L(s) = 1 | + 1.56·2-s + 1.44·4-s + 1.80·7-s + 0.695·8-s + 2.81·14-s − 0.356·16-s − 1.94·17-s − 0.445·19-s + 25-s + 2.60·28-s − 1.56·29-s − 1.24·31-s − 1.25·32-s − 3.04·34-s − 1.80·37-s − 0.695·38-s + 0.867·41-s + 1.24·43-s + 1.94·47-s + 2.24·49-s + 1.56·50-s + 1.94·53-s + 1.25·56-s − 2.44·58-s − 1.94·62-s − 1.60·64-s − 2.81·68-s + ⋯ |
L(s) = 1 | + 1.56·2-s + 1.44·4-s + 1.80·7-s + 0.695·8-s + 2.81·14-s − 0.356·16-s − 1.94·17-s − 0.445·19-s + 25-s + 2.60·28-s − 1.56·29-s − 1.24·31-s − 1.25·32-s − 3.04·34-s − 1.80·37-s − 0.695·38-s + 0.867·41-s + 1.24·43-s + 1.94·47-s + 2.24·49-s + 1.56·50-s + 1.94·53-s + 1.25·56-s − 2.44·58-s − 1.94·62-s − 1.60·64-s − 2.81·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.934699740\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.934699740\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 223 | \( 1 + T \) |
good | 2 | \( 1 - 1.56T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 1.80T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.94T + T^{2} \) |
| 19 | \( 1 + 0.445T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.56T + T^{2} \) |
| 31 | \( 1 + 1.24T + T^{2} \) |
| 37 | \( 1 + 1.80T + T^{2} \) |
| 41 | \( 1 - 0.867T + T^{2} \) |
| 43 | \( 1 - 1.24T + T^{2} \) |
| 47 | \( 1 - 1.94T + T^{2} \) |
| 53 | \( 1 - 1.94T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.24T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.56T + T^{2} \) |
| 89 | \( 1 + 0.867T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−8.886023016965008785485268466189, −8.797756422027551721730625645650, −7.38735058158521401708952054324, −6.99311905115584951607569673683, −5.74071295641751482739198358916, −5.29165818315901761263778054581, −4.32991469020337801119180432547, −4.03053973870126994144975349656, −2.52020698230579533138124979793, −1.83815540745740199642200787035,
1.83815540745740199642200787035, 2.52020698230579533138124979793, 4.03053973870126994144975349656, 4.32991469020337801119180432547, 5.29165818315901761263778054581, 5.74071295641751482739198358916, 6.99311905115584951607569673683, 7.38735058158521401708952054324, 8.797756422027551721730625645650, 8.886023016965008785485268466189