| L(s) = 1 | − 4·2-s + 4·4-s − 6·5-s + 16·8-s + 24·10-s − 10·13-s − 64·16-s − 76·17-s − 24·20-s + 25·25-s + 40·26-s + 120·29-s + 64·32-s + 304·34-s + 72·37-s − 96·40-s − 178·41-s + 98·49-s − 100·50-s − 40·52-s + 34·53-s − 480·58-s − 120·61-s + 192·64-s + 60·65-s − 304·68-s + 192·73-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4-s − 6/5·5-s + 2·8-s + 12/5·10-s − 0.769·13-s − 4·16-s − 4.47·17-s − 6/5·20-s + 25-s + 1.53·26-s + 4.13·29-s + 2·32-s + 8.94·34-s + 1.94·37-s − 2.39·40-s − 4.34·41-s + 2·49-s − 2·50-s − 0.769·52-s + 0.641·53-s − 8.27·58-s − 1.96·61-s + 3·64-s + 0.923·65-s − 4.47·68-s + 2.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.08618593437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08618593437\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 6 T + 11 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 10 T - 69 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{2}( 1 + 16 T - 33 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 19 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2}( 1 - 40 T + 759 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 31 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2}( 1 - 24 T - 793 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{2}( 1 + 18 T - 1357 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 43 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 90 T + 5291 T^{2} - 90 p^{2} T^{3} + p^{4} T^{4} )( 1 + 56 T + 327 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 59 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + 120 T + p^{2} T^{2} )^{2}( 1 - 120 T + 10679 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 67 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 96 T + 3887 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 89 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 78 T - 1837 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} )( 1 + 160 T + 17679 T^{2} + 160 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 97 | $C_2^2$ | \( ( 1 + 130 T + 7491 T^{2} + 130 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.592529626843195517641131097265, −8.491702525079208544694179075856, −8.206349488092296590559764707044, −7.82068060844654212620717265053, −7.59192386058305798865854122394, −7.23182815728851569937926556963, −6.99586726637072933478689802733, −6.92652967682239054255323867007, −6.44309579921524177154041997761, −6.36440983225109227526573197603, −6.21539960531143491588450793644, −5.30124349252112713690515285321, −4.90942177475011067998528868270, −4.73296068704726153130393633104, −4.60171437815323629614446035981, −4.56354293460122777118952007729, −3.91690800153975762157580556232, −3.85132536926433269700303775872, −3.13385628641558242796329744221, −2.64358951726579427849587917682, −2.17415845585678830826578341070, −2.15399442774853252009177636313, −1.24043682434048752513944659185, −0.73415444949603489408325205740, −0.16961060363249287244400209833,
0.16961060363249287244400209833, 0.73415444949603489408325205740, 1.24043682434048752513944659185, 2.15399442774853252009177636313, 2.17415845585678830826578341070, 2.64358951726579427849587917682, 3.13385628641558242796329744221, 3.85132536926433269700303775872, 3.91690800153975762157580556232, 4.56354293460122777118952007729, 4.60171437815323629614446035981, 4.73296068704726153130393633104, 4.90942177475011067998528868270, 5.30124349252112713690515285321, 6.21539960531143491588450793644, 6.36440983225109227526573197603, 6.44309579921524177154041997761, 6.92652967682239054255323867007, 6.99586726637072933478689802733, 7.23182815728851569937926556963, 7.59192386058305798865854122394, 7.82068060844654212620717265053, 8.206349488092296590559764707044, 8.491702525079208544694179075856, 8.592529626843195517641131097265
