| 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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 1875102 T^{2} + 3279229040895 T^{4} + 3114657711972625988 T^{6} + 3279229040895 p^{6} T^{8} + 1875102 p^{12} T^{10} + p^{18} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.17101627563732050273031890385, −4.70876439749632924683662116374, −4.67220408538410098056239838092, −4.64395579578838272290958776765, −4.53548041606639328844912480345, −4.51163362876657725221749890614, −4.27739455200452039846331921678, −3.83801165751145651949596911029, −3.76132541774341801609763845719, −3.73041657947852908161071233173, −3.60503741400000970699331310004, −3.56961332431105487293715373420, −3.32929635904348836602273141533, −3.28962069901879926215964069627, −2.80216602775170896299156825174, −2.62538181192070578351506071362, −2.52155765618949768654956980630, −2.27241778753236472852158670748, −2.08432874768695212605560707234, −2.01642472667265861816244478019, −1.39054773791052678107152119524, −1.26073656780239051508675372815, −1.25890573515024238064292746586, −1.21962719848813228380854939322, −0.76009032803160580489356082912, 0, 0, 0, 0, 0, 0,
0.76009032803160580489356082912, 1.21962719848813228380854939322, 1.25890573515024238064292746586, 1.26073656780239051508675372815, 1.39054773791052678107152119524, 2.01642472667265861816244478019, 2.08432874768695212605560707234, 2.27241778753236472852158670748, 2.52155765618949768654956980630, 2.62538181192070578351506071362, 2.80216602775170896299156825174, 3.28962069901879926215964069627, 3.32929635904348836602273141533, 3.56961332431105487293715373420, 3.60503741400000970699331310004, 3.73041657947852908161071233173, 3.76132541774341801609763845719, 3.83801165751145651949596911029, 4.27739455200452039846331921678, 4.51163362876657725221749890614, 4.53548041606639328844912480345, 4.64395579578838272290958776765, 4.67220408538410098056239838092, 4.70876439749632924683662116374, 5.17101627563732050273031890385
Plot not available for L-functions of degree greater than 10.
