L(s) = 1 | − 3-s − 4·7-s + 9-s + 4·11-s + 4·13-s − 2·17-s − 4·19-s + 4·21-s − 8·23-s − 5·25-s − 27-s − 8·29-s − 4·31-s − 4·33-s − 4·37-s − 4·39-s + 6·41-s + 4·43-s − 8·47-s + 9·49-s + 2·51-s − 8·53-s + 4·57-s − 12·59-s + 12·61-s − 4·63-s + 12·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 1.10·13-s − 0.485·17-s − 0.917·19-s + 0.872·21-s − 1.66·23-s − 25-s − 0.192·27-s − 1.48·29-s − 0.718·31-s − 0.696·33-s − 0.657·37-s − 0.640·39-s + 0.937·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.280·51-s − 1.09·53-s + 0.529·57-s − 1.56·59-s + 1.53·61-s − 0.503·63-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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
−9.744599246919104634775567385973, −9.313485660447772294885319845996, −8.247152815550235840332214408381, −6.99687341654975900890511495598, −6.23593170660239061738441922167, −5.83902316255569103541752486301, −4.09550440201686426997590498271, −3.62331160644145024955043318233, −1.85805342505545187947666684739, 0,
1.85805342505545187947666684739, 3.62331160644145024955043318233, 4.09550440201686426997590498271, 5.83902316255569103541752486301, 6.23593170660239061738441922167, 6.99687341654975900890511495598, 8.247152815550235840332214408381, 9.313485660447772294885319845996, 9.744599246919104634775567385973