L(s) = 1 | − 3-s + 9-s + 11-s − 4·13-s + 2·17-s − 4·19-s − 2·23-s − 27-s + 2·31-s − 33-s − 8·37-s + 4·39-s + 6·41-s + 4·43-s − 2·51-s − 6·53-s + 4·57-s − 4·59-s − 6·61-s + 4·67-s + 2·69-s + 4·73-s − 14·79-s + 81-s + 6·83-s − 2·93-s + 10·97-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.485·17-s − 0.917·19-s − 0.417·23-s − 0.192·27-s + 0.359·31-s − 0.174·33-s − 1.31·37-s + 0.640·39-s + 0.937·41-s + 0.609·43-s − 0.280·51-s − 0.824·53-s + 0.529·57-s − 0.520·59-s − 0.768·61-s + 0.488·67-s + 0.240·69-s + 0.468·73-s − 1.57·79-s + 1/9·81-s + 0.658·83-s − 0.207·93-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−12.57868607118018, −12.49490043652559, −11.95450906022750, −11.56055670741832, −11.00162855555151, −10.57735558640393, −10.14901916454287, −9.753973607404307, −9.210456922721887, −8.804720454524982, −8.181508568434353, −7.662237792180708, −7.331802792322960, −6.728004660749290, −6.306930917919271, −5.841067997925315, −5.314363207491435, −4.767954304105114, −4.398939823718929, −3.828947332328183, −3.212154165030946, −2.578746304500326, −2.003961920761185, −1.443413314204069, −0.6371641105111448, 0,
0.6371641105111448, 1.443413314204069, 2.003961920761185, 2.578746304500326, 3.212154165030946, 3.828947332328183, 4.398939823718929, 4.767954304105114, 5.314363207491435, 5.841067997925315, 6.306930917919271, 6.728004660749290, 7.331802792322960, 7.662237792180708, 8.181508568434353, 8.804720454524982, 9.210456922721887, 9.753973607404307, 10.14901916454287, 10.57735558640393, 11.00162855555151, 11.56055670741832, 11.95450906022750, 12.49490043652559, 12.57868607118018