L(s) = 1 | − 2·5-s + 4·7-s + 4·11-s − 2·13-s + 6·17-s + 4·19-s − 25-s − 2·29-s − 4·31-s − 8·35-s − 2·37-s − 2·41-s − 4·43-s + 8·47-s + 9·49-s − 10·53-s − 8·55-s − 4·59-s + 6·61-s + 4·65-s − 4·67-s − 16·71-s − 6·73-s + 16·77-s − 4·79-s + 12·83-s − 12·85-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 1.51·7-s + 1.20·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s − 1/5·25-s − 0.371·29-s − 0.718·31-s − 1.35·35-s − 0.328·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s − 1.37·53-s − 1.07·55-s − 0.520·59-s + 0.768·61-s + 0.496·65-s − 0.488·67-s − 1.89·71-s − 0.702·73-s + 1.82·77-s − 0.450·79-s + 1.31·83-s − 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.352639401\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.352639401\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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
−11.78751958221467998185382370821, −11.17481638991951468650430169053, −9.937852275620133476013119550490, −8.839451555498636571243071250626, −7.81343717072417778826484889358, −7.28654230418818125978122396710, −5.64125009969007529682624024748, −4.57894092460876565195206012945, −3.48999177685615809985045793307, −1.46933061420707130494872098274,
1.46933061420707130494872098274, 3.48999177685615809985045793307, 4.57894092460876565195206012945, 5.64125009969007529682624024748, 7.28654230418818125978122396710, 7.81343717072417778826484889358, 8.839451555498636571243071250626, 9.937852275620133476013119550490, 11.17481638991951468650430169053, 11.78751958221467998185382370821