| L(s) = 1 | + 9-s + 12·13-s + 12·17-s − 10·25-s + 8·29-s + 4·37-s − 4·41-s − 10·49-s − 24·53-s + 4·61-s − 20·73-s + 81-s + 4·89-s − 12·97-s + 8·101-s − 4·109-s − 28·113-s + 12·117-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + 157-s + ⋯ |
| L(s) = 1 | + 1/3·9-s + 3.32·13-s + 2.91·17-s − 2·25-s + 1.48·29-s + 0.657·37-s − 0.624·41-s − 1.42·49-s − 3.29·53-s + 0.512·61-s − 2.34·73-s + 1/9·81-s + 0.423·89-s − 1.21·97-s + 0.796·101-s − 0.383·109-s − 2.63·113-s + 1.10·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.146294034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.146294034\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.464663439718228165986962411065, −8.658903060264415544055202217864, −8.272861522300808832842531458007, −7.83756825226352882885863464519, −7.70036027517747780731408099423, −6.52243750033071865852983380426, −6.40139715881748782453956361829, −5.75568326693151721019291253980, −5.52928834758807112839354938677, −4.57667813854686016923499181308, −3.94434944725762226728078180666, −3.36330701716196318281397292070, −3.12662782741322079682173901720, −1.50413098873776121963751941726, −1.27109672072212401682622254955,
1.27109672072212401682622254955, 1.50413098873776121963751941726, 3.12662782741322079682173901720, 3.36330701716196318281397292070, 3.94434944725762226728078180666, 4.57667813854686016923499181308, 5.52928834758807112839354938677, 5.75568326693151721019291253980, 6.40139715881748782453956361829, 6.52243750033071865852983380426, 7.70036027517747780731408099423, 7.83756825226352882885863464519, 8.272861522300808832842531458007, 8.658903060264415544055202217864, 9.464663439718228165986962411065
