| L(s) = 1 | + 4·2-s + 16·4-s − 54·5-s + 64·8-s − 216·10-s − 216·11-s − 998·13-s + 256·16-s + 1.30e3·17-s − 884·19-s − 864·20-s − 864·22-s + 2.26e3·23-s − 209·25-s − 3.99e3·26-s + 1.48e3·29-s − 8.36e3·31-s + 1.02e3·32-s + 5.20e3·34-s − 4.71e3·37-s − 3.53e3·38-s − 3.45e3·40-s − 9.78e3·41-s + 1.94e4·43-s − 3.45e3·44-s + 9.07e3·46-s + 2.22e4·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.965·5-s + 0.353·8-s − 0.683·10-s − 0.538·11-s − 1.63·13-s + 1/4·16-s + 1.09·17-s − 0.561·19-s − 0.482·20-s − 0.380·22-s + 0.893·23-s − 0.0668·25-s − 1.15·26-s + 0.327·29-s − 1.56·31-s + 0.176·32-s + 0.772·34-s − 0.566·37-s − 0.397·38-s − 0.341·40-s − 0.909·41-s + 1.60·43-s − 0.269·44-s + 0.632·46-s + 1.46·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.027939288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.027939288\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 54 T + p^{5} T^{2} \) |
| 11 | \( 1 + 216 T + p^{5} T^{2} \) |
| 13 | \( 1 + 998 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1302 T + p^{5} T^{2} \) |
| 19 | \( 1 + 884 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2268 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1482 T + p^{5} T^{2} \) |
| 31 | \( 1 + 8360 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4714 T + p^{5} T^{2} \) |
| 41 | \( 1 + 9786 T + p^{5} T^{2} \) |
| 43 | \( 1 - 452 p T + p^{5} T^{2} \) |
| 47 | \( 1 - 22200 T + p^{5} T^{2} \) |
| 53 | \( 1 + 26790 T + p^{5} T^{2} \) |
| 59 | \( 1 - 28092 T + p^{5} T^{2} \) |
| 61 | \( 1 - 38866 T + p^{5} T^{2} \) |
| 67 | \( 1 - 23948 T + p^{5} T^{2} \) |
| 71 | \( 1 - 20628 T + p^{5} T^{2} \) |
| 73 | \( 1 + 290 T + p^{5} T^{2} \) |
| 79 | \( 1 + 99544 T + p^{5} T^{2} \) |
| 83 | \( 1 - 19308 T + p^{5} T^{2} \) |
| 89 | \( 1 - 36390 T + p^{5} T^{2} \) |
| 97 | \( 1 - 79078 T + p^{5} T^{2} \) |
| 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
−9.481673445293607525968744034897, −8.329073044194372055585367037436, −7.45178414625982777562523428247, −7.07255066061484694759657086969, −5.65238473017955470153430238735, −4.97375153927564712717965884008, −4.03247901963468113240627206993, −3.12057138598792914481258413149, −2.12904257120314563746148647551, −0.54752015922779193572616104679,
0.54752015922779193572616104679, 2.12904257120314563746148647551, 3.12057138598792914481258413149, 4.03247901963468113240627206993, 4.97375153927564712717965884008, 5.65238473017955470153430238735, 7.07255066061484694759657086969, 7.45178414625982777562523428247, 8.329073044194372055585367037436, 9.481673445293607525968744034897
