L(s) = 1 | + 3-s + 7-s + 9-s + 4·13-s − 2·19-s + 21-s + 23-s + 27-s + 6·29-s − 2·31-s + 10·37-s + 4·39-s + 6·41-s − 4·43-s + 6·47-s + 49-s + 6·53-s − 2·57-s − 12·59-s − 10·61-s + 63-s − 4·67-s + 69-s − 14·73-s − 8·79-s + 81-s + 6·83-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.10·13-s − 0.458·19-s + 0.218·21-s + 0.208·23-s + 0.192·27-s + 1.11·29-s − 0.359·31-s + 1.64·37-s + 0.640·39-s + 0.937·41-s − 0.609·43-s + 0.875·47-s + 1/7·49-s + 0.824·53-s − 0.264·57-s − 1.56·59-s − 1.28·61-s + 0.125·63-s − 0.488·67-s + 0.120·69-s − 1.63·73-s − 0.900·79-s + 1/9·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.38183197810119, −12.90788972624299, −12.50576399797795, −11.86052847285955, −11.51058845482002, −10.77469231760151, −10.68396402076490, −10.05751810303823, −9.428327160689193, −8.971887928525142, −8.651736907146218, −8.095418788027977, −7.670246926199164, −7.215067801798198, −6.511842619366175, −6.069511836596866, −5.684788127934715, −4.828482205902838, −4.399257138973059, −3.985581089945416, −3.296330329848740, −2.740252811894362, −2.270509713401279, −1.342542234607226, −1.117044921041818, 0,
1.117044921041818, 1.342542234607226, 2.270509713401279, 2.740252811894362, 3.296330329848740, 3.985581089945416, 4.399257138973059, 4.828482205902838, 5.684788127934715, 6.069511836596866, 6.511842619366175, 7.215067801798198, 7.670246926199164, 8.095418788027977, 8.651736907146218, 8.971887928525142, 9.428327160689193, 10.05751810303823, 10.68396402076490, 10.77469231760151, 11.51058845482002, 11.86052847285955, 12.50576399797795, 12.90788972624299, 13.38183197810119