L(s) = 1 | + 5·13-s − 19-s − 5·25-s + 11·31-s + 11·37-s − 13·43-s + 14·61-s + 5·67-s + 17·73-s + 17·79-s + 14·97-s − 13·103-s − 19·109-s + ⋯ |
L(s) = 1 | + 1.38·13-s − 0.229·19-s − 25-s + 1.97·31-s + 1.80·37-s − 1.98·43-s + 1.79·61-s + 0.610·67-s + 1.98·73-s + 1.91·79-s + 1.42·97-s − 1.28·103-s − 1.81·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.824793164\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.824793164\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 17 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−9.371369340475474030335733727421, −8.228115246436835029120734828381, −8.087898060104447287606612070089, −6.68988040532618663572401167961, −6.23769545148704938178746476280, −5.25343964239659451061264663366, −4.23748187348637361897475637498, −3.43898861300086424137501327609, −2.26111515808195169942016527355, −0.965089271400007404454750534601,
0.965089271400007404454750534601, 2.26111515808195169942016527355, 3.43898861300086424137501327609, 4.23748187348637361897475637498, 5.25343964239659451061264663366, 6.23769545148704938178746476280, 6.68988040532618663572401167961, 8.087898060104447287606612070089, 8.228115246436835029120734828381, 9.371369340475474030335733727421