| L(s) = 1 | − 3-s − 3·7-s + 9-s + 2·13-s − 2·17-s + 2·19-s + 3·21-s − 3·23-s − 27-s + 8·29-s − 3·31-s − 8·37-s − 2·39-s + 7·41-s − 6·43-s − 11·47-s + 2·49-s + 2·51-s − 2·57-s + 6·59-s − 3·63-s − 10·67-s + 3·69-s − 3·73-s + 8·79-s + 81-s − 8·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.554·13-s − 0.485·17-s + 0.458·19-s + 0.654·21-s − 0.625·23-s − 0.192·27-s + 1.48·29-s − 0.538·31-s − 1.31·37-s − 0.320·39-s + 1.09·41-s − 0.914·43-s − 1.60·47-s + 2/7·49-s + 0.280·51-s − 0.264·57-s + 0.781·59-s − 0.377·63-s − 1.22·67-s + 0.361·69-s − 0.351·73-s + 0.900·79-s + 1/9·81-s − 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6395869546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6395869546\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
| 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
−13.42335035732254, −12.90182039992168, −12.26955310765102, −12.13712881496004, −11.38376261009565, −11.09671225504526, −10.33675795831821, −10.14394981215522, −9.610756967978493, −9.068947946262432, −8.569165248895074, −8.042226483402473, −7.408779939960055, −6.736125299405479, −6.555047219500340, −6.025358893745923, −5.460566931366639, −4.914788554506152, −4.304192710097877, −3.693428927883583, −3.210309848865113, −2.626072707439145, −1.786792143009472, −1.150174632626723, −0.2627131436941634,
0.2627131436941634, 1.150174632626723, 1.786792143009472, 2.626072707439145, 3.210309848865113, 3.693428927883583, 4.304192710097877, 4.914788554506152, 5.460566931366639, 6.025358893745923, 6.555047219500340, 6.736125299405479, 7.408779939960055, 8.042226483402473, 8.569165248895074, 9.068947946262432, 9.610756967978493, 10.14394981215522, 10.33675795831821, 11.09671225504526, 11.38376261009565, 12.13712881496004, 12.26955310765102, 12.90182039992168, 13.42335035732254
