| L(s) = 1 | + 5-s − 7-s − 2·11-s + 13-s + 6·17-s + 6·19-s − 23-s − 4·25-s − 5·29-s − 8·31-s − 35-s − 3·37-s + 9·41-s − 3·43-s − 9·47-s + 49-s + 6·53-s − 2·55-s + 8·59-s − 2·61-s + 65-s − 4·67-s + 10·71-s + 2·73-s + 2·77-s − 4·79-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.603·11-s + 0.277·13-s + 1.45·17-s + 1.37·19-s − 0.208·23-s − 4/5·25-s − 0.928·29-s − 1.43·31-s − 0.169·35-s − 0.493·37-s + 1.40·41-s − 0.457·43-s − 1.31·47-s + 1/7·49-s + 0.824·53-s − 0.269·55-s + 1.04·59-s − 0.256·61-s + 0.124·65-s − 0.488·67-s + 1.18·71-s + 0.234·73-s + 0.227·77-s − 0.450·79-s − 0.439·83-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 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
| 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
−14.18197854667652, −13.52036153209509, −13.12995599595493, −12.68173703314970, −12.16447568605393, −11.59073619250528, −11.18556346896324, −10.56154146225121, −9.958084493659100, −9.706630247181483, −9.275485998571078, −8.576882860669770, −7.983156167145881, −7.402318988846934, −7.254965695552319, −6.302683005772223, −5.796490369409648, −5.401682103841488, −5.039842820524471, −3.989553790046823, −3.550515257163759, −3.071794321428808, −2.287699262206306, −1.642082658808555, −0.9306013534199568, 0,
0.9306013534199568, 1.642082658808555, 2.287699262206306, 3.071794321428808, 3.550515257163759, 3.989553790046823, 5.039842820524471, 5.401682103841488, 5.796490369409648, 6.302683005772223, 7.254965695552319, 7.402318988846934, 7.983156167145881, 8.576882860669770, 9.275485998571078, 9.706630247181483, 9.958084493659100, 10.56154146225121, 11.18556346896324, 11.59073619250528, 12.16447568605393, 12.68173703314970, 13.12995599595493, 13.52036153209509, 14.18197854667652
