| L(s) = 1 | + 4·3-s + 4·7-s + 8·9-s + 16·21-s − 8·25-s + 12·27-s + 8·31-s + 24·37-s − 24·43-s + 8·49-s − 24·61-s + 32·63-s + 16·67-s − 20·73-s − 32·75-s + 23·81-s + 32·93-s + 12·97-s − 4·103-s + 96·111-s + 40·121-s + 127-s − 96·129-s + 131-s + 137-s + 139-s + 32·147-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.51·7-s + 8/3·9-s + 3.49·21-s − 8/5·25-s + 2.30·27-s + 1.43·31-s + 3.94·37-s − 3.65·43-s + 8/7·49-s − 3.07·61-s + 4.03·63-s + 1.95·67-s − 2.34·73-s − 3.69·75-s + 23/9·81-s + 3.31·93-s + 1.21·97-s − 0.394·103-s + 9.11·111-s + 3.63·121-s + 0.0887·127-s − 8.45·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.63·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.884029338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.884029338\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )( 1 + 56 T^{2} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 170 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.656557568382396809254784069579, −8.567044678880102270845229872779, −8.220307151502032302784688510121, −8.138030654360897630184246494722, −8.087185093384237320099461870712, −7.57450160831616388846090142798, −7.41233023072329660843655268632, −7.38028564105894202736701374826, −6.84125817143758795961452044454, −6.37190787957464485731840504226, −6.23713535916788691914735255083, −5.80466371356607943799907301478, −5.79013113471455402614509553785, −4.95608830693979098888685878477, −4.69688962225048418572061635101, −4.64530619211594333345999103072, −4.45151706131114117735597423237, −3.80313014105359759317226301696, −3.52772229754449689331233736598, −3.31221585966140152183897399587, −2.74865019178506398584206151122, −2.48219697651783341661105068535, −2.18348590428698455706011520867, −1.61753522619537926076848143016, −1.22294184313253671895092966857,
1.22294184313253671895092966857, 1.61753522619537926076848143016, 2.18348590428698455706011520867, 2.48219697651783341661105068535, 2.74865019178506398584206151122, 3.31221585966140152183897399587, 3.52772229754449689331233736598, 3.80313014105359759317226301696, 4.45151706131114117735597423237, 4.64530619211594333345999103072, 4.69688962225048418572061635101, 4.95608830693979098888685878477, 5.79013113471455402614509553785, 5.80466371356607943799907301478, 6.23713535916788691914735255083, 6.37190787957464485731840504226, 6.84125817143758795961452044454, 7.38028564105894202736701374826, 7.41233023072329660843655268632, 7.57450160831616388846090142798, 8.087185093384237320099461870712, 8.138030654360897630184246494722, 8.220307151502032302784688510121, 8.567044678880102270845229872779, 8.656557568382396809254784069579
