| L(s) = 1 | + 3-s − 7-s + 9-s − 4·11-s − 2·13-s + 2·17-s − 21-s + 4·23-s + 27-s − 2·29-s − 4·31-s − 4·33-s + 10·37-s − 2·39-s + 6·41-s + 4·43-s + 8·47-s + 49-s + 2·51-s + 6·53-s − 12·59-s + 6·61-s − 63-s + 12·67-s + 4·69-s − 6·73-s + 4·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.218·21-s + 0.834·23-s + 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.696·33-s + 1.64·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 1.56·59-s + 0.768·61-s − 0.125·63-s + 1.46·67-s + 0.481·69-s − 0.702·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.995316244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.995316244\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.337629870602224006970901002284, −7.56560190620263711280706077345, −7.28252381072658386850916181311, −6.15062224371690357792511062892, −5.43131744477888829716404797346, −4.64198226047062275753775598540, −3.71503119010477163969535739726, −2.82048213703798745440354330591, −2.24105615412980341112289329168, −0.76221131510495772572789627370,
0.76221131510495772572789627370, 2.24105615412980341112289329168, 2.82048213703798745440354330591, 3.71503119010477163969535739726, 4.64198226047062275753775598540, 5.43131744477888829716404797346, 6.15062224371690357792511062892, 7.28252381072658386850916181311, 7.56560190620263711280706077345, 8.337629870602224006970901002284
