L(s) = 1 | − 3-s − 2·7-s − 2·9-s + 5·11-s + 6·13-s + 3·17-s + 19-s + 2·21-s + 4·23-s + 5·27-s − 6·29-s + 8·31-s − 5·33-s + 2·37-s − 6·39-s − 7·41-s − 4·43-s + 2·47-s − 3·49-s − 3·51-s + 4·53-s − 57-s + 4·59-s − 10·61-s + 4·63-s − 3·67-s − 4·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s − 2/3·9-s + 1.50·11-s + 1.66·13-s + 0.727·17-s + 0.229·19-s + 0.436·21-s + 0.834·23-s + 0.962·27-s − 1.11·29-s + 1.43·31-s − 0.870·33-s + 0.328·37-s − 0.960·39-s − 1.09·41-s − 0.609·43-s + 0.291·47-s − 3/7·49-s − 0.420·51-s + 0.549·53-s − 0.132·57-s + 0.520·59-s − 1.28·61-s + 0.503·63-s − 0.366·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.692308748\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.692308748\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 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
−8.168104256734507287237994635293, −7.07438500900839454557697914439, −6.44617809115758155935863264212, −6.04821310831038940018225108179, −5.37047454998105679638677226909, −4.35120026526139650574710042933, −3.50037192862885049253424443630, −3.06628266143783683534364797410, −1.55229331021857410177719605116, −0.74991314266939098790189474290,
0.74991314266939098790189474290, 1.55229331021857410177719605116, 3.06628266143783683534364797410, 3.50037192862885049253424443630, 4.35120026526139650574710042933, 5.37047454998105679638677226909, 6.04821310831038940018225108179, 6.44617809115758155935863264212, 7.07438500900839454557697914439, 8.168104256734507287237994635293