| L(s) = 1 | + 5-s − 2·7-s − 4·13-s + 2·17-s + 2·19-s − 4·23-s + 25-s + 6·29-s − 2·35-s + 10·37-s + 2·41-s + 2·43-s − 3·49-s − 6·53-s − 4·61-s − 4·65-s + 4·67-s + 2·79-s + 2·85-s − 10·89-s + 8·91-s + 2·95-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 1.10·13-s + 0.485·17-s + 0.458·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 0.338·35-s + 1.64·37-s + 0.312·41-s + 0.304·43-s − 3/7·49-s − 0.824·53-s − 0.512·61-s − 0.496·65-s + 0.488·67-s + 0.225·79-s + 0.216·85-s − 1.05·89-s + 0.838·91-s + 0.205·95-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.804993337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.804993337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.63102578078452, −14.02775356187380, −13.86705754379050, −12.98530034635506, −12.66223123044243, −12.21143477543739, −11.62338576263863, −11.06560140910586, −10.29537763070177, −9.891229242381357, −9.591091986590180, −9.046980979777885, −8.223869292039187, −7.759200076466167, −7.199779482286860, −6.507297193710076, −6.088608188988365, −5.494587098072641, −4.794076194843055, −4.283311567245118, −3.423268865117186, −2.810163407938494, −2.316884992921861, −1.369591431472191, −0.4924981658695197,
0.4924981658695197, 1.369591431472191, 2.316884992921861, 2.810163407938494, 3.423268865117186, 4.283311567245118, 4.794076194843055, 5.494587098072641, 6.088608188988365, 6.507297193710076, 7.199779482286860, 7.759200076466167, 8.223869292039187, 9.046980979777885, 9.591091986590180, 9.891229242381357, 10.29537763070177, 11.06560140910586, 11.62338576263863, 12.21143477543739, 12.66223123044243, 12.98530034635506, 13.86705754379050, 14.02775356187380, 14.63102578078452
