| L(s) = 1 | − 128·4-s − 747·7-s − 1.26e4·13-s − 2.87e4·19-s + 1.56e5·25-s + 9.56e4·28-s + 9.51e5·37-s + 6.25e5·43-s − 4.51e5·49-s + 1.61e6·52-s + 1.99e6·61-s + 2.09e6·64-s + 3.67e6·67-s + 3.67e6·76-s + 9.03e6·79-s + 9.41e6·91-s − 6.96e6·97-s − 2.00e7·100-s − 4.38e7·103-s − 1.94e7·121-s + 127-s + 131-s + 2.14e7·133-s + 137-s + 139-s − 1.21e8·148-s + 149-s + ⋯ |
| L(s) = 1 | − 4-s − 0.823·7-s − 1.59·13-s − 0.961·19-s + 2·25-s + 0.823·28-s + 3.08·37-s + 1.20·43-s − 0.548·49-s + 1.59·52-s + 1.12·61-s + 64-s + 1.49·67-s + 0.961·76-s + 2.06·79-s + 1.30·91-s − 0.775·97-s − 2·100-s − 3.95·103-s − 121-s + 0.791·133-s − 3.08·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.003204466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003204466\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 12605 T + p^{7} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p^{7} T^{2} + p^{14} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 508 T + p^{7} T^{2} )( 1 + 1255 T + p^{7} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + p^{7} T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 14357 T + p^{7} T^{2} )( 1 + 43091 T + p^{7} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 331387 T + p^{7} T^{2} )( 1 + 331387 T + p^{7} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 615373 T + p^{7} T^{2} )( 1 - 335663 T + p^{7} T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + p^{7} T^{2} + p^{14} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 1035224 T + p^{7} T^{2} )( 1 + 409495 T + p^{7} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + p^{7} T^{2} + p^{14} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 3535546 T + p^{7} T^{2} )( 1 + 1537199 T + p^{7} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4058455 T + p^{7} T^{2} )( 1 + 385072 T + p^{7} T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + p^{7} T^{2} + p^{14} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 5038001 T + p^{7} T^{2} )( 1 + 5038001 T + p^{7} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4517617 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + p^{7} T^{2} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5276357 T + p^{7} T^{2} )( 1 + 12245198 T + p^{7} T^{2} ) \) |
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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
−12.50537235127048263621346377449, −12.22849615360748594918826065258, −11.11155671137419229563289787408, −11.04715926239979278749714819725, −10.09348327400842980376725975495, −9.703428147229641288311631539316, −9.326341156660182885323110504278, −8.841858363260790146017325456409, −8.111206896315279756626368887268, −7.64449312141497114068209710006, −6.65004209172928397271148431750, −6.60878127087026097147110801325, −5.53366124256611610491547182887, −4.92360898989963043852843222789, −4.40444367401195270292210220726, −3.82308345938422321106635010582, −2.67025846443924858524417698522, −2.47616365700808603244943549223, −0.985968983004219189339005592674, −0.36671585554338431717686259896,
0.36671585554338431717686259896, 0.985968983004219189339005592674, 2.47616365700808603244943549223, 2.67025846443924858524417698522, 3.82308345938422321106635010582, 4.40444367401195270292210220726, 4.92360898989963043852843222789, 5.53366124256611610491547182887, 6.60878127087026097147110801325, 6.65004209172928397271148431750, 7.64449312141497114068209710006, 8.111206896315279756626368887268, 8.841858363260790146017325456409, 9.326341156660182885323110504278, 9.703428147229641288311631539316, 10.09348327400842980376725975495, 11.04715926239979278749714819725, 11.11155671137419229563289787408, 12.22849615360748594918826065258, 12.50537235127048263621346377449
