| L(s) = 1 | + 32·25-s + 8·37-s + 16·49-s + 24·61-s − 56·73-s + 32·97-s − 32·109-s − 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 32/5·25-s + 1.31·37-s + 16/7·49-s + 3.07·61-s − 6.55·73-s + 3.24·97-s − 3.06·109-s − 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(21.23878802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(21.23878802\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( ( 1 + T^{2} )^{6} \) |
| good | 5 | \( ( 1 - 16 T^{2} + 112 T^{4} - 568 T^{6} + 112 p^{2} T^{8} - 16 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 7 | \( ( 1 - 8 T^{2} + 88 T^{4} - 734 T^{6} + 88 p^{2} T^{8} - 8 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 + 6 T^{2} + 200 T^{4} + 610 T^{6} + 200 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 + 11 T^{2} - 32 T^{3} + 11 p T^{4} + p^{3} T^{6} )^{4} \) |
| 17 | \( ( 1 - 56 T^{2} + 1336 T^{4} - 22832 T^{6} + 1336 p^{2} T^{8} - 56 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 + 86 T^{2} + 3731 T^{4} + 104172 T^{6} + 3731 p^{2} T^{8} + 86 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 - 140 T^{2} + 8783 T^{4} - 322680 T^{6} + 8783 p^{2} T^{8} - 140 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 82 T^{2} + 3839 T^{4} - 133596 T^{6} + 3839 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - 2 T + 47 T^{2} - 180 T^{3} + 47 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 41 | \( ( 1 - 4 p T^{2} + 13303 T^{4} - 669128 T^{6} + 13303 p^{2} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 - 124 T^{2} + 9112 T^{4} - 444094 T^{6} + 9112 p^{2} T^{8} - 124 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 + 30 T^{2} + 112 p T^{4} + 58 p^{2} T^{6} + 112 p^{3} T^{8} + 30 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 - 76 T^{2} + 6367 T^{4} - 345976 T^{6} + 6367 p^{2} T^{8} - 76 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 + 82 T^{2} + 7543 T^{4} + 525148 T^{6} + 7543 p^{2} T^{8} + 82 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 - 6 T + 188 T^{2} - 722 T^{3} + 188 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 67 | \( ( 1 - 226 T^{2} + 26887 T^{4} - 2149756 T^{6} + 26887 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 + 218 T^{2} + 25823 T^{4} + 2133612 T^{6} + 25823 p^{2} T^{8} + 218 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 + 14 T + 268 T^{2} + 2060 T^{3} + 268 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 79 | \( ( 1 + 6 T^{2} + 7535 T^{4} + 308500 T^{6} + 7535 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 + 446 T^{2} + 86651 T^{4} + 9381852 T^{6} + 86651 p^{2} T^{8} + 446 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 - 228 T^{2} + 34503 T^{4} - 3356296 T^{6} + 34503 p^{2} T^{8} - 228 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( ( 1 - 8 T + 231 T^{2} - 1088 T^{3} + 231 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
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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
−2.62742486919491608503829380071, −2.58610348832327831837848310615, −2.56380837113565193380784083861, −2.43363409367303537087961116381, −2.39706324109275703007554516488, −2.30675557477197933864423597087, −2.23198429022216757063696137202, −2.20029965995739865112854486463, −2.13404355094132444202439920151, −1.86688738267607260649461388316, −1.60351589746959004142211061647, −1.55145759084189284227040603549, −1.55102708950185427130789848334, −1.51411235488639860060150225230, −1.46935983266311121854571825014, −1.15852917487542069099578525480, −1.10175439944812617385728847983, −1.02795777773014362923491531583, −0.983495118200961749660523739667, −0.925805280093667973519141595685, −0.836740506879374561629083131231, −0.58275200371131209268395488430, −0.33135807270013148906587928588, −0.28993748648632634447606972130, −0.25211806297622808022463856519,
0.25211806297622808022463856519, 0.28993748648632634447606972130, 0.33135807270013148906587928588, 0.58275200371131209268395488430, 0.836740506879374561629083131231, 0.925805280093667973519141595685, 0.983495118200961749660523739667, 1.02795777773014362923491531583, 1.10175439944812617385728847983, 1.15852917487542069099578525480, 1.46935983266311121854571825014, 1.51411235488639860060150225230, 1.55102708950185427130789848334, 1.55145759084189284227040603549, 1.60351589746959004142211061647, 1.86688738267607260649461388316, 2.13404355094132444202439920151, 2.20029965995739865112854486463, 2.23198429022216757063696137202, 2.30675557477197933864423597087, 2.39706324109275703007554516488, 2.43363409367303537087961116381, 2.56380837113565193380784083861, 2.58610348832327831837848310615, 2.62742486919491608503829380071
Plot not available for L-functions of degree greater than 10.
