| L(s) = 1 | − 6·2-s + 21·4-s − 2·5-s + 4·7-s − 56·8-s − 3·9-s + 12·10-s − 8·13-s − 24·14-s + 126·16-s + 18·18-s − 42·20-s + 25-s + 48·26-s + 84·28-s + 4·29-s − 252·32-s − 8·35-s − 63·36-s + 24·37-s + 112·40-s + 6·45-s − 32·47-s − 14·49-s − 6·50-s − 168·52-s − 224·56-s + ⋯ |
| L(s) = 1 | − 4.24·2-s + 21/2·4-s − 0.894·5-s + 1.51·7-s − 19.7·8-s − 9-s + 3.79·10-s − 2.21·13-s − 6.41·14-s + 63/2·16-s + 4.24·18-s − 9.39·20-s + 1/5·25-s + 9.41·26-s + 15.8·28-s + 0.742·29-s − 44.5·32-s − 1.35·35-s − 10.5·36-s + 3.94·37-s + 17.7·40-s + 0.894·45-s − 4.66·47-s − 2·49-s − 0.848·50-s − 23.2·52-s − 29.9·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1505490576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1505490576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T )^{6} \) |
| 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 + 2 T + 3 T^{2} + 12 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | \( 1 + 8 T + 7 T^{2} - 64 T^{3} + 7 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 7 | \( ( 1 - 2 T + 13 T^{2} - 32 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 - 2 p T^{2} + 375 T^{4} - 3988 T^{6} + 375 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 86 T^{2} + 3279 T^{4} - 71588 T^{6} + 3279 p^{2} T^{8} - 86 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 - 82 T^{2} + 3175 T^{4} - 75148 T^{6} + 3175 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 126 T^{2} + 6863 T^{4} - 206708 T^{6} + 6863 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 2 T + 43 T^{2} - 96 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 78 T^{2} + 3887 T^{4} - 149060 T^{6} + 3887 p^{2} T^{8} - 78 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( ( 1 - 12 T + 143 T^{2} - 872 T^{3} + 143 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - 170 T^{2} + 13695 T^{4} - 687692 T^{6} + 13695 p^{2} T^{8} - 170 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( 1 - 130 T^{2} + 11095 T^{4} - 558460 T^{6} + 11095 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 + 16 T + 189 T^{2} + 1536 T^{3} + 189 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( 1 - 82 T^{2} - 297 T^{4} + 234404 T^{6} - 297 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( 1 - 102 T^{2} + 7383 T^{4} - 510324 T^{6} + 7383 p^{2} T^{8} - 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 2 T + 147 T^{2} + 140 T^{3} + 147 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( ( 1 - 2 T + 149 T^{2} - 84 T^{3} + 149 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( 1 - 314 T^{2} + 46431 T^{4} - 4136876 T^{6} + 46431 p^{2} T^{8} - 314 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( ( 1 - 20 T + 255 T^{2} - 2372 T^{3} + 255 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 + 8 T + 21 T^{2} - 736 T^{3} + 21 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 + 105 T^{2} - 432 T^{3} + 105 p T^{4} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 282 T^{2} + 46175 T^{4} - 4874348 T^{6} + 46175 p^{2} T^{8} - 282 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( ( 1 + 191 T^{2} - 268 T^{3} + 191 p T^{4} + p^{3} T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.29525143779207504216013199398, −6.21241406943998267554991789679, −6.16256376006583740012697109310, −5.74693926360488739577909309124, −5.33066338832679045462462377031, −5.27817941130839632766697663946, −5.23035589700015478233555688959, −4.89091823680772441049149018717, −4.62937998853158358638475562307, −4.58026749649490482774007913749, −4.43858772875803094844806083539, −3.95279575150939540275416391622, −3.61374111508433110182600707394, −3.45406581769046821272458414643, −3.16175894264660280978586602361, −2.95874401367370896424782214228, −2.74065682697962436023162922668, −2.47033827728668067269220462768, −2.38392006250036269772555952355, −2.02713770982913507663827332085, −1.70217467885167089520530718337, −1.54114778183158116704486577621, −1.19264159371482447054391284703, −0.56287584350859621638684160636, −0.38129261353468290071053407799,
0.38129261353468290071053407799, 0.56287584350859621638684160636, 1.19264159371482447054391284703, 1.54114778183158116704486577621, 1.70217467885167089520530718337, 2.02713770982913507663827332085, 2.38392006250036269772555952355, 2.47033827728668067269220462768, 2.74065682697962436023162922668, 2.95874401367370896424782214228, 3.16175894264660280978586602361, 3.45406581769046821272458414643, 3.61374111508433110182600707394, 3.95279575150939540275416391622, 4.43858772875803094844806083539, 4.58026749649490482774007913749, 4.62937998853158358638475562307, 4.89091823680772441049149018717, 5.23035589700015478233555688959, 5.27817941130839632766697663946, 5.33066338832679045462462377031, 5.74693926360488739577909309124, 6.16256376006583740012697109310, 6.21241406943998267554991789679, 6.29525143779207504216013199398
Plot not available for L-functions of degree greater than 10.
