| L(s) = 1 | − 3-s + 9-s + 4·11-s − 2·13-s + 2·17-s + 8·19-s − 4·23-s − 27-s − 6·29-s − 4·33-s − 2·37-s + 2·39-s − 6·41-s − 4·43-s + 12·47-s − 7·49-s − 2·51-s + 6·53-s − 8·57-s + 12·59-s + 14·61-s + 12·67-s + 4·69-s − 2·73-s − 8·79-s + 81-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.485·17-s + 1.83·19-s − 0.834·23-s − 0.192·27-s − 1.11·29-s − 0.696·33-s − 0.328·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 1.75·47-s − 49-s − 0.280·51-s + 0.824·53-s − 1.05·57-s + 1.56·59-s + 1.79·61-s + 1.46·67-s + 0.481·69-s − 0.234·73-s − 0.900·79-s + 1/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.559654199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.559654199\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−9.107407990308486386378414573649, −8.133314117345812981838547690415, −7.25161785738870399619902449036, −6.76108755080622473818108847113, −5.67332768203163504810842963921, −5.23547397312422720656933696083, −4.07386566009613074586340066208, −3.38812013095656350596845988673, −1.98118336189287355394901064221, −0.852520270261560834076094235297,
0.852520270261560834076094235297, 1.98118336189287355394901064221, 3.38812013095656350596845988673, 4.07386566009613074586340066208, 5.23547397312422720656933696083, 5.67332768203163504810842963921, 6.76108755080622473818108847113, 7.25161785738870399619902449036, 8.133314117345812981838547690415, 9.107407990308486386378414573649
