| L(s) = 1 | − 3-s + 2·4-s − 7-s − 2·12-s − 13-s + 3·16-s + 19-s + 21-s − 25-s + 27-s − 2·28-s − 2·31-s − 2·37-s + 39-s + 43-s − 3·48-s − 2·52-s − 57-s + 61-s + 4·64-s − 2·67-s + 73-s + 75-s + 2·76-s − 2·79-s − 81-s + 2·84-s + ⋯ |
| L(s) = 1 | − 3-s + 2·4-s − 7-s − 2·12-s − 13-s + 3·16-s + 19-s + 21-s − 25-s + 27-s − 2·28-s − 2·31-s − 2·37-s + 39-s + 43-s − 3·48-s − 2·52-s − 57-s + 61-s + 4·64-s − 2·67-s + 73-s + 75-s + 2·76-s − 2·79-s − 81-s + 2·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5619882952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5619882952\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−12.06213236697114044589849094033, −11.96769113514174420928488581328, −11.38895877606578105374239008950, −11.16007466888114300179655769407, −10.44743648707332786626995305341, −10.29239079010745481225468279534, −9.759334153033061636597031405850, −9.180034034704280224597572975722, −8.494182551239677130502888253345, −7.62016775293943546341621076203, −7.26895720981183049590510254049, −7.07462944416210572592329253040, −6.34453796924869618127085803525, −5.96142704152549843878073701497, −5.50466788895028055692504694322, −5.03911227863593773966053577509, −3.75847791594342867408926511147, −3.23171219855847535167587717600, −2.54094456523031923158939771246, −1.69426267454846610406584398035,
1.69426267454846610406584398035, 2.54094456523031923158939771246, 3.23171219855847535167587717600, 3.75847791594342867408926511147, 5.03911227863593773966053577509, 5.50466788895028055692504694322, 5.96142704152549843878073701497, 6.34453796924869618127085803525, 7.07462944416210572592329253040, 7.26895720981183049590510254049, 7.62016775293943546341621076203, 8.494182551239677130502888253345, 9.180034034704280224597572975722, 9.759334153033061636597031405850, 10.29239079010745481225468279534, 10.44743648707332786626995305341, 11.16007466888114300179655769407, 11.38895877606578105374239008950, 11.96769113514174420928488581328, 12.06213236697114044589849094033
