·17-s − 0.801·18-s + 1.39·19-s − 4.13·20-s − 21-s + 1.73·22-s + 3.02·23-s + 2.69·24-s + 4.29·25-s + 27-s + 1.35·28-s + 2.47·29-s − 2.44·30-s + ⋯ |
| L(s) = 1 | − 0.567·2-s + 0.577·3-s − 0.678·4-s + 1.36·5-s − 0.327·6-s − 0.377·7-s + 0.951·8-s + 0.333·9-s − 0.773·10-s − 0.650·11-s − 0.391·12-s + 0.214·14-s + 0.787·15-s + 0.138·16-s + 0.194·17-s − 0.189·18-s + 0.320·19-s − 0.925·20-s − 0.218·21-s + 0.369·22-s + 0.631·23-s + 0.549·24-s + 0.859·25-s + 0.192·27-s + 0.256·28-s + 0.459·29-s − 0.446·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.815056917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.815056917\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.801T + 2T^{2} \) |
| 5 | \( 1 - 3.04T + 5T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 17 | \( 1 - 0.801T + 17T^{2} \) |
| 19 | \( 1 - 1.39T + 19T^{2} \) |
| 23 | \( 1 - 3.02T + 23T^{2} \) |
| 29 | \( 1 - 2.47T + 29T^{2} \) |
| 31 | \( 1 - 6.26T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 5.20T + 41T^{2} \) |
| 43 | \( 1 + 8.63T + 43T^{2} \) |
| 47 | \( 1 - 9.15T + 47T^{2} \) |
| 53 | \( 1 - 1.21T + 53T^{2} \) |
| 59 | \( 1 - 8.63T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 1.33T + 79T^{2} \) |
| 83 | \( 1 + 8.92T + 83T^{2} \) |
| 89 | \( 1 - 2.50T + 89T^{2} \) |
| 97 | \( 1 + 2.32T + 97T^{2} \) |
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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
−8.683210577952345949148328751497, −8.075029228935567754422599097606, −7.18598533651721862534514678408, −6.43190873497321378869570384499, −5.40464712626009580959877902530, −4.96062957038486844773241207431, −3.80850926568553136330869419433, −2.83813314837544373665263030647, −1.95212265669539883742762117786, −0.871377879668851869455126476560,
0.871377879668851869455126476560, 1.95212265669539883742762117786, 2.83813314837544373665263030647, 3.80850926568553136330869419433, 4.96062957038486844773241207431, 5.40464712626009580959877902530, 6.43190873497321378869570384499, 7.18598533651721862534514678408, 8.075029228935567754422599097606, 8.683210577952345949148328751497
