| L(s) = 1 | − 15·4-s + 24·5-s + 69·16-s + 42·17-s − 360·20-s + 24·25-s − 312·37-s + 360·41-s + 654·43-s + 1.81e3·47-s − 6·59-s + 141·64-s + 42·67-s − 630·68-s + 1.95e3·79-s + 1.65e3·80-s + 2.89e3·83-s + 1.00e3·85-s + 1.51e3·89-s − 360·100-s − 456·101-s + 2.55e3·109-s − 2.40e3·121-s − 1.23e3·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.87·4-s + 2.14·5-s + 1.07·16-s + 0.599·17-s − 4.02·20-s + 0.191·25-s − 1.38·37-s + 1.37·41-s + 2.31·43-s + 5.62·47-s − 0.0132·59-s + 0.275·64-s + 0.0765·67-s − 1.12·68-s + 2.78·79-s + 2.31·80-s + 3.82·83-s + 1.28·85-s + 1.80·89-s − 0.359·100-s − 0.449·101-s + 2.24·109-s − 1.80·121-s − 0.884·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(20.24341689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(20.24341689\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 15 T^{2} + 39 p^{2} T^{4} + 291 p^{2} T^{6} + 39 p^{8} T^{8} + 15 p^{12} T^{10} + p^{18} T^{12} \) |
| 5 | \( ( 1 - 12 T + 204 T^{2} - 654 p T^{3} + 204 p^{3} T^{4} - 12 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 11 | \( 1 + 2406 T^{2} + 2486343 T^{4} + 1930638516 T^{6} + 2486343 p^{6} T^{8} + 2406 p^{12} T^{10} + p^{18} T^{12} \) |
| 13 | \( 1 + 4677 T^{2} + 15059643 T^{4} + 218103758 p^{2} T^{6} + 15059643 p^{6} T^{8} + 4677 p^{12} T^{10} + p^{18} T^{12} \) |
| 17 | \( ( 1 - 21 T + 9654 T^{2} - 298065 T^{3} + 9654 p^{3} T^{4} - 21 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 19 | \( 1 + 10698 T^{2} + 33088839 T^{4} - 140201322100 T^{6} + 33088839 p^{6} T^{8} + 10698 p^{12} T^{10} + p^{18} T^{12} \) |
| 23 | \( 1 + 50265 T^{2} + 1230954627 T^{4} + 18571227575238 T^{6} + 1230954627 p^{6} T^{8} + 50265 p^{12} T^{10} + p^{18} T^{12} \) |
| 29 | \( 1 + 3537 p T^{2} + 5093278491 T^{4} + 155279545452270 T^{6} + 5093278491 p^{6} T^{8} + 3537 p^{13} T^{10} + p^{18} T^{12} \) |
| 31 | \( 1 + 113865 T^{2} + 6700832043 T^{4} + 247428066014534 T^{6} + 6700832043 p^{6} T^{8} + 113865 p^{12} T^{10} + p^{18} T^{12} \) |
| 37 | \( ( 1 + 156 T + 147900 T^{2} + 15097686 T^{3} + 147900 p^{3} T^{4} + 156 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 41 | \( ( 1 - 180 T + 149496 T^{2} - 14321250 T^{3} + 149496 p^{3} T^{4} - 180 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( ( 1 - 327 T + 196824 T^{2} - 36989763 T^{3} + 196824 p^{3} T^{4} - 327 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 47 | \( ( 1 - 906 T + 390030 T^{2} - 124576836 T^{3} + 390030 p^{3} T^{4} - 906 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 53 | \( 1 + 180597 T^{2} + 71733427059 T^{4} + 7770811583749470 T^{6} + 71733427059 p^{6} T^{8} + 180597 p^{12} T^{10} + p^{18} T^{12} \) |
| 59 | \( ( 1 + 3 T + 324096 T^{2} - 59668521 T^{3} + 324096 p^{3} T^{4} + 3 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 61 | \( 1 + 782898 T^{2} + 282242036535 T^{4} + 70824652394396060 T^{6} + 282242036535 p^{6} T^{8} + 782898 p^{12} T^{10} + p^{18} T^{12} \) |
| 67 | \( ( 1 - 21 T + 805641 T^{2} - 16881406 T^{3} + 805641 p^{3} T^{4} - 21 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 71 | \( 1 + 718233 T^{2} + 448945913091 T^{4} + 180928540927620678 T^{6} + 448945913091 p^{6} T^{8} + 718233 p^{12} T^{10} + p^{18} T^{12} \) |
| 73 | \( 1 + 859254 T^{2} + 509280578079 T^{4} + 226900074133018100 T^{6} + 509280578079 p^{6} T^{8} + 859254 p^{12} T^{10} + p^{18} T^{12} \) |
| 79 | \( ( 1 - 978 T + 1121190 T^{2} - 639733376 T^{3} + 1121190 p^{3} T^{4} - 978 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( ( 1 - 1446 T + 1946202 T^{2} - 1587114996 T^{3} + 1946202 p^{3} T^{4} - 1446 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 89 | \( ( 1 - 759 T + 1423131 T^{2} - 991050762 T^{3} + 1423131 p^{3} T^{4} - 759 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 + 1875102 T^{2} + 3279229040895 T^{4} + 3114657711972625988 T^{6} + 3279229040895 p^{6} T^{8} + 1875102 p^{12} T^{10} + p^{18} T^{12} \) |
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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
−4.71507928590176734735820650100, −4.35156331695660781084203995269, −4.32014485989271344474754956229, −4.21549869388628126670839612783, −4.00826063272024448316888000141, −3.94379316290337485469800444152, −3.89467604790867467242664013459, −3.52187322706236384613979336704, −3.33271733968894316057613519684, −3.27000335820518476310392028177, −3.10119958152249523559508703292, −2.62021621516129182697966224903, −2.52558297724396924021314255477, −2.40852046420157380343612050502, −2.22156312906962778224823103860, −2.19222594412830502245172157774, −1.92052261892594533998291353029, −1.70093403959821990369821741301, −1.59237771399940184716505464579, −1.07010449121753004343972994882, −1.03140266686704113044661630908, −0.68760119111977909916692903770, −0.62227424419970931131937199954, −0.47409032684847244214625537038, −0.37095865391295669053099305598,
0.37095865391295669053099305598, 0.47409032684847244214625537038, 0.62227424419970931131937199954, 0.68760119111977909916692903770, 1.03140266686704113044661630908, 1.07010449121753004343972994882, 1.59237771399940184716505464579, 1.70093403959821990369821741301, 1.92052261892594533998291353029, 2.19222594412830502245172157774, 2.22156312906962778224823103860, 2.40852046420157380343612050502, 2.52558297724396924021314255477, 2.62021621516129182697966224903, 3.10119958152249523559508703292, 3.27000335820518476310392028177, 3.33271733968894316057613519684, 3.52187322706236384613979336704, 3.89467604790867467242664013459, 3.94379316290337485469800444152, 4.00826063272024448316888000141, 4.21549869388628126670839612783, 4.32014485989271344474754956229, 4.35156331695660781084203995269, 4.71507928590176734735820650100
Plot not available for L-functions of degree greater than 10.
