| L(s) = 1 | + 3-s + 4-s − 2·9-s + 12-s − 4·13-s + 16-s − 25-s − 5·27-s − 2·36-s − 4·39-s − 2·43-s + 48-s + 5·49-s − 4·52-s + 16·61-s + 64-s − 75-s − 20·79-s + 81-s − 100-s + 28·103-s − 5·108-s + 8·117-s − 22·121-s + 127-s − 2·129-s + 131-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 2/3·9-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 1/5·25-s − 0.962·27-s − 1/3·36-s − 0.640·39-s − 0.304·43-s + 0.144·48-s + 5/7·49-s − 0.554·52-s + 2.04·61-s + 1/8·64-s − 0.115·75-s − 2.25·79-s + 1/9·81-s − 0.0999·100-s + 2.75·103-s − 0.481·108-s + 0.739·117-s − 2·121-s + 0.0887·127-s − 0.176·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.053781309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.053781309\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.95100331996814604368205357657, −11.50404979571082961800443018176, −11.01619189424883405349147291451, −10.06695645523144294016656879053, −9.913352699581214890373944199804, −8.959593629099469848691258073309, −8.575390845192196003667033941777, −7.76086759399205832375001571762, −7.32239322779430326128563245946, −6.55879167140954809453789000786, −5.73907409309440481030466769989, −5.06559856971705064956593654142, −3.99768339105414826196839079842, −3.00020209252807799601405239891, −2.19599255252903267519822480871,
2.19599255252903267519822480871, 3.00020209252807799601405239891, 3.99768339105414826196839079842, 5.06559856971705064956593654142, 5.73907409309440481030466769989, 6.55879167140954809453789000786, 7.32239322779430326128563245946, 7.76086759399205832375001571762, 8.575390845192196003667033941777, 8.959593629099469848691258073309, 9.913352699581214890373944199804, 10.06695645523144294016656879053, 11.01619189424883405349147291451, 11.50404979571082961800443018176, 11.95100331996814604368205357657
