| L(s) = 1 | + 2·4-s + 22·7-s + 58·13-s − 60·16-s + 58·19-s + 38·25-s + 44·28-s − 536·31-s + 166·37-s − 464·43-s − 323·49-s + 116·52-s + 1.53e3·61-s − 248·64-s − 1.02e3·67-s + 274·73-s + 116·76-s − 950·79-s + 1.27e3·91-s + 1.64e3·97-s + 76·100-s + 1.67e3·103-s + 436·109-s − 1.32e3·112-s − 2.37e3·121-s − 1.07e3·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/4·4-s + 1.18·7-s + 1.23·13-s − 0.937·16-s + 0.700·19-s + 0.303·25-s + 0.296·28-s − 3.10·31-s + 0.737·37-s − 1.64·43-s − 0.941·49-s + 0.309·52-s + 3.21·61-s − 0.484·64-s − 1.86·67-s + 0.439·73-s + 0.175·76-s − 1.35·79-s + 1.46·91-s + 1.71·97-s + 0.0759·100-s + 1.60·103-s + 0.383·109-s − 1.11·112-s − 1.78·121-s − 0.776·124-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.508273680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.508273680\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 38 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 11 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2374 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 29 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 7234 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 29 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 17134 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 24950 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 268 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 83 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 64114 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 232 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 55294 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 204442 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 327526 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 767 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 511 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 207790 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 137 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 475 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 810646 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 1345138 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 821 T + p^{3} T^{2} )^{2} \) |
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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
−17.18177942962759303036800099359, −16.25388206363723425517205308376, −16.20618956446640162892156869561, −15.31631999601821986768075624593, −14.46197015393425287073510693822, −14.41720579861908857245683955394, −13.17584804052637597225859534675, −13.11534342138308376740884024778, −11.85337484826259846492692337696, −11.18817318095793592425228336693, −11.12803599573148043873345394602, −10.03376611420772481445750937158, −9.057905406340226516475858000550, −8.496769943942561159034675497768, −7.61562976592539579606755120734, −6.83934534148495890283129173884, −5.73096983177186866516324061486, −4.83734370790959764913816121600, −3.58813664606728046092224331868, −1.76189977349892985089275079008,
1.76189977349892985089275079008, 3.58813664606728046092224331868, 4.83734370790959764913816121600, 5.73096983177186866516324061486, 6.83934534148495890283129173884, 7.61562976592539579606755120734, 8.496769943942561159034675497768, 9.057905406340226516475858000550, 10.03376611420772481445750937158, 11.12803599573148043873345394602, 11.18817318095793592425228336693, 11.85337484826259846492692337696, 13.11534342138308376740884024778, 13.17584804052637597225859534675, 14.41720579861908857245683955394, 14.46197015393425287073510693822, 15.31631999601821986768075624593, 16.20618956446640162892156869561, 16.25388206363723425517205308376, 17.18177942962759303036800099359
