| L(s) = 1 | − 3·5-s − 3·7-s − 6·11-s + 3·13-s + 6·19-s − 6·23-s + 9·25-s − 15·29-s + 3·31-s + 9·35-s − 6·37-s − 6·41-s − 3·43-s − 15·47-s + 3·49-s + 36·53-s + 18·55-s − 3·59-s + 6·61-s − 9·65-s + 6·67-s + 30·71-s + 18·73-s + 18·77-s − 3·79-s − 18·83-s − 12·89-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.13·7-s − 1.80·11-s + 0.832·13-s + 1.37·19-s − 1.25·23-s + 9/5·25-s − 2.78·29-s + 0.538·31-s + 1.52·35-s − 0.986·37-s − 0.937·41-s − 0.457·43-s − 2.18·47-s + 3/7·49-s + 4.94·53-s + 2.42·55-s − 0.390·59-s + 0.768·61-s − 1.11·65-s + 0.733·67-s + 3.56·71-s + 2.10·73-s + 2.05·77-s − 0.337·79-s − 1.97·83-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4694354368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4694354368\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + T + T^{2} )^{3} \) |
| good | 5 | \( 1 + 3 T - 3 p T^{3} - 27 T^{4} - 6 T^{5} + 61 T^{6} - 6 p T^{7} - 27 p^{2} T^{8} - 3 p^{4} T^{9} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 + 6 T - 30 T^{3} + 162 T^{4} + 402 T^{5} - 821 T^{6} + 402 p T^{7} + 162 p^{2} T^{8} - 30 p^{3} T^{9} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \) |
| 17 | \( ( 1 + 18 T^{2} - 9 T^{3} + 18 p T^{4} + p^{3} T^{6} )^{2} \) |
| 19 | \( ( 1 - 3 T + 21 T^{2} - 65 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 + 6 T - 18 T^{2} - 30 T^{3} + 612 T^{4} - 2172 T^{5} - 30449 T^{6} - 2172 p T^{7} + 612 p^{2} T^{8} - 30 p^{3} T^{9} - 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 15 T + 90 T^{2} + 411 T^{3} + 2205 T^{4} + 4110 T^{5} - 17723 T^{6} + 4110 p T^{7} + 2205 p^{2} T^{8} + 411 p^{3} T^{9} + 90 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 - 3 T - 6 T^{2} + 221 T^{3} - 639 T^{4} - 2088 T^{5} + 61647 T^{6} - 2088 p T^{7} - 639 p^{2} T^{8} + 221 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 3 T + 33 T^{2} + 115 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 6 T - 90 T^{2} - 210 T^{3} + 7812 T^{4} + 8952 T^{5} - 340301 T^{6} + 8952 p T^{7} + 7812 p^{2} T^{8} - 210 p^{3} T^{9} - 90 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 3 T - 12 T^{2} + 11 p T^{3} + 153 T^{4} - 4176 T^{5} + 165435 T^{6} - 4176 p T^{7} + 153 p^{2} T^{8} + 11 p^{4} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 + 15 T + 48 T^{2} - 3 T^{3} + 3075 T^{4} + 18798 T^{5} + 4399 T^{6} + 18798 p T^{7} + 3075 p^{2} T^{8} - 3 p^{3} T^{9} + 48 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 18 T + 198 T^{2} - 1521 T^{3} + 198 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 + 3 T - 114 T^{2} - 501 T^{3} + 6567 T^{4} + 20406 T^{5} - 323957 T^{6} + 20406 p T^{7} + 6567 p^{2} T^{8} - 501 p^{3} T^{9} - 114 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 6 T - 102 T^{2} + 698 T^{3} + 6048 T^{4} - 26604 T^{5} - 259509 T^{6} - 26604 p T^{7} + 6048 p^{2} T^{8} + 698 p^{3} T^{9} - 102 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 6 T - 138 T^{2} + 446 T^{3} + 14148 T^{4} - 16668 T^{5} - 1033545 T^{6} - 16668 p T^{7} + 14148 p^{2} T^{8} + 446 p^{3} T^{9} - 138 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 15 T + 231 T^{2} - 1833 T^{3} + 231 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 9 T + 207 T^{2} - 1235 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 + 3 T + 6 T^{2} + 1787 T^{3} + 2439 T^{4} + 14634 T^{5} + 1719519 T^{6} + 14634 p T^{7} + 2439 p^{2} T^{8} + 1787 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 + 18 T - 198 T^{3} + 23814 T^{4} + 132750 T^{5} - 711245 T^{6} + 132750 p T^{7} + 23814 p^{2} T^{8} - 198 p^{3} T^{9} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 6 T + 72 T^{2} - 21 T^{3} + 72 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 15 T - 84 T^{2} + 1139 T^{3} + 22203 T^{4} - 134028 T^{5} - 1218567 T^{6} - 134028 p T^{7} + 22203 p^{2} T^{8} + 1139 p^{3} T^{9} - 84 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−5.52064804373471982670011037144, −5.23525044695191013907422486005, −5.22245464917787869270493972152, −5.10166165030897111845489729637, −5.02162545401017479031138444263, −4.64316494496575662151854331775, −4.46456138682150318556744421009, −4.27351400976625973180300252341, −4.05241152674397150649007101481, −3.86035194981524524969622361774, −3.57619716120214753988244862522, −3.56012379800345751299086077632, −3.52953455567551157675310852909, −3.45251340906088493338573930749, −2.94284488232426332824125288468, −2.83202415652949347191092519160, −2.78223821277526226933979136015, −2.20948986306317523231398171639, −2.20555669123274766998825540203, −2.00685068889002740864156113407, −1.80525521284261654280776356592, −1.16255006564115699392039085438, −1.01942786560611794143905654828, −0.58579216665042322515851451507, −0.18485275232889787345606650173,
0.18485275232889787345606650173, 0.58579216665042322515851451507, 1.01942786560611794143905654828, 1.16255006564115699392039085438, 1.80525521284261654280776356592, 2.00685068889002740864156113407, 2.20555669123274766998825540203, 2.20948986306317523231398171639, 2.78223821277526226933979136015, 2.83202415652949347191092519160, 2.94284488232426332824125288468, 3.45251340906088493338573930749, 3.52953455567551157675310852909, 3.56012379800345751299086077632, 3.57619716120214753988244862522, 3.86035194981524524969622361774, 4.05241152674397150649007101481, 4.27351400976625973180300252341, 4.46456138682150318556744421009, 4.64316494496575662151854331775, 5.02162545401017479031138444263, 5.10166165030897111845489729637, 5.22245464917787869270493972152, 5.23525044695191013907422486005, 5.52064804373471982670011037144
Plot not available for L-functions of degree greater than 10.
