L(s) = 1 | − 3-s − 2·9-s + 6·11-s + 3·13-s + 17-s + 7·19-s + 8·23-s + 5·27-s − 5·29-s − 5·31-s − 6·33-s + 8·37-s − 3·39-s + 4·43-s − 3·47-s − 7·49-s − 51-s + 9·53-s − 7·57-s − 5·59-s − 3·61-s + 2·67-s − 8·69-s + 15·71-s − 11·73-s − 8·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 2/3·9-s + 1.80·11-s + 0.832·13-s + 0.242·17-s + 1.60·19-s + 1.66·23-s + 0.962·27-s − 0.928·29-s − 0.898·31-s − 1.04·33-s + 1.31·37-s − 0.480·39-s + 0.609·43-s − 0.437·47-s − 49-s − 0.140·51-s + 1.23·53-s − 0.927·57-s − 0.650·59-s − 0.384·61-s + 0.244·67-s − 0.963·69-s + 1.78·71-s − 1.28·73-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.030785949\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.030785949\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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
−7.926601570360659905393686576299, −7.11033404411954517136920129013, −6.56703116243100281481974387791, −5.78194421338694940250400840342, −5.35747152531346205554458017033, −4.36625093855695586860216711865, −3.55460953572539867577454990542, −2.93953932873213555178552008607, −1.49166526297850565867500440545, −0.842399540651489755749403345325,
0.842399540651489755749403345325, 1.49166526297850565867500440545, 2.93953932873213555178552008607, 3.55460953572539867577454990542, 4.36625093855695586860216711865, 5.35747152531346205554458017033, 5.78194421338694940250400840342, 6.56703116243100281481974387791, 7.11033404411954517136920129013, 7.926601570360659905393686576299