| L(s) = 1 | + 7.56e3·5-s − 5.36e5·13-s + 3.85e6·17-s + 2.86e7·25-s − 1.12e7·29-s + 3.05e7·37-s + 2.37e8·41-s + 1.07e8·49-s − 8.35e8·53-s + 2.61e9·61-s − 4.05e9·65-s − 2.37e8·73-s + 2.91e10·85-s + 1.11e10·89-s + 9.66e9·97-s + 3.73e10·101-s + 1.10e10·109-s + 7.13e10·113-s + 6.79e10·121-s + 1.28e11·125-s + 127-s + 131-s + 137-s + 139-s − 8.52e10·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2.41·5-s − 1.44·13-s + 2.71·17-s + 2.93·25-s − 0.549·29-s + 0.439·37-s + 2.05·41-s + 0.382·49-s − 1.99·53-s + 3.09·61-s − 3.49·65-s − 0.114·73-s + 6.57·85-s + 2.00·89-s + 1.12·97-s + 3.55·101-s + 0.717·109-s + 3.87·113-s + 2.62·121-s + 4.20·125-s − 1.33·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+5)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(23.06727064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(23.06727064\) |
| \(L(6)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 - 756 p T + 1424446 p T^{2} - 756 p^{11} T^{3} + p^{20} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 107909092 T^{2} + 515752461638502 p^{2} T^{4} - 107909092 p^{20} T^{6} + p^{40} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 67987804420 T^{2} + \)\(23\!\cdots\!82\)\( T^{4} - 67987804420 p^{20} T^{6} + p^{40} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 268156 T + 247092947862 T^{2} + 268156 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 1928556 T + 4674862871462 T^{2} - 1928556 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 11396009075908 T^{2} + \)\(67\!\cdots\!98\)\( T^{4} - 11396009075908 p^{20} T^{6} + p^{40} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 57560728155460 T^{2} + \)\(42\!\cdots\!22\)\( T^{4} - 57560728155460 p^{20} T^{6} + p^{40} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 5639436 T + 551257283207606 T^{2} + 5639436 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1785509547336484 T^{2} + \)\(18\!\cdots\!66\)\( T^{4} - 1785509547336484 p^{20} T^{6} + p^{40} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 15255220 T + 9227793694532118 T^{2} - 15255220 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 118854540 T + 28110027532085702 T^{2} - 118854540 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 3337921746848260 T^{2} + \)\(68\!\cdots\!22\)\( T^{4} - 3337921746848260 p^{20} T^{6} + p^{40} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 22958275505088772 T^{2} - \)\(31\!\cdots\!82\)\( T^{4} - 22958275505088772 p^{20} T^{6} + p^{40} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 417942828 T + 380158218111049814 T^{2} + 417942828 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1717674283247984260 T^{2} + \)\(12\!\cdots\!82\)\( T^{4} - 1717674283247984260 p^{20} T^{6} + p^{40} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 1306619188 T + 1842940593015440118 T^{2} - 1306619188 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 3067696319988122692 T^{2} + \)\(51\!\cdots\!98\)\( T^{4} - 3067696319988122692 p^{20} T^{6} + p^{40} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 9494363384388213700 T^{2} + \)\(43\!\cdots\!22\)\( T^{4} - 9494363384388213700 p^{20} T^{6} + p^{40} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 118687324 T + 7332861243362857062 T^{2} + 118687324 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 15414374288754462628 T^{2} + \)\(19\!\cdots\!78\)\( T^{4} - 15414374288754462628 p^{20} T^{6} + p^{40} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 27382880992584417220 T^{2} + \)\(47\!\cdots\!22\)\( T^{4} - 27382880992584417220 p^{20} T^{6} + p^{40} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 5592224988 T + 68730968898102971558 T^{2} - 5592224988 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 4834774532 T - 44713135597567105146 T^{2} - 4834774532 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.66530384253803224387071662674, −7.37945999377435045276480851716, −7.05191880438100009130423091302, −6.88852926750276600933575730456, −6.51290670753997074050724314528, −5.87538831867380472622530399243, −5.79805228256637532862429042657, −5.77797201028198966450230155180, −5.76457106772429900922661681197, −5.11916578100454938328938268862, −4.72281015182043445937838277931, −4.64700104342173174540855190677, −4.46648201578325874889189558991, −3.55709197938909106825013764940, −3.36968748015583399216884092127, −3.33891841869237105072549033106, −2.86380150037034382402870725307, −2.28004335442637784393507097859, −2.17459420818848691738329480235, −2.03571734874368050794131972772, −1.73956351060655560465892461585, −1.13166932700236846216230653797, −0.911702848481230427097753031090, −0.57414379002770929237026674949, −0.52886966157891232436398395259,
0.52886966157891232436398395259, 0.57414379002770929237026674949, 0.911702848481230427097753031090, 1.13166932700236846216230653797, 1.73956351060655560465892461585, 2.03571734874368050794131972772, 2.17459420818848691738329480235, 2.28004335442637784393507097859, 2.86380150037034382402870725307, 3.33891841869237105072549033106, 3.36968748015583399216884092127, 3.55709197938909106825013764940, 4.46648201578325874889189558991, 4.64700104342173174540855190677, 4.72281015182043445937838277931, 5.11916578100454938328938268862, 5.76457106772429900922661681197, 5.77797201028198966450230155180, 5.79805228256637532862429042657, 5.87538831867380472622530399243, 6.51290670753997074050724314528, 6.88852926750276600933575730456, 7.05191880438100009130423091302, 7.37945999377435045276480851716, 7.66530384253803224387071662674
