| L(s) = 1 | − 4·4-s − 6·7-s − 4·13-s + 8·16-s − 16·25-s + 24·28-s + 4·31-s + 8·37-s − 20·43-s + 21·49-s + 16·52-s − 16·61-s − 15·64-s + 24·67-s − 44·73-s − 8·79-s + 24·91-s − 16·97-s + 64·100-s − 8·103-s − 20·109-s − 48·112-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 2·4-s − 2.26·7-s − 1.10·13-s + 2·16-s − 3.19·25-s + 4.53·28-s + 0.718·31-s + 1.31·37-s − 3.04·43-s + 3·49-s + 2.21·52-s − 2.04·61-s − 1.87·64-s + 2.93·67-s − 5.14·73-s − 0.900·79-s + 2.51·91-s − 1.62·97-s + 32/5·100-s − 0.788·103-s − 1.91·109-s − 4.53·112-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 + T )^{6} \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p^{2} T^{2} + p^{3} T^{4} + 15 T^{6} + p^{5} T^{8} + p^{6} T^{10} + p^{6} T^{12} \) |
| 5 | \( 1 + 16 T^{2} + 28 p T^{4} + 834 T^{6} + 28 p^{3} T^{8} + 16 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 2 T + 20 T^{2} + 48 T^{3} + 20 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 + 54 T^{2} + 1199 T^{4} + 19316 T^{6} + 1199 p^{2} T^{8} + 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 20 T^{2} + 16 T^{3} + 20 p T^{4} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 + 30 T^{2} + 767 T^{4} + 11492 T^{6} + 767 p^{2} T^{8} + 30 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 + 52 T^{2} + 2892 T^{4} + 86614 T^{6} + 2892 p^{2} T^{8} + 52 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 2 T + 17 T^{2} + 180 T^{3} + 17 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 4 T + 96 T^{2} - 262 T^{3} + 96 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 74 T^{2} + 4479 T^{4} + 233228 T^{6} + 4479 p^{2} T^{8} + 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 10 T + 125 T^{2} + 828 T^{3} + 125 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 124 T^{2} + 11060 T^{4} + 591870 T^{6} + 11060 p^{2} T^{8} + 124 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 + 98 T^{2} + 8247 T^{4} + 518972 T^{6} + 8247 p^{2} T^{8} + 98 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( 1 + 116 T^{2} + 6924 T^{4} + 354662 T^{6} + 6924 p^{2} T^{8} + 116 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 8 T + 123 T^{2} + 704 T^{3} + 123 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( ( 1 - 12 T + 212 T^{2} - 1540 T^{3} + 212 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( 1 + 18 T^{2} + 6671 T^{4} - 156772 T^{6} + 6671 p^{2} T^{8} + 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( ( 1 + 22 T + 360 T^{2} + 3448 T^{3} + 360 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 + 4 T + 93 T^{2} - 8 T^{3} + 93 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 262 T^{2} + 27815 T^{4} + 2144628 T^{6} + 27815 p^{2} T^{8} + 262 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 + 430 T^{2} + 85071 T^{4} + 9709540 T^{6} + 85071 p^{2} T^{8} + 430 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( ( 1 + 8 T + 235 T^{2} + 1112 T^{3} + 235 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−4.50828283615525444723039872182, −4.18308472774341442118880275701, −4.03515566024006228144023148137, −3.95752538257071210501812986912, −3.92367820366977138491530830245, −3.77041440429644529380752963325, −3.76217520382456529349727985270, −3.37668536365208490872264852948, −3.30935333068302278114677354304, −3.18516327942281102377470237241, −3.17448734016587102989347309742, −3.14000006668427440973112615805, −2.71591268077093425033555685692, −2.59206584808179450413215240631, −2.46429841429380939934748907626, −2.40251344406843557970290076524, −2.29389571999042068484699310266, −2.17322665307442663212965261263, −1.87438598457899038394933658217, −1.64755656639938604479487576495, −1.30212401097663864130389979639, −1.29361731405575613553715967331, −1.17814293089732507111409359764, −1.11252404908101992539910229405, −0.805223434286125553378838092839, 0, 0, 0, 0, 0, 0,
0.805223434286125553378838092839, 1.11252404908101992539910229405, 1.17814293089732507111409359764, 1.29361731405575613553715967331, 1.30212401097663864130389979639, 1.64755656639938604479487576495, 1.87438598457899038394933658217, 2.17322665307442663212965261263, 2.29389571999042068484699310266, 2.40251344406843557970290076524, 2.46429841429380939934748907626, 2.59206584808179450413215240631, 2.71591268077093425033555685692, 3.14000006668427440973112615805, 3.17448734016587102989347309742, 3.18516327942281102377470237241, 3.30935333068302278114677354304, 3.37668536365208490872264852948, 3.76217520382456529349727985270, 3.77041440429644529380752963325, 3.92367820366977138491530830245, 3.95752538257071210501812986912, 4.03515566024006228144023148137, 4.18308472774341442118880275701, 4.50828283615525444723039872182
Plot not available for L-functions of degree greater than 10.
