| L(s) = 1 | + 2·7-s + 3·11-s − 2·13-s − 3·17-s + 19-s − 6·23-s − 5·25-s − 6·29-s − 4·31-s + 4·37-s + 9·41-s + 43-s − 6·47-s − 3·49-s − 12·53-s − 3·59-s − 8·61-s − 5·67-s − 12·71-s + 11·73-s + 6·77-s − 4·79-s − 12·83-s + 6·89-s − 4·91-s + 5·97-s + 14·103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.904·11-s − 0.554·13-s − 0.727·17-s + 0.229·19-s − 1.25·23-s − 25-s − 1.11·29-s − 0.718·31-s + 0.657·37-s + 1.40·41-s + 0.152·43-s − 0.875·47-s − 3/7·49-s − 1.64·53-s − 0.390·59-s − 1.02·61-s − 0.610·67-s − 1.42·71-s + 1.28·73-s + 0.683·77-s − 0.450·79-s − 1.31·83-s + 0.635·89-s − 0.419·91-s + 0.507·97-s + 1.37·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−7.72025132764032720699159371966, −7.36864117697809528975094371324, −6.25296606336542427270750931270, −5.84014438525125889787299722655, −4.75060394033924964563703545448, −4.24345858681212892536929813216, −3.38361884307164439809700256262, −2.16985049193462021634699913439, −1.54192685329519068608929989781, 0,
1.54192685329519068608929989781, 2.16985049193462021634699913439, 3.38361884307164439809700256262, 4.24345858681212892536929813216, 4.75060394033924964563703545448, 5.84014438525125889787299722655, 6.25296606336542427270750931270, 7.36864117697809528975094371324, 7.72025132764032720699159371966
