| L(s) = 1 | + 3-s + 2.85·5-s + 4.23·7-s + 9-s + 1.76·13-s + 2.85·15-s − 4.61·17-s + 6.09·19-s + 4.23·21-s − 4.23·23-s + 3.14·25-s + 27-s + 4.47·29-s + 8.61·31-s + 12.0·35-s − 8.23·37-s + 1.76·39-s − 0.527·41-s + 0.527·43-s + 2.85·45-s − 1.38·47-s + 10.9·49-s − 4.61·51-s − 13.5·53-s + 6.09·57-s − 8.85·59-s + 0.381·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.27·5-s + 1.60·7-s + 0.333·9-s + 0.489·13-s + 0.736·15-s − 1.12·17-s + 1.39·19-s + 0.924·21-s − 0.883·23-s + 0.629·25-s + 0.192·27-s + 0.830·29-s + 1.54·31-s + 2.04·35-s − 1.35·37-s + 0.282·39-s − 0.0824·41-s + 0.0804·43-s + 0.425·45-s − 0.201·47-s + 1.56·49-s − 0.646·51-s − 1.86·53-s + 0.806·57-s − 1.15·59-s + 0.0489·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.259129964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.259129964\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2.85T + 5T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 + 4.61T + 17T^{2} \) |
| 19 | \( 1 - 6.09T + 19T^{2} \) |
| 23 | \( 1 + 4.23T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 8.61T + 31T^{2} \) |
| 37 | \( 1 + 8.23T + 37T^{2} \) |
| 41 | \( 1 + 0.527T + 41T^{2} \) |
| 43 | \( 1 - 0.527T + 43T^{2} \) |
| 47 | \( 1 + 1.38T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 + 8.85T + 59T^{2} \) |
| 61 | \( 1 - 0.381T + 61T^{2} \) |
| 67 | \( 1 - 6.85T + 67T^{2} \) |
| 71 | \( 1 - 3.61T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 - 9.76T + 79T^{2} \) |
| 83 | \( 1 - 6.52T + 83T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 - 6.09T + 97T^{2} \) |
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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
−8.181182132872386575625230609059, −7.58852183466220987938246228417, −6.59299333708480298433216367895, −6.03253519832929230463042143898, −5.00716835657789161052398499596, −4.73923028091202161496117583029, −3.60818146771917266348490086330, −2.56111781369464597263741278414, −1.85844511111894555186141197849, −1.20503737985211782268765787417,
1.20503737985211782268765787417, 1.85844511111894555186141197849, 2.56111781369464597263741278414, 3.60818146771917266348490086330, 4.73923028091202161496117583029, 5.00716835657789161052398499596, 6.03253519832929230463042143898, 6.59299333708480298433216367895, 7.58852183466220987938246228417, 8.181182132872386575625230609059
