| L(s) = 1 | + 2.86·3-s − 0.462·5-s + 7-s + 5.18·9-s + 2·11-s + 6.64·13-s − 1.32·15-s − 17-s − 4.64·19-s + 2.86·21-s + 5.72·23-s − 4.78·25-s + 6.24·27-s + 0.796·29-s − 0.184·31-s + 5.72·33-s − 0.462·35-s − 7.44·37-s + 19.0·39-s − 6.71·41-s − 5.50·43-s − 2.39·45-s + 5.72·47-s + 49-s − 2.86·51-s − 4.33·53-s − 0.925·55-s + ⋯ |
| L(s) = 1 | + 1.65·3-s − 0.206·5-s + 0.377·7-s + 1.72·9-s + 0.603·11-s + 1.84·13-s − 0.341·15-s − 0.242·17-s − 1.06·19-s + 0.624·21-s + 1.19·23-s − 0.957·25-s + 1.20·27-s + 0.147·29-s − 0.0330·31-s + 0.996·33-s − 0.0781·35-s − 1.22·37-s + 3.04·39-s − 1.04·41-s − 0.839·43-s − 0.357·45-s + 0.834·47-s + 0.142·49-s − 0.400·51-s − 0.595·53-s − 0.124·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.503143784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.503143784\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2.86T + 3T^{2} \) |
| 5 | \( 1 + 0.462T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 6.64T + 13T^{2} \) |
| 19 | \( 1 + 4.64T + 19T^{2} \) |
| 23 | \( 1 - 5.72T + 23T^{2} \) |
| 29 | \( 1 - 0.796T + 29T^{2} \) |
| 31 | \( 1 + 0.184T + 31T^{2} \) |
| 37 | \( 1 + 7.44T + 37T^{2} \) |
| 41 | \( 1 + 6.71T + 41T^{2} \) |
| 43 | \( 1 + 5.50T + 43T^{2} \) |
| 47 | \( 1 - 5.72T + 47T^{2} \) |
| 53 | \( 1 + 4.33T + 53T^{2} \) |
| 59 | \( 1 + 0.646T + 59T^{2} \) |
| 61 | \( 1 - 4.86T + 61T^{2} \) |
| 67 | \( 1 - 15.7T + 67T^{2} \) |
| 71 | \( 1 - 0.556T + 71T^{2} \) |
| 73 | \( 1 + 4.73T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + 6.36T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.798406694818457864514145900815, −8.640770688533828938352060755689, −7.940446176997107900318309469779, −6.96631861836683142157249959152, −6.26441397115944840908464459761, −4.93785022072233047760233828547, −3.78503736753561187335789771494, −3.56147095837315352697262306257, −2.25075522896510227282001964734, −1.36758085670400387803488523093,
1.36758085670400387803488523093, 2.25075522896510227282001964734, 3.56147095837315352697262306257, 3.78503736753561187335789771494, 4.93785022072233047760233828547, 6.26441397115944840908464459761, 6.96631861836683142157249959152, 7.940446176997107900318309469779, 8.640770688533828938352060755689, 8.798406694818457864514145900815
