| L(s) = 1 | + 2·3-s + 5·4-s + 9-s + 10·12-s + 11·16-s − 4·19-s + 2·27-s + 5·36-s − 40·43-s + 22·48-s + 24·49-s − 8·57-s + 15·64-s − 60·67-s + 12·73-s − 20·76-s + 2·81-s − 32·97-s + 10·108-s + 94·121-s + 127-s − 80·129-s + 131-s + 137-s + 139-s + 11·144-s + 48·147-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 5/2·4-s + 1/3·9-s + 2.88·12-s + 11/4·16-s − 0.917·19-s + 0.384·27-s + 5/6·36-s − 6.09·43-s + 3.17·48-s + 24/7·49-s − 1.05·57-s + 15/8·64-s − 7.33·67-s + 1.40·73-s − 2.29·76-s + 2/9·81-s − 3.24·97-s + 0.962·108-s + 8.54·121-s + 0.0887·127-s − 7.04·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.916·144-s + 3.95·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8364391890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8364391890\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 5 T^{2} + 7 p T^{4} - 15 p T^{6} + 7 p^{3} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 3 | \( ( 1 - T + T^{2} - 2 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 12 T^{2} + 120 T^{4} - 906 T^{6} + 120 p^{2} T^{8} - 12 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 - 47 T^{2} + 1091 T^{4} - 15090 T^{6} + 1091 p^{2} T^{8} - 47 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 - 20 T^{2} + 360 T^{4} - 6710 T^{6} + 360 p^{2} T^{8} - 20 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 - 67 T^{2} + 2131 T^{4} - 43546 T^{6} + 2131 p^{2} T^{8} - 67 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 + T + 2 p T^{2} + 63 T^{3} + 2 p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 23 | \( ( 1 + 34 T^{2} + 503 T^{4} + 2412 T^{6} + 503 p^{2} T^{8} + 34 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 + 34 T^{2} + 2127 T^{4} + 58156 T^{6} + 2127 p^{2} T^{8} + 34 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 88 T^{2} + 5232 T^{4} - 186430 T^{6} + 5232 p^{2} T^{8} - 88 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - 114 T^{2} + 7751 T^{4} - 344572 T^{6} + 7751 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 - 91 T^{2} + 6403 T^{4} - 299146 T^{6} + 6403 p^{2} T^{8} - 91 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 + 10 T + 100 T^{2} + 712 T^{3} + 100 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 47 | \( ( 1 + 70 T^{2} + 4799 T^{4} + 313140 T^{6} + 4799 p^{2} T^{8} + 70 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 + 38 T^{2} + 6351 T^{4} + 175844 T^{6} + 6351 p^{2} T^{8} + 38 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 - 214 T^{2} + 21991 T^{4} - 1515316 T^{6} + 21991 p^{2} T^{8} - 214 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 - 212 T^{2} + 24168 T^{4} - 1817654 T^{6} + 24168 p^{2} T^{8} - 212 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 + 15 T + 266 T^{2} + 2093 T^{3} + 266 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 71 | \( ( 1 + 190 T^{2} + 23367 T^{4} + 25964 p T^{6} + 23367 p^{2} T^{8} + 190 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 - 3 T + 131 T^{2} - 682 T^{3} + 131 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 79 | \( ( 1 - 126 T^{2} + 21023 T^{4} - 1571716 T^{6} + 21023 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 - 23 T + 209 T^{2} - 1486 T^{3} + 209 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} )^{2}( 1 + 23 T + 209 T^{2} + 1486 T^{3} + 209 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( ( 1 - 347 T^{2} + 57331 T^{4} - 6080690 T^{6} + 57331 p^{2} T^{8} - 347 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( ( 1 + 8 T + 192 T^{2} + 1310 T^{3} + 192 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.30759521036824084896309195642, −3.25596577512938004787154871854, −3.20095950132259066413469768720, −3.14181632568653571432203369754, −3.11688661049304838525538616355, −2.99878289579363774861597315236, −2.86297690941946191305457273602, −2.82436525113404361556022604229, −2.67024557781757052972530825506, −2.60270675642621597050104604013, −2.51123898959639264419875583446, −2.28129762558718865253730068691, −2.02766869069327549103996091332, −2.02384253399343085356714966890, −1.97131419488216359021610439019, −1.86698780667373821314737884337, −1.80328622107225989924124052729, −1.68958963855640806490396489584, −1.57385905653866789352655337793, −1.42509976847702703621453727802, −1.12266409584375841745951544379, −0.972909857315987642003599393187, −0.826948284132675442116671838772, −0.48960063557071366626937892985, −0.05919363940459307048160349351,
0.05919363940459307048160349351, 0.48960063557071366626937892985, 0.826948284132675442116671838772, 0.972909857315987642003599393187, 1.12266409584375841745951544379, 1.42509976847702703621453727802, 1.57385905653866789352655337793, 1.68958963855640806490396489584, 1.80328622107225989924124052729, 1.86698780667373821314737884337, 1.97131419488216359021610439019, 2.02384253399343085356714966890, 2.02766869069327549103996091332, 2.28129762558718865253730068691, 2.51123898959639264419875583446, 2.60270675642621597050104604013, 2.67024557781757052972530825506, 2.82436525113404361556022604229, 2.86297690941946191305457273602, 2.99878289579363774861597315236, 3.11688661049304838525538616355, 3.14181632568653571432203369754, 3.20095950132259066413469768720, 3.25596577512938004787154871854, 3.30759521036824084896309195642
Plot not available for L-functions of degree greater than 10.
