| L(s) = 1 | + 3·7-s + 8-s + 18·11-s + 8·27-s + 12·31-s + 12·37-s + 24·49-s + 3·56-s + 54·77-s + 18·83-s + 18·88-s − 42·103-s + 27·107-s + 36·113-s + 141·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 0.353·8-s + 5.42·11-s + 1.53·27-s + 2.15·31-s + 1.97·37-s + 24/7·49-s + 0.400·56-s + 6.15·77-s + 1.97·83-s + 1.91·88-s − 4.13·103-s + 2.61·107-s + 3.38·113-s + 12.8·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.56231170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.56231170\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{3} + T^{6} \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 8 T^{3} + 37 T^{6} - 8 p^{3} T^{9} + p^{6} T^{12} \) |
| 5 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 7 | \( ( 1 - 5 T + p T^{2} )^{3}( 1 + 4 T + p T^{2} )^{3} \) |
| 11 | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{3} \) |
| 13 | \( ( 1 - 89 T^{3} + p^{3} T^{6} )( 1 + 19 T^{3} + p^{3} T^{6} ) \) |
| 17 | \( 1 - 126 T^{3} + 10963 T^{6} - 126 p^{3} T^{9} + p^{6} T^{12} \) |
| 23 | \( 1 - 180 T^{3} + 20233 T^{6} - 180 p^{3} T^{9} + p^{6} T^{12} \) |
| 29 | \( 1 - 54 T^{3} - 21473 T^{6} - 54 p^{3} T^{9} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 11 T + p T^{2} )^{3}( 1 + 7 T + p T^{2} )^{3} \) |
| 37 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 41 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 449 T^{3} + p^{3} T^{6} )( 1 - 71 T^{3} + p^{3} T^{6} ) \) |
| 47 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 53 | \( 1 + 450 T^{3} + 53623 T^{6} + 450 p^{3} T^{9} + p^{6} T^{12} \) |
| 59 | \( 1 - 864 T^{3} + 541117 T^{6} - 864 p^{3} T^{9} + p^{6} T^{12} \) |
| 61 | \( 1 + 830 T^{3} + 461919 T^{6} + 830 p^{3} T^{9} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 1007 T^{3} + p^{3} T^{6} )( 1 + 127 T^{3} + p^{3} T^{6} ) \) |
| 71 | \( 1 + 1062 T^{3} + 769933 T^{6} + 1062 p^{3} T^{9} + p^{6} T^{12} \) |
| 73 | \( ( 1 + 271 T^{3} + p^{3} T^{6} )( 1 + 919 T^{3} + p^{3} T^{6} ) \) |
| 79 | \( 1 + 1370 T^{3} + 1383861 T^{6} + 1370 p^{3} T^{9} + p^{6} T^{12} \) |
| 83 | \( ( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{3} \) |
| 89 | \( 1 + 1476 T^{3} + 1473607 T^{6} + 1476 p^{3} T^{9} + p^{6} T^{12} \) |
| 97 | \( 1 + 1910 T^{3} + 2735427 T^{6} + 1910 p^{3} T^{9} + p^{6} T^{12} \) |
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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
−5.76256742501649678668128197245, −5.26048106056144150817686825500, −5.08690080424406260279723076106, −4.95952966688914636675416574951, −4.89989448932821614647582959161, −4.58873909472711908625287774867, −4.42820155366933605898206969031, −4.41375252937378254555112539149, −4.19821164256336056053636932056, −3.96171804786003108649136739563, −3.83427279337972830970364107300, −3.77765938451211476147794712054, −3.54176391018470446930257296287, −3.30010251058060033451872534468, −3.19386627895935498053782102512, −2.60994540278981947906347185091, −2.51046720756697611570296529813, −2.43278765033373228012444590842, −2.27413665242578004865924568394, −1.67451830837371868014513434309, −1.55818474055833538056788055642, −1.31200864098846274265031601308, −1.05798373296979343332913414203, −0.994987878445128122932214103324, −0.76654460221100648959069070568,
0.76654460221100648959069070568, 0.994987878445128122932214103324, 1.05798373296979343332913414203, 1.31200864098846274265031601308, 1.55818474055833538056788055642, 1.67451830837371868014513434309, 2.27413665242578004865924568394, 2.43278765033373228012444590842, 2.51046720756697611570296529813, 2.60994540278981947906347185091, 3.19386627895935498053782102512, 3.30010251058060033451872534468, 3.54176391018470446930257296287, 3.77765938451211476147794712054, 3.83427279337972830970364107300, 3.96171804786003108649136739563, 4.19821164256336056053636932056, 4.41375252937378254555112539149, 4.42820155366933605898206969031, 4.58873909472711908625287774867, 4.89989448932821614647582959161, 4.95952966688914636675416574951, 5.08690080424406260279723076106, 5.26048106056144150817686825500, 5.76256742501649678668128197245
Plot not available for L-functions of degree greater than 10.
