| L(s) = 1 | − 2·5-s + 7-s − 4·11-s − 4·13-s + 8·17-s + 2·19-s − 23-s − 25-s + 2·29-s − 6·31-s − 2·35-s + 10·37-s − 6·41-s + 8·43-s − 6·47-s + 49-s + 2·53-s + 8·55-s − 10·61-s + 8·65-s − 8·67-s + 12·71-s + 6·73-s − 4·77-s + 2·83-s − 16·85-s − 12·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 1.20·11-s − 1.10·13-s + 1.94·17-s + 0.458·19-s − 0.208·23-s − 1/5·25-s + 0.371·29-s − 1.07·31-s − 0.338·35-s + 1.64·37-s − 0.937·41-s + 1.21·43-s − 0.875·47-s + 1/7·49-s + 0.274·53-s + 1.07·55-s − 1.28·61-s + 0.992·65-s − 0.977·67-s + 1.42·71-s + 0.702·73-s − 0.455·77-s + 0.219·83-s − 1.73·85-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
| 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
−14.07044347734954, −13.71366704567893, −12.93242889338174, −12.52278932150370, −12.15054815303238, −11.67781000408589, −11.20247212792616, −10.59319431584490, −10.16295894269922, −9.626939846166066, −9.242623099395244, −8.299110729660464, −7.876297087696069, −7.678595313427348, −7.297362138039080, −6.494841052035425, −5.622399099797511, −5.414017263652105, −4.791403918533788, −4.226135253872640, −3.538122632020381, −2.988784255117605, −2.454399854474897, −1.604726984896727, −0.7604112020528307, 0,
0.7604112020528307, 1.604726984896727, 2.454399854474897, 2.988784255117605, 3.538122632020381, 4.226135253872640, 4.791403918533788, 5.414017263652105, 5.622399099797511, 6.494841052035425, 7.297362138039080, 7.678595313427348, 7.876297087696069, 8.299110729660464, 9.242623099395244, 9.626939846166066, 10.16295894269922, 10.59319431584490, 11.20247212792616, 11.67781000408589, 12.15054815303238, 12.52278932150370, 12.93242889338174, 13.71366704567893, 14.07044347734954
