L(s) = 1 | − 2·5-s + 4·7-s − 2·11-s + 2·13-s + 2·17-s − 2·19-s + 4·23-s − 25-s + 6·29-s − 8·35-s + 10·37-s + 6·41-s − 6·43-s − 8·47-s + 9·49-s + 6·53-s + 4·55-s + 14·59-s + 2·61-s − 4·65-s − 10·67-s + 12·71-s + 14·73-s − 8·77-s + 8·79-s − 6·83-s − 4·85-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 1.51·7-s − 0.603·11-s + 0.554·13-s + 0.485·17-s − 0.458·19-s + 0.834·23-s − 1/5·25-s + 1.11·29-s − 1.35·35-s + 1.64·37-s + 0.937·41-s − 0.914·43-s − 1.16·47-s + 9/7·49-s + 0.824·53-s + 0.539·55-s + 1.82·59-s + 0.256·61-s − 0.496·65-s − 1.22·67-s + 1.42·71-s + 1.63·73-s − 0.911·77-s + 0.900·79-s − 0.658·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.647878569\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.647878569\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−9.878602275413527189048173292252, −8.675368536957604275874860747441, −8.088659191412945349881968287052, −7.63242500478736670765641851926, −6.52256885432452015704757780960, −5.34050418818343429420051907291, −4.62995407509459210959268189869, −3.73824894896581631226090928700, −2.44235807183272353930685105042, −1.02724387350145696940295704376,
1.02724387350145696940295704376, 2.44235807183272353930685105042, 3.73824894896581631226090928700, 4.62995407509459210959268189869, 5.34050418818343429420051907291, 6.52256885432452015704757780960, 7.63242500478736670765641851926, 8.088659191412945349881968287052, 8.675368536957604275874860747441, 9.878602275413527189048173292252