| L(s) = 1 | + 4-s − 4·13-s − 12·19-s − 2·25-s + 6·37-s + 2·43-s − 14·49-s − 4·52-s + 16·61-s − 64-s − 12·67-s − 12·76-s − 44·79-s − 12·97-s − 2·100-s + 56·103-s − 13·121-s + 127-s + 131-s + 137-s + 139-s + 6·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.10·13-s − 2.75·19-s − 2/5·25-s + 0.986·37-s + 0.304·43-s − 2·49-s − 0.554·52-s + 2.04·61-s − 1/8·64-s − 1.46·67-s − 1.37·76-s − 4.95·79-s − 1.21·97-s − 1/5·100-s + 5.51·103-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.493·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8608079893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8608079893\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 43 T^{2} + 1320 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 55 T^{2} + 2184 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 46 T^{2} + 435 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 37 T^{2} - 2112 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 71 | $C_2^3$ | \( 1 + 106 T^{2} + 6195 T^{4} + 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 6 T + 109 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.90348050415290383759791082399, −6.86409590290084049904587064450, −6.63133891886517095599378897726, −6.29846824628144305493188505908, −5.99043954139401541718245953443, −5.98517544037203685004062885159, −5.83941196901512329257800384829, −5.44867083845582109644410344733, −5.22760902524571996164334225796, −4.81122654184525279681477889232, −4.69827183866046425205551955935, −4.45728789059022702286325449096, −4.26221583108399137086613038056, −4.10249977911475551203351956351, −3.79701017572963325831829949385, −3.38115471330529954600344171391, −3.03838800239601637790430338817, −2.94481903669729730426138014031, −2.60203675634136363534359677463, −2.21140690244419458182422680365, −2.06859880090404520935780201036, −1.76261844795353970379534763981, −1.46441954361548520109945058475, −0.78381860323698292894134219631, −0.21928326095332361961473682522,
0.21928326095332361961473682522, 0.78381860323698292894134219631, 1.46441954361548520109945058475, 1.76261844795353970379534763981, 2.06859880090404520935780201036, 2.21140690244419458182422680365, 2.60203675634136363534359677463, 2.94481903669729730426138014031, 3.03838800239601637790430338817, 3.38115471330529954600344171391, 3.79701017572963325831829949385, 4.10249977911475551203351956351, 4.26221583108399137086613038056, 4.45728789059022702286325449096, 4.69827183866046425205551955935, 4.81122654184525279681477889232, 5.22760902524571996164334225796, 5.44867083845582109644410344733, 5.83941196901512329257800384829, 5.98517544037203685004062885159, 5.99043954139401541718245953443, 6.29846824628144305493188505908, 6.63133891886517095599378897726, 6.86409590290084049904587064450, 6.90348050415290383759791082399
