| L(s) = 1 | + 3-s − 2·7-s + 9-s + 13-s − 4·17-s + 2·19-s − 2·21-s − 6·23-s + 27-s + 4·31-s + 2·37-s + 39-s − 6·41-s − 4·43-s − 4·47-s − 3·49-s − 4·51-s + 10·53-s + 2·57-s + 8·59-s + 6·61-s − 2·63-s + 8·67-s − 6·69-s − 16·73-s + 81-s + 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.970·17-s + 0.458·19-s − 0.436·21-s − 1.25·23-s + 0.192·27-s + 0.718·31-s + 0.328·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s − 0.583·47-s − 3/7·49-s − 0.560·51-s + 1.37·53-s + 0.264·57-s + 1.04·59-s + 0.768·61-s − 0.251·63-s + 0.977·67-s − 0.722·69-s − 1.87·73-s + 1/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.939023899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.939023899\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−14.98445882698899, −14.66468020682852, −13.92148979864019, −13.42913794864004, −13.19251287178842, −12.55006180563305, −11.76998884771917, −11.58870104683481, −10.68416957037354, −10.07892528547145, −9.801322287072763, −9.132970197697340, −8.515028197744948, −8.162673565254554, −7.415302075067210, −6.724338865427682, −6.415992082215791, −5.639290066547683, −4.942999749667855, −4.145763679079168, −3.694627016201551, −2.965622598303937, −2.326152714193210, −1.586041746557500, −0.5010679882464749,
0.5010679882464749, 1.586041746557500, 2.326152714193210, 2.965622598303937, 3.694627016201551, 4.145763679079168, 4.942999749667855, 5.639290066547683, 6.415992082215791, 6.724338865427682, 7.415302075067210, 8.162673565254554, 8.515028197744948, 9.132970197697340, 9.801322287072763, 10.07892528547145, 10.68416957037354, 11.58870104683481, 11.76998884771917, 12.55006180563305, 13.19251287178842, 13.42913794864004, 13.92148979864019, 14.66468020682852, 14.98445882698899
