| L(s) = 1 | + 3-s − 2·4-s + 5-s − 7-s + 9-s − 2·11-s − 2·12-s − 5·13-s + 15-s + 4·16-s + 2·19-s − 2·20-s − 21-s + 23-s + 25-s + 27-s + 2·28-s − 8·29-s − 31-s − 2·33-s − 35-s − 2·36-s + 3·37-s − 5·39-s + 7·41-s + 4·44-s + 45-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.577·12-s − 1.38·13-s + 0.258·15-s + 16-s + 0.458·19-s − 0.447·20-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 1.48·29-s − 0.179·31-s − 0.348·33-s − 0.169·35-s − 1/3·36-s + 0.493·37-s − 0.800·39-s + 1.09·41-s + 0.603·44-s + 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.36239904861023, −14.64651543845644, −14.26063226497850, −13.97335029610309, −13.11352995893142, −12.84393451856283, −12.60610212465596, −11.71820341088437, −11.02735296217078, −10.35341771824640, −9.777855245333608, −9.406485147103156, −9.187019561983146, −8.207023713355286, −7.869006039480311, −7.269107010952537, −6.625989666936906, −5.696041124617278, −5.274782367995728, −4.732565153582354, −3.973336956429519, −3.369615262441099, −2.611096824962537, −2.041695542874978, −0.9252390693074910, 0,
0.9252390693074910, 2.041695542874978, 2.611096824962537, 3.369615262441099, 3.973336956429519, 4.732565153582354, 5.274782367995728, 5.696041124617278, 6.625989666936906, 7.269107010952537, 7.869006039480311, 8.207023713355286, 9.187019561983146, 9.406485147103156, 9.777855245333608, 10.35341771824640, 11.02735296217078, 11.71820341088437, 12.60610212465596, 12.84393451856283, 13.11352995893142, 13.97335029610309, 14.26063226497850, 14.64651543845644, 15.36239904861023
