| L(s) = 1 | + 4·4-s − 12·9-s + 112·13-s + 592·16-s + 1.18e4·25-s − 48·36-s + 6.44e3·37-s + 5.64e4·49-s + 448·52-s + 5.96e4·61-s + 8.76e3·64-s − 1.29e5·73-s + 5.02e4·81-s − 3.92e5·97-s + 4.72e4·100-s + 6.03e5·109-s − 1.34e3·117-s − 1.01e6·121-s + 127-s + 131-s + 137-s + 139-s − 7.10e3·144-s + 2.57e4·148-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 1/8·4-s − 0.0493·9-s + 0.183·13-s + 0.578·16-s + 3.78·25-s − 0.00617·36-s + 0.774·37-s + 3.35·49-s + 0.0229·52-s + 2.05·61-s + 0.267·64-s − 2.83·73-s + 0.850·81-s − 4.23·97-s + 0.472·100-s + 4.86·109-s − 0.00907·117-s − 6.27·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 0.0285·144-s + 0.0967·148-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.957870245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.957870245\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T^{2} - 9 p^{6} T^{4} - p^{12} T^{6} + p^{20} T^{8} \) |
| 3 | \( 1 + 4 p T^{2} - 206 p^{5} T^{4} + 4 p^{11} T^{6} + p^{20} T^{8} \) |
| good | 5 | \( ( 1 - 5908 T^{2} + 24052566 T^{4} - 5908 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 7 | \( ( 1 - 28228 T^{2} + 481424742 T^{4} - 28228 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 11 | \( ( 1 + 505004 T^{2} + 110941402518 T^{4} + 505004 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 13 | \( ( 1 - 28 T + 574590 T^{2} - 28 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 17 | \( ( 1 - 2078020 T^{2} + 5084618860998 T^{4} - 2078020 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 19 | \( ( 1 - 8024116 T^{2} + 1455617573682 p T^{4} - 8024116 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 23 | \( ( 1 + 14885084 T^{2} + 138058315089894 T^{4} + 14885084 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 29 | \( ( 1 - 54660340 T^{2} + 1583587051366134 T^{4} - 54660340 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 31 | \( ( 1 - 79362916 T^{2} + 3197045389933638 T^{4} - 79362916 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 37 | \( ( 1 - 1612 T + 3761334 p T^{2} - 1612 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 41 | \( ( 1 - 348130852 T^{2} + 56556946490999526 T^{4} - 348130852 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 43 | \( ( 1 - 6765820 p T^{2} + 49997815939989270 T^{4} - 6765820 p^{11} T^{6} + p^{20} T^{8} )^{2} \) |
| 47 | \( ( 1 + 88846268 T^{2} + 86182082358004422 T^{4} + 88846268 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 53 | \( ( 1 - 1535086804 T^{2} + 934675065335266710 T^{4} - 1535086804 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 59 | \( ( 1 + 979982444 T^{2} + 491357832465102678 T^{4} + 979982444 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 61 | \( ( 1 - 14908 T + 23967606 p T^{2} - 14908 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 67 | \( ( 1 - 3568432372 T^{2} + 1461291205104198 p^{2} T^{4} - 3568432372 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 71 | \( ( 1 + 5687616668 T^{2} + 14475080465515182246 T^{4} + 5687616668 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 73 | \( ( 1 + 32300 T + 2297165238 T^{2} + 32300 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 79 | \( ( 1 - 7492853668 T^{2} + 30010687456161264390 T^{4} - 7492853668 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 83 | \( ( 1 + 15651882572 T^{2} + 92275066617585920694 T^{4} + 15651882572 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 89 | \( ( 1 - 4696163812 T^{2} + 36381430695704929830 T^{4} - 4696163812 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 97 | \( ( 1 + 98012 T + 18865657350 T^{2} + 98012 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.151777804060623101682587669533, −9.049919436288037503348170062703, −9.026773222648669868502847748085, −8.652398431836808972887432529098, −8.413237574211803470486473337022, −8.182448829984718223718398639380, −8.022932933086208939709107198468, −7.50078580798860230093326067977, −7.34573815542678746260068059937, −6.95991806357111451970752064714, −6.85291047060060154795200766231, −6.76109795532483583164615825524, −6.15609171820708310619378131278, −5.88452464376975547628511978710, −5.63974679889459828618839435776, −5.29014300799777620325700486549, −4.95691342704886059388678804355, −4.60148148880221237721442937796, −4.17191068651267564912068914668, −3.82697122851037828727746900248, −3.26438044507686794124104050459, −2.65981930108040445813116710220, −2.55273244054702814622203562385, −1.30326050094125470645358086998, −0.797356925996867022011891916546,
0.797356925996867022011891916546, 1.30326050094125470645358086998, 2.55273244054702814622203562385, 2.65981930108040445813116710220, 3.26438044507686794124104050459, 3.82697122851037828727746900248, 4.17191068651267564912068914668, 4.60148148880221237721442937796, 4.95691342704886059388678804355, 5.29014300799777620325700486549, 5.63974679889459828618839435776, 5.88452464376975547628511978710, 6.15609171820708310619378131278, 6.76109795532483583164615825524, 6.85291047060060154795200766231, 6.95991806357111451970752064714, 7.34573815542678746260068059937, 7.50078580798860230093326067977, 8.022932933086208939709107198468, 8.182448829984718223718398639380, 8.413237574211803470486473337022, 8.652398431836808972887432529098, 9.026773222648669868502847748085, 9.049919436288037503348170062703, 9.151777804060623101682587669533
Plot not available for L-functions of degree greater than 10.
