| L(s) = 1 | + 6·9-s − 6·13-s + 12·29-s − 24·37-s − 24·47-s − 18·49-s − 12·61-s − 24·67-s + 48·73-s + 24·79-s + 9·81-s − 24·83-s + 36·97-s + 12·101-s − 36·117-s + 30·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9·169-s + 173-s + ⋯ |
| L(s) = 1 | + 2·9-s − 1.66·13-s + 2.22·29-s − 3.94·37-s − 3.50·47-s − 2.57·49-s − 1.53·61-s − 2.93·67-s + 5.61·73-s + 2.70·79-s + 81-s − 2.63·83-s + 3.65·97-s + 1.19·101-s − 3.32·117-s + 2.72·121-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 + 9/13·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.564405119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.564405119\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 6 T + 27 T^{2} + 132 T^{3} + 27 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 3 | \( 1 - 2 p T^{2} + p^{3} T^{4} - 104 T^{6} + p^{5} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 7 | \( ( 1 + 9 T^{2} - 12 T^{3} + 9 p T^{4} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 - 30 T^{2} + 447 T^{4} - 4912 T^{6} + 447 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 - p T^{2} )^{6} \) |
| 19 | \( 1 - 18 T^{2} + 423 T^{4} - 200 p T^{6} + 423 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 42 T^{2} + 2091 T^{4} - 45808 T^{6} + 2091 p^{2} T^{8} - 42 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 6 T + 75 T^{2} - 264 T^{3} + 75 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 126 T^{2} + 7911 T^{4} - 303536 T^{6} + 7911 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 12 T + 3 p T^{2} + 636 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - 138 T^{2} + 9663 T^{4} - 461068 T^{6} + 9663 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( 1 - 138 T^{2} + 9411 T^{4} - 455920 T^{6} + 9411 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 + 12 T + 105 T^{2} + 660 T^{3} + 105 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( 1 - 114 T^{2} + 5655 T^{4} - 256156 T^{6} + 5655 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( 1 - 138 T^{2} + 13767 T^{4} - 903112 T^{6} + 13767 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 6 T + 87 T^{2} + 200 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( ( 1 + 12 T + 153 T^{2} + 1020 T^{3} + 153 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( 1 + 6 T^{2} - 417 T^{4} - 224296 T^{6} - 417 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( ( 1 - 24 T + 363 T^{2} - 3756 T^{3} + 363 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 - 12 T + 93 T^{2} - 200 T^{3} + 93 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 + 12 T + 3 p T^{2} + 1740 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 + 42 T^{2} + 10527 T^{4} + 115148 T^{6} + 10527 p^{2} T^{8} + 42 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( ( 1 - 18 T + 351 T^{2} - 3516 T^{3} + 351 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−5.00499929596449118904564439067, −4.82570427786468677879966646169, −4.73100533158260182350020353396, −4.70943761193993475291737210290, −4.61453352908165580185550950457, −4.45182463084605406932170030987, −4.21844171348275325713469279904, −3.75020927033707863401052893757, −3.73825496080945422225693934581, −3.59827462987004916157621817741, −3.44932128369986632656163607875, −3.32660745836864565218338221717, −3.19889958733815877689541497328, −2.96222359943498307246660196235, −2.68989525763417544475380019086, −2.43814277599021283746440751149, −2.24704616834037225355164745271, −2.11677515441819475717118098002, −1.88563042287382903623469608333, −1.63202583425899656706702994114, −1.39105679110539805771836358157, −1.38700541683735393516857829277, −1.05399794786229604523363617308, −0.47209481027513463578094777977, −0.21462817197938629799587231115,
0.21462817197938629799587231115, 0.47209481027513463578094777977, 1.05399794786229604523363617308, 1.38700541683735393516857829277, 1.39105679110539805771836358157, 1.63202583425899656706702994114, 1.88563042287382903623469608333, 2.11677515441819475717118098002, 2.24704616834037225355164745271, 2.43814277599021283746440751149, 2.68989525763417544475380019086, 2.96222359943498307246660196235, 3.19889958733815877689541497328, 3.32660745836864565218338221717, 3.44932128369986632656163607875, 3.59827462987004916157621817741, 3.73825496080945422225693934581, 3.75020927033707863401052893757, 4.21844171348275325713469279904, 4.45182463084605406932170030987, 4.61453352908165580185550950457, 4.70943761193993475291737210290, 4.73100533158260182350020353396, 4.82570427786468677879966646169, 5.00499929596449118904564439067
Plot not available for L-functions of degree greater than 10.
