| L(s) = 1 | + 3·9-s − 6·11-s + 14·19-s − 20·29-s + 6·31-s + 8·41-s + 11·49-s − 2·61-s − 34·71-s + 22·79-s − 7·81-s − 60·89-s − 18·99-s + 16·101-s − 36·109-s + 21·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 75·169-s + 42·171-s + ⋯ |
| L(s) = 1 | + 9-s − 1.80·11-s + 3.21·19-s − 3.71·29-s + 1.07·31-s + 1.24·41-s + 11/7·49-s − 0.256·61-s − 4.03·71-s + 2.47·79-s − 7/9·81-s − 6.35·89-s − 1.80·99-s + 1.59·101-s − 3.44·109-s + 1.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.76·169-s + 3.21·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.199317273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.199317273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( ( 1 + T )^{6} \) |
| good | 3 | \( 1 - p T^{2} + 16 T^{4} - 20 p T^{6} + 16 p^{2} T^{8} - p^{5} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 11 T^{2} + 180 T^{4} - 1104 T^{6} + 180 p^{2} T^{8} - 11 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{3} \) |
| 17 | \( 1 - 79 T^{2} + 2812 T^{4} - 59756 T^{6} + 2812 p^{2} T^{8} - 79 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 7 T + 44 T^{2} - 239 T^{3} + 44 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 90 T^{2} + 4015 T^{4} - 113723 T^{6} + 4015 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 10 T + 79 T^{2} + 379 T^{3} + 79 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 3 T + 40 T^{2} + 21 T^{3} + 40 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 156 T^{2} + 11472 T^{4} - 521170 T^{6} + 11472 p^{2} T^{8} - 156 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 4 T + 106 T^{2} - 304 T^{3} + 106 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 65 T^{2} + 6819 T^{4} - 247014 T^{6} + 6819 p^{2} T^{8} - 65 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 128 T^{2} + 10112 T^{4} - 555806 T^{6} + 10112 p^{2} T^{8} - 128 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 11 T^{2} - 1648 T^{4} - 41104 T^{6} - 1648 p^{2} T^{8} - 11 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 100 T^{2} + 192 T^{3} + 100 p T^{4} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + T + 142 T^{2} + 8 T^{3} + 142 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 174 T^{2} + 9175 T^{4} - 259588 T^{6} + 9175 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 17 T + 287 T^{2} + 2466 T^{3} + 287 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 39 T^{2} + 8052 T^{4} - 431260 T^{6} + 8052 p^{2} T^{8} - 39 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 11 T + 216 T^{2} - 1594 T^{3} + 216 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 110 T^{2} + 7715 T^{4} - 221443 T^{6} + 7715 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 30 T + 553 T^{2} + 6219 T^{3} + 553 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 418 T^{2} + 78987 T^{4} - 9282371 T^{6} + 78987 p^{2} T^{8} - 418 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.20876767843263932215344618773, −4.14098630866838841507693662101, −4.09243605855051445211902848660, −4.03533169277165593586457842204, −3.68592920835996253678116013585, −3.54117965806974895970938042529, −3.40504395423764516851178667167, −3.32027873370364428275945290571, −3.14116854626314402209580024666, −2.92117273973427781767713476079, −2.85878822126954387550321228036, −2.72023893005642295635903899752, −2.42783282516369637136825252636, −2.42004474059131924207054102446, −2.31797468548193660927195766532, −2.23725586533180640272723912658, −1.62421498171095563732141284565, −1.56866698180086515302174629208, −1.44777389603307342419452146209, −1.42598080874532912201091398744, −1.16525307819811943786882023230, −1.12308700942583563045841123194, −0.44713889193513969813333676381, −0.36777052186744322152624833373, −0.34877819004811475506153239958,
0.34877819004811475506153239958, 0.36777052186744322152624833373, 0.44713889193513969813333676381, 1.12308700942583563045841123194, 1.16525307819811943786882023230, 1.42598080874532912201091398744, 1.44777389603307342419452146209, 1.56866698180086515302174629208, 1.62421498171095563732141284565, 2.23725586533180640272723912658, 2.31797468548193660927195766532, 2.42004474059131924207054102446, 2.42783282516369637136825252636, 2.72023893005642295635903899752, 2.85878822126954387550321228036, 2.92117273973427781767713476079, 3.14116854626314402209580024666, 3.32027873370364428275945290571, 3.40504395423764516851178667167, 3.54117965806974895970938042529, 3.68592920835996253678116013585, 4.03533169277165593586457842204, 4.09243605855051445211902848660, 4.14098630866838841507693662101, 4.20876767843263932215344618773
Plot not available for L-functions of degree greater than 10.
