| L(s) = 1 | − 2·2-s − 4-s − 2·7-s + 6·8-s + 4·14-s − 9·16-s − 7·25-s + 2·28-s − 16·29-s + 2·32-s − 34·41-s − 26·43-s − 19·49-s + 14·50-s − 36·53-s − 12·56-s + 32·58-s + 18·59-s − 6·61-s + 18·64-s + 16·71-s + 14·73-s + 68·82-s + 52·86-s − 24·89-s + 38·98-s + 7·100-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s − 0.755·7-s + 2.12·8-s + 1.06·14-s − 9/4·16-s − 7/5·25-s + 0.377·28-s − 2.97·29-s + 0.353·32-s − 5.30·41-s − 3.96·43-s − 2.71·49-s + 1.97·50-s − 4.94·53-s − 1.60·56-s + 4.20·58-s + 2.34·59-s − 0.768·61-s + 9/4·64-s + 1.89·71-s + 1.63·73-s + 7.50·82-s + 5.60·86-s − 2.54·89-s + 3.83·98-s + 7/10·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( ( 1 + T + p T^{2} + T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 5 | \( 1 + 7 T^{2} + 67 T^{4} + 269 T^{6} + 67 p^{2} T^{8} + 7 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( ( 1 + T + 11 T^{2} + 23 T^{3} + 11 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 + 14 T^{2} + 170 T^{4} + 2713 T^{6} + 170 p^{2} T^{8} + 14 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 - 5 T^{2} - 7 T^{4} + 3193 T^{6} - 7 p^{2} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 19 T^{2} + 514 T^{4} + 8795 T^{6} + 514 p^{2} T^{8} + 19 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 + 5 p T^{2} + 5971 T^{4} + 177029 T^{6} + 5971 p^{2} T^{8} + 5 p^{5} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 8 T + 104 T^{2} + 473 T^{3} + 104 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 + 54 T^{2} + 1966 T^{4} + 53305 T^{6} + 1966 p^{2} T^{8} + 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 + 141 T^{2} + 9679 T^{4} + 428785 T^{6} + 9679 p^{2} T^{8} + 141 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 17 T + 197 T^{2} + 1415 T^{3} + 197 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + 13 T + 159 T^{2} + 1109 T^{3} + 159 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 143 T^{2} + 230 p T^{4} + 567631 T^{6} + 230 p^{3} T^{8} + 143 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 18 T + 262 T^{2} + 2091 T^{3} + 262 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( ( 1 - 9 T + 159 T^{2} - 873 T^{3} + 159 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 3 T + 111 T^{2} + 285 T^{3} + 111 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 + 184 T^{2} + 16274 T^{4} + 1062565 T^{6} + 16274 p^{2} T^{8} + 184 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 8 T + 110 T^{2} - 305 T^{3} + 110 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 7 T + 102 T^{2} - 707 T^{3} + 102 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 + 213 T^{2} + 25459 T^{4} + 2130505 T^{6} + 25459 p^{2} T^{8} + 213 p^{4} T^{10} + p^{6} T^{12} \) |
| 83 | \( 1 + 359 T^{2} + 60994 T^{4} + 6297895 T^{6} + 60994 p^{2} T^{8} + 359 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 12 T + 166 T^{2} + 2091 T^{3} + 166 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 23 T^{2} + 19499 T^{4} - 463343 T^{6} + 19499 p^{2} T^{8} - 23 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.34352602584784208321745677188, −3.96544604772043268841255762144, −3.93655230807697657696002111121, −3.93354071676634866947817373272, −3.72183755348121810165542868684, −3.52720203376669324904647660208, −3.44832394580403494699963935622, −3.43670512157800846258977378415, −3.20198855738152936127545506090, −3.13237832977399780503177608414, −3.11052659019735919788954235531, −2.95966738470660116090975052792, −2.66952252154884128247882539786, −2.38357089930026607493303528096, −2.34435445272704022998928993880, −2.00874920233634041340819947977, −1.95440159274728351704004281567, −1.93701933710894850905468072146, −1.70180990726944340359554409610, −1.66370417949049815529026816360, −1.45690265960086518288850818768, −1.44918712450285266154355175516, −1.13176376167595163621203561586, −0.943718917888662980396155980888, −0.64752456077565852396771428068, 0, 0, 0, 0, 0, 0,
0.64752456077565852396771428068, 0.943718917888662980396155980888, 1.13176376167595163621203561586, 1.44918712450285266154355175516, 1.45690265960086518288850818768, 1.66370417949049815529026816360, 1.70180990726944340359554409610, 1.93701933710894850905468072146, 1.95440159274728351704004281567, 2.00874920233634041340819947977, 2.34435445272704022998928993880, 2.38357089930026607493303528096, 2.66952252154884128247882539786, 2.95966738470660116090975052792, 3.11052659019735919788954235531, 3.13237832977399780503177608414, 3.20198855738152936127545506090, 3.43670512157800846258977378415, 3.44832394580403494699963935622, 3.52720203376669324904647660208, 3.72183755348121810165542868684, 3.93354071676634866947817373272, 3.93655230807697657696002111121, 3.96544604772043268841255762144, 4.34352602584784208321745677188
Plot not available for L-functions of degree greater than 10.
