| L(s) = 1 | + 2·9-s + 32·23-s − 4·25-s + 12·49-s + 32·71-s − 40·73-s − 5·81-s − 24·97-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 64·207-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 6.67·23-s − 4/5·25-s + 12/7·49-s + 3.79·71-s − 4.68·73-s − 5/9·81-s − 2.43·97-s + 3.27·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 4.44·207-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.965121222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.965121222\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
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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
−8.207351237007476044536365432384, −8.120392734803400601703102035981, −7.51722692481107305037871800976, −7.30018491953753815298702269467, −7.15802566357469675757158148371, −7.08832167594432746985729948718, −6.93173108089170759841363258284, −6.56216569056071767764736280298, −6.27151545042709666373914242059, −5.73925680093140389530464301458, −5.71176362597283345245879739875, −5.38132132023293670107484485591, −4.94690964170703788921002533785, −4.90262128472470923969857010055, −4.62376623683424415542603396228, −4.35263455807574320277553178882, −3.83511327565108663264206937062, −3.61995776869795313834644009391, −3.17345858220063849604074900420, −2.89369658756497020588932642867, −2.73251475822297227269246547105, −2.27558610678440678205869454755, −1.54725662438713011073558818112, −1.15882525896736858807680651954, −0.832003487129236713181405303910,
0.832003487129236713181405303910, 1.15882525896736858807680651954, 1.54725662438713011073558818112, 2.27558610678440678205869454755, 2.73251475822297227269246547105, 2.89369658756497020588932642867, 3.17345858220063849604074900420, 3.61995776869795313834644009391, 3.83511327565108663264206937062, 4.35263455807574320277553178882, 4.62376623683424415542603396228, 4.90262128472470923969857010055, 4.94690964170703788921002533785, 5.38132132023293670107484485591, 5.71176362597283345245879739875, 5.73925680093140389530464301458, 6.27151545042709666373914242059, 6.56216569056071767764736280298, 6.93173108089170759841363258284, 7.08832167594432746985729948718, 7.15802566357469675757158148371, 7.30018491953753815298702269467, 7.51722692481107305037871800976, 8.120392734803400601703102035981, 8.207351237007476044536365432384
