L(s) = 1 | + 3-s + 2·5-s − 7-s + 9-s − 2·13-s + 2·15-s + 6·17-s − 4·19-s − 21-s − 4·23-s − 25-s + 27-s + 6·29-s − 8·31-s − 2·35-s − 10·37-s − 2·39-s − 10·41-s + 12·43-s + 2·45-s − 8·47-s + 49-s + 6·51-s + 6·53-s − 4·57-s + 4·59-s − 10·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.917·19-s − 0.218·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.338·35-s − 1.64·37-s − 0.320·39-s − 1.56·41-s + 1.82·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s − 0.529·57-s + 0.520·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.451262370\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.451262370\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.82992732310357645044415807593, −12.06652674974971659339169474440, −10.42205409739275983523104171923, −9.848554569012739739807460123677, −8.820314983809012661425051705328, −7.65994621019403063529610936786, −6.42136658293515809263418102895, −5.25019213638990204722565722077, −3.56921408081238054859871707771, −2.06515692945354055956624655608,
2.06515692945354055956624655608, 3.56921408081238054859871707771, 5.25019213638990204722565722077, 6.42136658293515809263418102895, 7.65994621019403063529610936786, 8.820314983809012661425051705328, 9.848554569012739739807460123677, 10.42205409739275983523104171923, 12.06652674974971659339169474440, 12.82992732310357645044415807593