| L(s) = 1 | − 3·3-s − 3·7-s + 3·9-s − 6·13-s − 6·17-s − 3·19-s + 9·21-s + 6·23-s + 6·25-s + 2·27-s + 24·29-s − 3·31-s − 3·37-s + 18·39-s − 12·41-s + 30·43-s − 12·47-s + 9·49-s + 18·51-s − 6·53-s + 9·57-s − 12·59-s + 18·61-s − 9·63-s − 9·67-s − 18·69-s − 33·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.13·7-s + 9-s − 1.66·13-s − 1.45·17-s − 0.688·19-s + 1.96·21-s + 1.25·23-s + 6/5·25-s + 0.384·27-s + 4.45·29-s − 0.538·31-s − 0.493·37-s + 2.88·39-s − 1.87·41-s + 4.57·43-s − 1.75·47-s + 9/7·49-s + 2.52·51-s − 0.824·53-s + 1.19·57-s − 1.56·59-s + 2.30·61-s − 1.13·63-s − 1.09·67-s − 2.16·69-s − 3.86·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.258619785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.258619785\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T + T^{2} )^{3} \) |
| 7 | \( 1 + 3 T - 5 T^{3} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 5 | \( 1 - 6 T^{2} + 8 T^{3} + 6 T^{4} - 24 T^{5} + 86 T^{6} - 24 p T^{7} + 6 p^{2} T^{8} + 8 p^{3} T^{9} - 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 6 T^{2} + 76 T^{3} - 30 T^{4} - 228 T^{5} + 3710 T^{6} - 228 p T^{7} - 30 p^{2} T^{8} + 76 p^{3} T^{9} - 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 3 T + 3 T^{2} - 34 T^{3} + 3 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 + 6 T + 9 T^{2} - 54 T^{3} - 378 T^{4} - 858 T^{5} - 1307 T^{6} - 858 p T^{7} - 378 p^{2} T^{8} - 54 p^{3} T^{9} + 9 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 + 3 T - 12 T^{2} + 59 T^{3} + 36 T^{4} - 1269 T^{5} + 2094 T^{6} - 1269 p T^{7} + 36 p^{2} T^{8} + 59 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 6 T - 9 T^{2} + 90 T^{3} - 90 T^{4} + 1146 T^{5} - 8885 T^{6} + 1146 p T^{7} - 90 p^{2} T^{8} + 90 p^{3} T^{9} - 9 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 12 T + 126 T^{2} - 728 T^{3} + 126 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 + 3 T - 63 T^{2} - 2 p T^{3} + 2535 T^{4} - 501 T^{5} - 90450 T^{6} - 501 p T^{7} + 2535 p^{2} T^{8} - 2 p^{4} T^{9} - 63 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 + 3 T - 18 T^{2} + 373 T^{3} + 168 T^{4} - 5829 T^{5} + 83328 T^{6} - 5829 p T^{7} + 168 p^{2} T^{8} + 373 p^{3} T^{9} - 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 6 T + 27 T^{2} - 20 T^{3} + 27 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 - 15 T + 177 T^{2} - 1318 T^{3} + 177 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 12 T - 9 T^{2} - 196 T^{3} + 4026 T^{4} + 9372 T^{5} - 178417 T^{6} + 9372 p T^{7} + 4026 p^{2} T^{8} - 196 p^{3} T^{9} - 9 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + 6 T - 42 T^{2} - 1136 T^{3} - 54 p T^{4} + 29802 T^{5} + 517382 T^{6} + 29802 p T^{7} - 54 p^{3} T^{8} - 1136 p^{3} T^{9} - 42 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 12 T - 54 T^{2} - 292 T^{3} + 11514 T^{4} + 25536 T^{5} - 623986 T^{6} + 25536 p T^{7} + 11514 p^{2} T^{8} - 292 p^{3} T^{9} - 54 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 18 T + 129 T^{2} - 222 T^{3} - 3462 T^{4} + 40662 T^{5} - 368431 T^{6} + 40662 p T^{7} - 3462 p^{2} T^{8} - 222 p^{3} T^{9} + 129 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 9 T - 108 T^{2} - 383 T^{3} + 13680 T^{4} + 9405 T^{5} - 1069626 T^{6} + 9405 p T^{7} + 13680 p^{2} T^{8} - 383 p^{3} T^{9} - 108 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 177 T^{2} - 32 T^{3} + 177 p T^{4} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 33 T + 534 T^{2} + 6687 T^{3} + 75648 T^{4} + 728637 T^{5} + 6291524 T^{6} + 728637 p T^{7} + 75648 p^{2} T^{8} + 6687 p^{3} T^{9} + 534 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 27 T + 297 T^{2} - 2426 T^{3} + 27783 T^{4} - 321003 T^{5} + 3028254 T^{6} - 321003 p T^{7} + 27783 p^{2} T^{8} - 2426 p^{3} T^{9} + 297 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( ( 1 - 18 T + 234 T^{2} - 2040 T^{3} + 234 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 12 T - 135 T^{2} + 700 T^{3} + 28722 T^{4} - 63804 T^{5} - 2561959 T^{6} - 63804 p T^{7} + 28722 p^{2} T^{8} + 700 p^{3} T^{9} - 135 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( ( 1 + 264 T^{2} + 38 T^{3} + 264 p T^{4} + p^{3} T^{6} )^{2} \) |
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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
−5.66920526721769046640309414376, −5.41678743265006061890962401122, −5.31633730872631031441914339665, −5.00845291865478558062044036231, −4.79129624247512716307319495920, −4.78046629980915590893143624098, −4.68282486761253215789688117910, −4.64282702262763600430699875355, −4.48327230957096945319915433144, −4.06182769229584032342078039341, −3.98123892043118129213566176476, −3.56365676583528729550983293537, −3.48923396564009907179769146765, −3.35607417998774801754694280030, −2.88584702042626497665860878084, −2.73117626653799877063791609869, −2.63402218813191748297696399293, −2.58745876350191324964081149448, −2.43443398385433632526537736872, −1.85241022554694451685908184267, −1.64392056228792586546819303860, −1.36593705110594339848532864332, −0.72098421832338440150283413748, −0.56509583830694083975825042947, −0.50764102534434127834495587329,
0.50764102534434127834495587329, 0.56509583830694083975825042947, 0.72098421832338440150283413748, 1.36593705110594339848532864332, 1.64392056228792586546819303860, 1.85241022554694451685908184267, 2.43443398385433632526537736872, 2.58745876350191324964081149448, 2.63402218813191748297696399293, 2.73117626653799877063791609869, 2.88584702042626497665860878084, 3.35607417998774801754694280030, 3.48923396564009907179769146765, 3.56365676583528729550983293537, 3.98123892043118129213566176476, 4.06182769229584032342078039341, 4.48327230957096945319915433144, 4.64282702262763600430699875355, 4.68282486761253215789688117910, 4.78046629980915590893143624098, 4.79129624247512716307319495920, 5.00845291865478558062044036231, 5.31633730872631031441914339665, 5.41678743265006061890962401122, 5.66920526721769046640309414376
Plot not available for L-functions of degree greater than 10.
