L(s) = 1 | + 2·5-s + 6·13-s − 4·17-s + 5·25-s + 10·29-s − 32·37-s + 4·41-s + 49-s − 32·53-s + 26·61-s + 12·65-s + 60·73-s − 8·85-s − 56·89-s − 36·97-s + 26·101-s − 48·109-s + 58·113-s + 8·121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 20·145-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 1.66·13-s − 0.970·17-s + 25-s + 1.85·29-s − 5.26·37-s + 0.624·41-s + 1/7·49-s − 4.39·53-s + 3.32·61-s + 1.48·65-s + 7.02·73-s − 0.867·85-s − 5.93·89-s − 3.65·97-s + 2.58·101-s − 4.59·109-s + 5.45·113-s + 8/11·121-s − 0.536·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.66·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(9.393823686\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.393823686\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( ( 1 - T - T^{2} + 8 T^{3} - 26 T^{4} + 8 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 - T^{2} - 23 T^{4} + 74 T^{6} - 1874 T^{8} + 74 p^{2} T^{10} - 23 p^{4} T^{12} - p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 8 T^{2} + 103 T^{4} + 2248 T^{6} - 20864 T^{8} + 2248 p^{2} T^{10} + 103 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 3 T - 11 T^{2} + 18 T^{3} + 114 T^{4} + 18 p T^{5} - 11 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + T + 26 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 31 T^{2} + 888 T^{4} + 31 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 65 T^{2} + 95 p T^{4} - 63830 T^{6} + 1646734 T^{8} - 63830 p^{2} T^{10} + 95 p^{5} T^{12} - 65 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 5 T - 31 T^{2} + 10 T^{3} + 1570 T^{4} + 10 p T^{5} - 31 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 + 11 T^{2} - 1163 T^{4} - 7018 T^{6} + 608854 T^{8} - 7018 p^{2} T^{10} - 1163 p^{4} T^{12} + 11 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 - T - 40 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 - 136 T^{2} + 10471 T^{4} - 588472 T^{6} + 26782720 T^{8} - 588472 p^{2} T^{10} + 10471 p^{4} T^{12} - 136 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 - 161 T^{2} + 15097 T^{4} - 1031366 T^{6} + 55019806 T^{8} - 1031366 p^{2} T^{10} + 15097 p^{4} T^{12} - 161 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 59 | \( 1 - 56 T^{2} - 4313 T^{4} - 27272 T^{6} + 32453824 T^{8} - 27272 p^{2} T^{10} - 4313 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 13 T + 79 T^{2} + 416 T^{3} - 5930 T^{4} + 416 p T^{5} + 79 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 88 T^{2} - 2873 T^{4} - 144232 T^{6} + 57806752 T^{8} - 144232 p^{2} T^{10} - 2873 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 176 T^{2} + 16638 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 15 T + 194 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( 1 - 181 T^{2} + 12757 T^{4} - 1361482 T^{6} + 156748534 T^{8} - 1361482 p^{2} T^{10} + 12757 p^{4} T^{12} - 181 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 - 125 T^{2} + 6925 T^{4} + 634750 T^{6} - 79408946 T^{8} + 634750 p^{2} T^{10} + 6925 p^{4} T^{12} - 125 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 14 T + 194 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.98674837456652559194771318690, −3.88438407513076570682291826669, −3.63733701063192806089646621835, −3.40572949544772737927450904017, −3.34744672942999697510313345598, −3.33228499072756416399018815698, −3.31093499798757949900716491875, −3.09130314457877190594183750976, −2.91996682471425819857578078915, −2.76648596671658269251234551829, −2.76396737136100062443867475743, −2.37341402824408821246459674337, −2.25445038957899373445358037708, −2.25215757902240464830299626602, −2.01254726180545942120668644812, −2.00718647677469567645625053038, −1.85107608860105508847457713832, −1.48761920906037437307952990886, −1.37976285560405631397523295244, −1.32239604121223499014212463136, −1.27992393557980266644772216734, −0.974661062406748184115686956954, −0.53206111431234991727724713400, −0.45407993916457973362391304019, −0.32848739650209368152553800138,
0.32848739650209368152553800138, 0.45407993916457973362391304019, 0.53206111431234991727724713400, 0.974661062406748184115686956954, 1.27992393557980266644772216734, 1.32239604121223499014212463136, 1.37976285560405631397523295244, 1.48761920906037437307952990886, 1.85107608860105508847457713832, 2.00718647677469567645625053038, 2.01254726180545942120668644812, 2.25215757902240464830299626602, 2.25445038957899373445358037708, 2.37341402824408821246459674337, 2.76396737136100062443867475743, 2.76648596671658269251234551829, 2.91996682471425819857578078915, 3.09130314457877190594183750976, 3.31093499798757949900716491875, 3.33228499072756416399018815698, 3.34744672942999697510313345598, 3.40572949544772737927450904017, 3.63733701063192806089646621835, 3.88438407513076570682291826669, 3.98674837456652559194771318690
Plot not available for L-functions of degree greater than 10.