| L(s) = 1 | + 4-s + 6·7-s − 6·13-s + 4·16-s + 12·19-s − 11·25-s + 6·28-s + 8·31-s + 36·37-s + 8·43-s + 17·49-s − 6·52-s + 6·61-s − 4·64-s + 18·67-s − 50·73-s + 12·76-s + 14·79-s − 36·91-s − 4·97-s − 11·100-s + 36·103-s − 22·109-s + 24·112-s − 51·121-s + 8·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 2.26·7-s − 1.66·13-s + 16-s + 2.75·19-s − 2.19·25-s + 1.13·28-s + 1.43·31-s + 5.91·37-s + 1.21·43-s + 17/7·49-s − 0.832·52-s + 0.768·61-s − 1/2·64-s + 2.19·67-s − 5.85·73-s + 1.37·76-s + 1.57·79-s − 3.77·91-s − 0.406·97-s − 1.09·100-s + 3.54·103-s − 2.10·109-s + 2.26·112-s − 4.63·121-s + 0.718·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 61^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 61^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.689815571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.689815571\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 61 | \( ( 1 - T )^{6} \) |
| good | 2 | \( 1 - T^{2} - 3 T^{4} + 11 T^{6} - 3 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \) |
| 5 | \( 1 + 11 T^{2} + 3 p^{2} T^{4} + 386 T^{6} + 3 p^{4} T^{8} + 11 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( ( 1 - 3 T + 5 T^{2} + 2 T^{3} + 5 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 + 51 T^{2} + 1211 T^{4} + 16958 T^{6} + 1211 p^{2} T^{8} + 51 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 3 T + 23 T^{2} + 86 T^{3} + 23 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 + 46 T^{2} + 587 T^{4} + 3180 T^{6} + 587 p^{2} T^{8} + 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 23 | \( 1 + 123 T^{2} + 6611 T^{4} + 198302 T^{6} + 6611 p^{2} T^{8} + 123 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 + 114 T^{2} + 223 p T^{4} + 228980 T^{6} + 223 p^{3} T^{8} + 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 4 T + 41 T^{2} - 24 T^{3} + 41 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 6 T + p T^{2} )^{6} \) |
| 41 | \( 1 + 43 T^{2} - 957 T^{4} - 92942 T^{6} - 957 p^{2} T^{8} + 43 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 4 T + 77 T^{2} - 120 T^{3} + 77 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 90 T^{2} + 3119 T^{4} + 53036 T^{6} + 3119 p^{2} T^{8} + 90 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 + 134 T^{2} + 10995 T^{4} + 654716 T^{6} + 10995 p^{2} T^{8} + 134 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( 1 + 131 T^{2} + 15003 T^{4} + 939614 T^{6} + 15003 p^{2} T^{8} + 131 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 9 T + 209 T^{2} - 1202 T^{3} + 209 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( 1 + 314 T^{2} + 45531 T^{4} + 4012772 T^{6} + 45531 p^{2} T^{8} + 314 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( ( 1 + 25 T + 387 T^{2} + 3958 T^{3} + 387 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 - 7 T + 69 T^{2} + 266 T^{3} + 69 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 306 T^{2} + 45671 T^{4} + 4458716 T^{6} + 45671 p^{2} T^{8} + 306 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 + 10 T^{2} + 7835 T^{4} + 663684 T^{6} + 7835 p^{2} T^{8} + 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( ( 1 + 2 T + 235 T^{2} + 500 T^{3} + 235 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−5.75748030404101824158143723580, −5.66928947186166096402872804378, −5.42838489138277192712809759142, −5.31131358727908220104930526428, −5.11655011286013937734885156030, −5.05854168850768964543661048637, −4.71289286007625089052634028048, −4.37196724091690988009834608521, −4.32024818426201178375212262852, −4.29732818955649029275427708404, −4.25427615030360921636752815688, −4.04073712703547150764795151579, −3.52323902085538982889895996716, −3.25215316991035744241816225756, −3.21322312304271486246116447816, −2.88626513497237303181102357605, −2.65293858688259224293994083927, −2.59978565193927232097037340727, −2.30479490588331288532918105893, −2.06837510657152450052061989974, −1.85668965637112826909460164724, −1.33121782324095828013283306741, −1.25660723029185307138829580758, −0.944416013963063434902403668002, −0.62882740502267072103530069447,
0.62882740502267072103530069447, 0.944416013963063434902403668002, 1.25660723029185307138829580758, 1.33121782324095828013283306741, 1.85668965637112826909460164724, 2.06837510657152450052061989974, 2.30479490588331288532918105893, 2.59978565193927232097037340727, 2.65293858688259224293994083927, 2.88626513497237303181102357605, 3.21322312304271486246116447816, 3.25215316991035744241816225756, 3.52323902085538982889895996716, 4.04073712703547150764795151579, 4.25427615030360921636752815688, 4.29732818955649029275427708404, 4.32024818426201178375212262852, 4.37196724091690988009834608521, 4.71289286007625089052634028048, 5.05854168850768964543661048637, 5.11655011286013937734885156030, 5.31131358727908220104930526428, 5.42838489138277192712809759142, 5.66928947186166096402872804378, 5.75748030404101824158143723580
Plot not available for L-functions of degree greater than 10.
