| L(s) = 1 | + 2·5-s + 4·11-s + 4·19-s + 25-s + 4·29-s + 4·31-s − 28·41-s − 3·49-s + 8·55-s − 16·59-s + 4·61-s − 12·71-s − 40·79-s + 4·89-s + 8·95-s − 60·101-s + 20·109-s − 14·121-s + 8·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 8·155-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s + 0.917·19-s + 1/5·25-s + 0.742·29-s + 0.718·31-s − 4.37·41-s − 3/7·49-s + 1.07·55-s − 2.08·59-s + 0.512·61-s − 1.42·71-s − 4.50·79-s + 0.423·89-s + 0.820·95-s − 5.97·101-s + 1.91·109-s − 1.27·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s + 0.642·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.245966885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.245966885\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | \( ( 1 + T^{2} )^{3} \) |
| good | 11 | \( ( 1 - 2 T + 13 T^{2} - 36 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( ( 1 - 8 T + 7 T^{2} + 64 T^{3} + 7 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )( 1 + 8 T + 7 T^{2} - 64 T^{3} + 7 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} ) \) |
| 17 | \( 1 - 70 T^{2} + 2415 T^{4} - 51220 T^{6} + 2415 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 2 T + 45 T^{2} - 68 T^{3} + 45 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 90 T^{2} + 3839 T^{4} - 105452 T^{6} + 3839 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 2 T + 81 T^{2} - 116 T^{3} + 81 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 10 T - 5 T^{2} + 356 T^{3} - 5 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )( 1 + 10 T - 5 T^{2} - 356 T^{3} - 5 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} ) \) |
| 41 | \( ( 1 + 14 T + 175 T^{2} + 1188 T^{3} + 175 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 82 T^{2} + 5399 T^{4} - 217692 T^{6} + 5399 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 90 T^{2} + 5231 T^{4} - 236204 T^{6} + 5231 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 66 T^{2} + 2711 T^{4} + 8452 T^{6} + 2711 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 8 T + 113 T^{2} + 688 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 2 T + 35 T^{2} - 780 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( ( 1 - p T^{2} )^{6} \) |
| 71 | \( ( 1 + 6 T + 113 T^{2} + 1052 T^{3} + 113 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 154 T^{2} + 21503 T^{4} - 1698540 T^{6} + 21503 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 20 T + 317 T^{2} + 3096 T^{3} + 317 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 47783 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 2 T + 207 T^{2} - 156 T^{3} + 207 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 394 T^{2} + 77839 T^{4} - 9393484 T^{6} + 77839 p^{2} T^{8} - 394 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.23745412274369756692447816449, −4.16803674188944107692167173230, −3.96257363586955074180395428124, −3.80530723647412184223168735575, −3.67286586839929888059526103492, −3.55639171219658295957004474112, −3.15302763512907222261324587447, −3.11292514693152223794055164383, −3.10076145798553163204916222820, −3.08336073777974469444590746386, −2.78598502913500036042658295919, −2.64387412583295144009033824384, −2.63811689677487952242117107850, −2.39406987055977735011620008069, −1.94245463844161543277341908029, −1.88071727369130462449103292935, −1.80008523886097250863541947771, −1.55799116004474242025304180930, −1.45849441114014214021733887501, −1.45473302498275342756281985128, −1.29719185731468485723872521634, −0.859945658357297931804624746460, −0.72174761720976282092262123330, −0.39496115836632745959942262345, −0.097133414712482843201546568408,
0.097133414712482843201546568408, 0.39496115836632745959942262345, 0.72174761720976282092262123330, 0.859945658357297931804624746460, 1.29719185731468485723872521634, 1.45473302498275342756281985128, 1.45849441114014214021733887501, 1.55799116004474242025304180930, 1.80008523886097250863541947771, 1.88071727369130462449103292935, 1.94245463844161543277341908029, 2.39406987055977735011620008069, 2.63811689677487952242117107850, 2.64387412583295144009033824384, 2.78598502913500036042658295919, 3.08336073777974469444590746386, 3.10076145798553163204916222820, 3.11292514693152223794055164383, 3.15302763512907222261324587447, 3.55639171219658295957004474112, 3.67286586839929888059526103492, 3.80530723647412184223168735575, 3.96257363586955074180395428124, 4.16803674188944107692167173230, 4.23745412274369756692447816449
Plot not available for L-functions of degree greater than 10.
