| L(s) = 1 | + 3.24·3-s − 2.27·5-s − 0.587·7-s + 7.50·9-s + 0.562·11-s + 1.76·13-s − 7.36·15-s − 3.97·17-s + 5.97·19-s − 1.90·21-s − 0.0297·23-s + 0.168·25-s + 14.5·27-s + 2.02·29-s − 2.64·31-s + 1.82·33-s + 1.33·35-s + 6.81·37-s + 5.72·39-s + 8.90·41-s + 5.63·43-s − 17.0·45-s − 8.87·47-s − 6.65·49-s − 12.8·51-s + 11.8·53-s − 1.27·55-s + ⋯ |
| L(s) = 1 | + 1.87·3-s − 1.01·5-s − 0.221·7-s + 2.50·9-s + 0.169·11-s + 0.490·13-s − 1.90·15-s − 0.964·17-s + 1.36·19-s − 0.415·21-s − 0.00620·23-s + 0.0337·25-s + 2.80·27-s + 0.375·29-s − 0.474·31-s + 0.317·33-s + 0.225·35-s + 1.12·37-s + 0.917·39-s + 1.39·41-s + 0.860·43-s − 2.54·45-s − 1.29·47-s − 0.950·49-s − 1.80·51-s + 1.62·53-s − 0.172·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.814518970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.814518970\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 547 | \( 1 - T \) |
| good | 3 | \( 1 - 3.24T + 3T^{2} \) |
| 5 | \( 1 + 2.27T + 5T^{2} \) |
| 7 | \( 1 + 0.587T + 7T^{2} \) |
| 11 | \( 1 - 0.562T + 11T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 + 3.97T + 17T^{2} \) |
| 19 | \( 1 - 5.97T + 19T^{2} \) |
| 23 | \( 1 + 0.0297T + 23T^{2} \) |
| 29 | \( 1 - 2.02T + 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 - 6.81T + 37T^{2} \) |
| 41 | \( 1 - 8.90T + 41T^{2} \) |
| 43 | \( 1 - 5.63T + 43T^{2} \) |
| 47 | \( 1 + 8.87T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 7.50T + 61T^{2} \) |
| 67 | \( 1 + 3.77T + 67T^{2} \) |
| 71 | \( 1 + 7.23T + 71T^{2} \) |
| 73 | \( 1 - 4.75T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 6.66T + 83T^{2} \) |
| 89 | \( 1 + 4.60T + 89T^{2} \) |
| 97 | \( 1 - 0.593T + 97T^{2} \) |
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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
−7.82771094041144167265606649834, −7.34299416770485137442933865781, −6.74905898647777062377798469222, −5.72972843337887414258490930015, −4.47348725251098414091467543761, −4.13856194903758154732423424770, −3.35727242391667508432281646737, −2.85884943642250878569518294143, −1.95231004566548590391577616983, −0.886070747832566101278270930829,
0.886070747832566101278270930829, 1.95231004566548590391577616983, 2.85884943642250878569518294143, 3.35727242391667508432281646737, 4.13856194903758154732423424770, 4.47348725251098414091467543761, 5.72972843337887414258490930015, 6.74905898647777062377798469222, 7.34299416770485137442933865781, 7.82771094041144167265606649834
