| L(s) = 1 | − 8·4-s + 52·13-s − 69·16-s − 62·19-s − 398·25-s − 82·31-s − 1.13e3·37-s − 1.56e3·43-s − 416·52-s + 886·61-s + 766·64-s − 2.08e3·67-s − 2.39e3·73-s + 496·76-s + 984·79-s − 682·97-s + 3.18e3·100-s − 3.02e3·103-s − 2.47e3·109-s − 1.94e3·121-s + 656·124-s + 127-s + 131-s + 137-s + 139-s + 9.05e3·148-s + 149-s + ⋯ |
| L(s) = 1 | − 4-s + 1.10·13-s − 1.07·16-s − 0.748·19-s − 3.18·25-s − 0.475·31-s − 5.02·37-s − 5.55·43-s − 1.10·52-s + 1.85·61-s + 1.49·64-s − 3.80·67-s − 3.84·73-s + 0.748·76-s + 1.40·79-s − 0.713·97-s + 3.18·100-s − 2.89·103-s − 2.17·109-s − 1.46·121-s + 0.475·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 5.02·148-s + 0.000549·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p^{3} T^{2} + 133 T^{4} + 425 p T^{6} + 133 p^{6} T^{8} + p^{15} T^{10} + p^{18} T^{12} \) |
| 5 | \( 1 + 398 T^{2} + 88267 T^{4} + 13515052 T^{6} + 88267 p^{6} T^{8} + 398 p^{12} T^{10} + p^{18} T^{12} \) |
| 11 | \( 1 + 1946 T^{2} + 1291819 T^{4} - 782053340 T^{6} + 1291819 p^{6} T^{8} + 1946 p^{12} T^{10} + p^{18} T^{12} \) |
| 13 | \( ( 1 - 2 p T + 20 p^{2} T^{2} - 150062 T^{3} + 20 p^{5} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} )^{2} \) |
| 17 | \( 1 + 19638 T^{2} + 193457043 T^{4} + 1185167951260 T^{6} + 193457043 p^{6} T^{8} + 19638 p^{12} T^{10} + p^{18} T^{12} \) |
| 19 | \( ( 1 + 31 T + 11177 T^{2} + 354214 T^{3} + 11177 p^{3} T^{4} + 31 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 + 26874 T^{2} + 137157615 T^{4} - 1070796212084 T^{6} + 137157615 p^{6} T^{8} + 26874 p^{12} T^{10} + p^{18} T^{12} \) |
| 29 | \( 1 + 107486 T^{2} + 5558833147 T^{4} + 171068655050380 T^{6} + 5558833147 p^{6} T^{8} + 107486 p^{12} T^{10} + p^{18} T^{12} \) |
| 31 | \( ( 1 + 41 T + 55666 T^{2} + 237883 T^{3} + 55666 p^{3} T^{4} + 41 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( ( 1 + 566 T + 246688 T^{2} + 61584934 T^{3} + 246688 p^{3} T^{4} + 566 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 41 | \( 1 + 201446 T^{2} + 24090872899 T^{4} + 1992163622083516 T^{6} + 24090872899 p^{6} T^{8} + 201446 p^{12} T^{10} + p^{18} T^{12} \) |
| 43 | \( ( 1 + 783 T + 433368 T^{2} + 140027659 T^{3} + 433368 p^{3} T^{4} + 783 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 47 | \( 1 + 127818 T^{2} + 18065263059 T^{4} + 2679796621113892 T^{6} + 18065263059 p^{6} T^{8} + 127818 p^{12} T^{10} + p^{18} T^{12} \) |
| 53 | \( 1 + 680526 T^{2} + 219567635127 T^{4} + 41561598578306884 T^{6} + 219567635127 p^{6} T^{8} + 680526 p^{12} T^{10} + p^{18} T^{12} \) |
| 59 | \( 1 + 252810 T^{2} + 118459786923 T^{4} + 17779226383468420 T^{6} + 118459786923 p^{6} T^{8} + 252810 p^{12} T^{10} + p^{18} T^{12} \) |
| 61 | \( ( 1 - 443 T + 203324 T^{2} + 18833347 T^{3} + 203324 p^{3} T^{4} - 443 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 67 | \( ( 1 + 1042 T + 867758 T^{2} + 445253716 T^{3} + 867758 p^{3} T^{4} + 1042 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 71 | \( 1 + 1054578 T^{2} + 491902090755 T^{4} + 2420618974688108 p T^{6} + 491902090755 p^{6} T^{8} + 1054578 p^{12} T^{10} + p^{18} T^{12} \) |
| 73 | \( ( 1 + 1199 T + 1316443 T^{2} + 794269678 T^{3} + 1316443 p^{3} T^{4} + 1199 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( ( 1 - 492 T + 608946 T^{2} - 546599468 T^{3} + 608946 p^{3} T^{4} - 492 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( 1 - 834486 T^{2} + 1207789948167 T^{4} - 565854790583996660 T^{6} + 1207789948167 p^{6} T^{8} - 834486 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( 1 + 949814 T^{2} + 691701373123 T^{4} + 216133792389242716 T^{6} + 691701373123 p^{6} T^{8} + 949814 p^{12} T^{10} + p^{18} T^{12} \) |
| 97 | \( ( 1 + 341 T + 490090 T^{2} - 960147395 T^{3} + 490090 p^{3} T^{4} + 341 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
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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
−5.26294724029638900132206161947, −4.85451331151224034712813048845, −4.73078665708286700704571438026, −4.65583043705429639970561919243, −4.49871053815137030683181201711, −4.46231177138341515217118411778, −4.25078958207615032378551319388, −3.82758166150088820172536582103, −3.78121247416154817245925447508, −3.76514500587472660682676713347, −3.59123432968280780572248451908, −3.38065834222808802547182986241, −3.36458530371851516773485364460, −3.13131109833473487007797049253, −2.86373177777671029302926768517, −2.59047028250162005532818507251, −2.48602901327919660557542862417, −2.03878263989753877803479575372, −2.01668180266059204247380194139, −1.95461313810555395240566264092, −1.60981621940311158152830876499, −1.52633736250609528190850799819, −1.33146412243848291458767386932, −1.19670716109598163394732490012, −0.900100721732320072677139453255, 0, 0, 0, 0, 0, 0,
0.900100721732320072677139453255, 1.19670716109598163394732490012, 1.33146412243848291458767386932, 1.52633736250609528190850799819, 1.60981621940311158152830876499, 1.95461313810555395240566264092, 2.01668180266059204247380194139, 2.03878263989753877803479575372, 2.48602901327919660557542862417, 2.59047028250162005532818507251, 2.86373177777671029302926768517, 3.13131109833473487007797049253, 3.36458530371851516773485364460, 3.38065834222808802547182986241, 3.59123432968280780572248451908, 3.76514500587472660682676713347, 3.78121247416154817245925447508, 3.82758166150088820172536582103, 4.25078958207615032378551319388, 4.46231177138341515217118411778, 4.49871053815137030683181201711, 4.65583043705429639970561919243, 4.73078665708286700704571438026, 4.85451331151224034712813048845, 5.26294724029638900132206161947
Plot not available for L-functions of degree greater than 10.
