| L(s) = 1 | − 2·4-s − 4·7-s − 3·9-s + 4·16-s − 8·19-s − 5·25-s + 8·28-s + 6·36-s − 8·43-s + 7·49-s + 14·61-s + 12·63-s − 8·64-s + 10·73-s + 16·76-s + 9·81-s + 10·100-s − 16·112-s − 11·121-s + 127-s + 131-s + 32·133-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s − 1.51·7-s − 9-s + 16-s − 1.83·19-s − 25-s + 1.51·28-s + 36-s − 1.21·43-s + 49-s + 1.79·61-s + 1.51·63-s − 64-s + 1.17·73-s + 1.83·76-s + 81-s + 100-s − 1.51·112-s − 121-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-s + 0.0854·137-s + 0.0848·139-s − 144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−10.97455297816291874087994848900, −10.20830983466443674930040750481, −9.931973061929162445570994946539, −9.327452062130334903699137040015, −8.751710603875262169019204511252, −8.380538887301880119381122963176, −7.73153329402525560534738717037, −6.70668525227380276025520048225, −6.30224972650556647667110253190, −5.67317917241321247132869453278, −4.93850310469037768835065637697, −3.95912613542587085600475854132, −3.47628810644293837018089876045, −2.42487600846985920555473281476, 0,
2.42487600846985920555473281476, 3.47628810644293837018089876045, 3.95912613542587085600475854132, 4.93850310469037768835065637697, 5.67317917241321247132869453278, 6.30224972650556647667110253190, 6.70668525227380276025520048225, 7.73153329402525560534738717037, 8.380538887301880119381122963176, 8.751710603875262169019204511252, 9.327452062130334903699137040015, 9.931973061929162445570994946539, 10.20830983466443674930040750481, 10.97455297816291874087994848900
