L(s) = 1 | + 2·2-s + 2·4-s − 5-s − 2·10-s − 11-s − 3·13-s − 4·16-s − 3·17-s − 6·19-s − 2·20-s − 2·22-s + 4·23-s + 25-s − 6·26-s + 29-s − 6·31-s − 8·32-s − 6·34-s − 12·38-s + 6·41-s − 6·43-s − 2·44-s + 8·46-s − 9·47-s + 2·50-s − 6·52-s + 10·53-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 0.632·10-s − 0.301·11-s − 0.832·13-s − 16-s − 0.727·17-s − 1.37·19-s − 0.447·20-s − 0.426·22-s + 0.834·23-s + 1/5·25-s − 1.17·26-s + 0.185·29-s − 1.07·31-s − 1.41·32-s − 1.02·34-s − 1.94·38-s + 0.937·41-s − 0.914·43-s − 0.301·44-s + 1.17·46-s − 1.31·47-s + 0.282·50-s − 0.832·52-s + 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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
−8.687547197615304460449731233256, −7.67498052105787514083614275616, −6.86443729079700164782083564306, −6.22556914735544177770859299371, −5.21052441685335331437171683372, −4.64183738354361936328487143401, −3.90569825188449594842439463254, −2.95540115752479584920936053969, −2.09497145721286844219096031032, 0,
2.09497145721286844219096031032, 2.95540115752479584920936053969, 3.90569825188449594842439463254, 4.64183738354361936328487143401, 5.21052441685335331437171683372, 6.22556914735544177770859299371, 6.86443729079700164782083564306, 7.67498052105787514083614275616, 8.687547197615304460449731233256