| L(s) = 1 | + 3-s + 2·5-s − 2·7-s + 9-s − 3·13-s + 2·15-s − 17-s + 5·19-s − 2·21-s − 25-s + 27-s − 6·29-s − 4·35-s + 6·37-s − 3·39-s + 4·41-s + 11·43-s + 2·45-s + 9·47-s − 3·49-s − 51-s + 6·53-s + 5·57-s + 4·59-s + 4·61-s − 2·63-s − 6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.832·13-s + 0.516·15-s − 0.242·17-s + 1.14·19-s − 0.436·21-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.676·35-s + 0.986·37-s − 0.480·39-s + 0.624·41-s + 1.67·43-s + 0.298·45-s + 1.31·47-s − 3/7·49-s − 0.140·51-s + 0.824·53-s + 0.662·57-s + 0.520·59-s + 0.512·61-s − 0.251·63-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.434833658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.434833658\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−12.61122175629499, −12.10519587992400, −11.53277238360812, −11.05420774911698, −10.57880013061333, −9.935257772508886, −9.673303768131853, −9.382425641266545, −9.069580925331255, −8.385234956218142, −7.776910644471248, −7.438858721266991, −7.021970685144422, −6.433324557994198, −5.967975334948263, −5.421722912427437, −5.196002446553495, −4.281209258995417, −3.968308122695185, −3.330591740064142, −2.784545625611457, −2.225949059088307, −2.038631455812430, −1.002031521135670, −0.5752676914469272,
0.5752676914469272, 1.002031521135670, 2.038631455812430, 2.225949059088307, 2.784545625611457, 3.330591740064142, 3.968308122695185, 4.281209258995417, 5.196002446553495, 5.421722912427437, 5.967975334948263, 6.433324557994198, 7.021970685144422, 7.438858721266991, 7.776910644471248, 8.385234956218142, 9.069580925331255, 9.382425641266545, 9.673303768131853, 9.935257772508886, 10.57880013061333, 11.05420774911698, 11.53277238360812, 12.10519587992400, 12.61122175629499
