| L(s) = 1 | + 3-s − 7-s − 2·9-s − 11-s − 6·13-s + 7·17-s + 19-s − 21-s − 8·23-s − 5·27-s − 6·29-s + 4·31-s − 33-s − 8·37-s − 6·39-s − 5·41-s − 6·47-s + 49-s + 7·51-s − 4·53-s + 57-s − 4·59-s + 6·61-s + 2·63-s + 5·67-s − 8·69-s + 14·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.301·11-s − 1.66·13-s + 1.69·17-s + 0.229·19-s − 0.218·21-s − 1.66·23-s − 0.962·27-s − 1.11·29-s + 0.718·31-s − 0.174·33-s − 1.31·37-s − 0.960·39-s − 0.780·41-s − 0.875·47-s + 1/7·49-s + 0.980·51-s − 0.549·53-s + 0.132·57-s − 0.520·59-s + 0.768·61-s + 0.251·63-s + 0.610·67-s − 0.963·69-s + 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.337230639065845945510700602168, −8.107566967722018383534406259839, −7.82450670059276365611628469847, −6.82404020705380367185989578980, −5.69795078942090655453125020752, −5.09521348715540545761725111624, −3.74075632776133593282338137080, −2.97081882189724539571709885137, −1.98299141225506520923628653836, 0,
1.98299141225506520923628653836, 2.97081882189724539571709885137, 3.74075632776133593282338137080, 5.09521348715540545761725111624, 5.69795078942090655453125020752, 6.82404020705380367185989578980, 7.82450670059276365611628469847, 8.107566967722018383534406259839, 9.337230639065845945510700602168
