| L(s) = 1 | + 5-s + 4.77·7-s + 3.77·11-s + 3·13-s − 1.77·17-s − 2.77·19-s − 7.77·23-s + 25-s − 3.77·29-s + 2.22·31-s + 4.77·35-s − 0.772·37-s + 9.54·41-s − 1.77·43-s − 5.77·47-s + 15.7·49-s + 9.54·53-s + 3.77·55-s − 12·59-s + 0.772·61-s + 3·65-s + 14.7·67-s − 13.5·71-s + 4.77·73-s + 18·77-s − 5·79-s − 6·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.80·7-s + 1.13·11-s + 0.832·13-s − 0.429·17-s − 0.635·19-s − 1.62·23-s + 0.200·25-s − 0.700·29-s + 0.400·31-s + 0.806·35-s − 0.126·37-s + 1.49·41-s − 0.270·43-s − 0.841·47-s + 2.25·49-s + 1.31·53-s + 0.508·55-s − 1.56·59-s + 0.0988·61-s + 0.372·65-s + 1.80·67-s − 1.60·71-s + 0.558·73-s + 2.05·77-s − 0.562·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.261914900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.261914900\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 4.77T + 7T^{2} \) |
| 11 | \( 1 - 3.77T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 + 1.77T + 17T^{2} \) |
| 19 | \( 1 + 2.77T + 19T^{2} \) |
| 23 | \( 1 + 7.77T + 23T^{2} \) |
| 29 | \( 1 + 3.77T + 29T^{2} \) |
| 31 | \( 1 - 2.22T + 31T^{2} \) |
| 37 | \( 1 + 0.772T + 37T^{2} \) |
| 41 | \( 1 - 9.54T + 41T^{2} \) |
| 43 | \( 1 + 1.77T + 43T^{2} \) |
| 47 | \( 1 + 5.77T + 47T^{2} \) |
| 53 | \( 1 - 9.54T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 0.772T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 4.77T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−9.881342831571658890458184278689, −8.871396403497008980934232853459, −8.359516381603993741077479895122, −7.49836179084295472784312208454, −6.37892643896390471232880589886, −5.68017052219723964540283539195, −4.53806753810589408016431165344, −3.90977358029582823112183821314, −2.16390211514549087645485351507, −1.36682501051982937142302674576,
1.36682501051982937142302674576, 2.16390211514549087645485351507, 3.90977358029582823112183821314, 4.53806753810589408016431165344, 5.68017052219723964540283539195, 6.37892643896390471232880589886, 7.49836179084295472784312208454, 8.359516381603993741077479895122, 8.871396403497008980934232853459, 9.881342831571658890458184278689
