| L(s) = 1 | − 3·3-s + 18·7-s + 9·9-s − 30·11-s + 38·13-s − 70·17-s − 12·19-s − 54·21-s − 72·23-s − 27·27-s − 64·29-s + 312·31-s + 90·33-s + 138·37-s − 114·39-s − 374·41-s + 468·43-s − 132·47-s − 19·49-s + 210·51-s + 446·53-s + 36·57-s − 510·59-s + 754·61-s + 162·63-s − 384·67-s + 216·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.971·7-s + 1/3·9-s − 0.822·11-s + 0.810·13-s − 0.998·17-s − 0.144·19-s − 0.561·21-s − 0.652·23-s − 0.192·27-s − 0.409·29-s + 1.80·31-s + 0.474·33-s + 0.613·37-s − 0.468·39-s − 1.42·41-s + 1.65·43-s − 0.409·47-s − 0.0553·49-s + 0.576·51-s + 1.15·53-s + 0.0836·57-s − 1.12·59-s + 1.58·61-s + 0.323·63-s − 0.700·67-s + 0.376·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.749854636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.749854636\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 70 T + p^{3} T^{2} \) |
| 19 | \( 1 + 12 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 64 T + p^{3} T^{2} \) |
| 31 | \( 1 - 312 T + p^{3} T^{2} \) |
| 37 | \( 1 - 138 T + p^{3} T^{2} \) |
| 41 | \( 1 + 374 T + p^{3} T^{2} \) |
| 43 | \( 1 - 468 T + p^{3} T^{2} \) |
| 47 | \( 1 + 132 T + p^{3} T^{2} \) |
| 53 | \( 1 - 446 T + p^{3} T^{2} \) |
| 59 | \( 1 + 510 T + p^{3} T^{2} \) |
| 61 | \( 1 - 754 T + p^{3} T^{2} \) |
| 67 | \( 1 + 384 T + p^{3} T^{2} \) |
| 71 | \( 1 - 924 T + p^{3} T^{2} \) |
| 73 | \( 1 + 340 T + p^{3} T^{2} \) |
| 79 | \( 1 + 72 T + p^{3} T^{2} \) |
| 83 | \( 1 - 156 T + p^{3} T^{2} \) |
| 89 | \( 1 + 290 T + p^{3} T^{2} \) |
| 97 | \( 1 + 376 T + p^{3} T^{2} \) |
| 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
−8.354880184444482826116177339420, −8.045268704319522856427584235287, −7.00728400430366366443632994027, −6.23092159194688695340896848582, −5.46310438917271066114574173105, −4.66826773736196948087283634493, −4.00637894518212286076243255154, −2.66165998487663736412369453963, −1.71374460497457036734076453514, −0.61278389418410103819441849497,
0.61278389418410103819441849497, 1.71374460497457036734076453514, 2.66165998487663736412369453963, 4.00637894518212286076243255154, 4.66826773736196948087283634493, 5.46310438917271066114574173105, 6.23092159194688695340896848582, 7.00728400430366366443632994027, 8.045268704319522856427584235287, 8.354880184444482826116177339420
