| L(s) = 1 | + 54·9-s − 500·25-s − 572·49-s + 4.76e3·73-s + 2.18e3·81-s + 5.32e3·97-s + 5.32e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.01e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 2.70e4·225-s + ⋯ |
| L(s) = 1 | + 2·9-s − 4·25-s − 1.66·49-s + 7.63·73-s + 3·81-s + 5.56·97-s + 4·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.460·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s − 8·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.524742251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.524742251\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2}( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 506 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 10582 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 308 T + p^{3} T^{2} )^{2}( 1 + 308 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 89206 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 111386 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 420838 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 172874 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 1190 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 884 T + p^{3} T^{2} )^{2}( 1 + 884 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1330 T + p^{3} T^{2} )^{4} \) |
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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
−7.16467530793755123608534981311, −6.72379042748880347893368257619, −6.44854651293924181005811722430, −6.30497639782092580294247892933, −6.19682136607361985120193838625, −5.97647440911135218878905238692, −5.55893651135371698107494499026, −5.28635984110818305702164819532, −5.06377479949282502051850852729, −4.79166039007804855196649063330, −4.67355234377016592877830906177, −4.36144709228684008276385914395, −3.93137647710591911381583804306, −3.77225937669149348761557334967, −3.62047601524181520318039335250, −3.49052771749545545092349896134, −3.12618668152856792452429721140, −2.35149854208433467230435262784, −2.31416284767667679521119821656, −1.96049114185205099858085734968, −1.73888187572639600384899261061, −1.62137991434244596865235610125, −0.817882817300581082198919422250, −0.63050624707714154825009934729, −0.43383315521406447586548889891,
0.43383315521406447586548889891, 0.63050624707714154825009934729, 0.817882817300581082198919422250, 1.62137991434244596865235610125, 1.73888187572639600384899261061, 1.96049114185205099858085734968, 2.31416284767667679521119821656, 2.35149854208433467230435262784, 3.12618668152856792452429721140, 3.49052771749545545092349896134, 3.62047601524181520318039335250, 3.77225937669149348761557334967, 3.93137647710591911381583804306, 4.36144709228684008276385914395, 4.67355234377016592877830906177, 4.79166039007804855196649063330, 5.06377479949282502051850852729, 5.28635984110818305702164819532, 5.55893651135371698107494499026, 5.97647440911135218878905238692, 6.19682136607361985120193838625, 6.30497639782092580294247892933, 6.44854651293924181005811722430, 6.72379042748880347893368257619, 7.16467530793755123608534981311
