| L(s) = 1 | + 654·7-s − 4.18e3·13-s + 6.89e3·19-s + 3.10e4·25-s − 7.47e4·31-s − 9.17e3·37-s − 1.25e5·43-s + 1.40e4·49-s − 7.27e5·61-s − 4.27e5·67-s − 9.36e5·73-s − 1.02e6·79-s − 2.73e6·91-s − 1.89e6·97-s − 2.15e6·103-s − 2.04e6·109-s + 3.79e5·121-s + 127-s + 131-s + 4.50e6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.90·7-s − 1.90·13-s + 1.00·19-s + 1.98·25-s − 2.51·31-s − 0.181·37-s − 1.57·43-s + 0.119·49-s − 3.20·61-s − 1.42·67-s − 2.40·73-s − 2.07·79-s − 3.62·91-s − 2.07·97-s − 1.97·103-s − 1.57·109-s + 0.214·121-s + 1.91·133-s − 1.87·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s+3)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.7364175727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7364175727\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 31062 T^{2} + 25218327 p^{2} T^{4} - 15192393972 p^{4} T^{6} + 25218327 p^{14} T^{8} - 31062 p^{24} T^{10} + p^{36} T^{12} \) |
| 7 | \( ( 1 - 327 T + 153390 T^{2} - 76542819 T^{3} + 153390 p^{6} T^{4} - 327 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 11 | \( 1 - 379926 T^{2} + 3802211274495 T^{4} - 6497818437808951092 T^{6} + 3802211274495 p^{12} T^{8} - 379926 p^{24} T^{10} + p^{36} T^{12} \) |
| 13 | \( ( 1 + 2091 T + 11094654 T^{2} + 18436384847 T^{3} + 11094654 p^{6} T^{4} + 2091 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 17 | \( 1 - 128480358 T^{2} + 7229572430861775 T^{4} - \)\(78\!\cdots\!28\)\( p^{2} T^{6} + 7229572430861775 p^{12} T^{8} - 128480358 p^{24} T^{10} + p^{36} T^{12} \) |
| 19 | \( ( 1 - 3447 T + 12940038 T^{2} - 332406328291 T^{3} + 12940038 p^{6} T^{4} - 3447 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 23 | \( 1 - 30468522 p T^{2} + 224222116255148847 T^{4} - \)\(42\!\cdots\!52\)\( T^{6} + 224222116255148847 p^{12} T^{8} - 30468522 p^{25} T^{10} + p^{36} T^{12} \) |
| 29 | \( 1 - 3119191062 T^{2} + 4278423914924260767 T^{4} - \)\(33\!\cdots\!64\)\( T^{6} + 4278423914924260767 p^{12} T^{8} - 3119191062 p^{24} T^{10} + p^{36} T^{12} \) |
| 31 | \( ( 1 + 37398 T + 1371542799 T^{2} + 18765283918196 T^{3} + 1371542799 p^{6} T^{4} + 37398 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 37 | \( ( 1 + 4587 T + 7664443950 T^{2} + 23428171829487 T^{3} + 7664443950 p^{6} T^{4} + 4587 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 41 | \( 1 - 13128421542 T^{2} + \)\(11\!\cdots\!19\)\( T^{4} - \)\(64\!\cdots\!64\)\( T^{6} + \)\(11\!\cdots\!19\)\( p^{12} T^{8} - 13128421542 p^{24} T^{10} + p^{36} T^{12} \) |
| 43 | \( ( 1 + 62766 T + 14752901799 T^{2} + 711073498397668 T^{3} + 14752901799 p^{6} T^{4} + 62766 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 47 | \( 1 - 15763620966 T^{2} + 14816039815170161871 T^{4} + \)\(84\!\cdots\!16\)\( T^{6} + 14816039815170161871 p^{12} T^{8} - 15763620966 p^{24} T^{10} + p^{36} T^{12} \) |
| 53 | \( 1 - 100172065398 T^{2} + \)\(47\!\cdots\!19\)\( T^{4} - \)\(13\!\cdots\!16\)\( T^{6} + \)\(47\!\cdots\!19\)\( p^{12} T^{8} - 100172065398 p^{24} T^{10} + p^{36} T^{12} \) |
| 59 | \( 1 - 140402290518 T^{2} + \)\(10\!\cdots\!67\)\( T^{4} - \)\(53\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!67\)\( p^{12} T^{8} - 140402290518 p^{24} T^{10} + p^{36} T^{12} \) |
| 61 | \( ( 1 + 5967 p T + 135811600110 T^{2} + 33701070358959815 T^{3} + 135811600110 p^{6} T^{4} + 5967 p^{13} T^{5} + p^{18} T^{6} )^{2} \) |
| 67 | \( ( 1 + 213825 T + 283998839862 T^{2} + 38909750763274373 T^{3} + 283998839862 p^{6} T^{4} + 213825 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 71 | \( 1 - 345299965542 T^{2} + \)\(69\!\cdots\!07\)\( T^{4} - \)\(97\!\cdots\!64\)\( T^{6} + \)\(69\!\cdots\!07\)\( p^{12} T^{8} - 345299965542 p^{24} T^{10} + p^{36} T^{12} \) |
| 73 | \( ( 1 + 468339 T + 446106013494 T^{2} + 125788239719789303 T^{3} + 446106013494 p^{6} T^{4} + 468339 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 79 | \( ( 1 + 510873 T + 778408287006 T^{2} + 244860792007184989 T^{3} + 778408287006 p^{6} T^{4} + 510873 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 83 | \( 1 - 818212602198 T^{2} + \)\(26\!\cdots\!79\)\( T^{4} - \)\(65\!\cdots\!36\)\( T^{6} + \)\(26\!\cdots\!79\)\( p^{12} T^{8} - 818212602198 p^{24} T^{10} + p^{36} T^{12} \) |
| 89 | \( 1 + 231976424442 T^{2} + \)\(68\!\cdots\!47\)\( T^{4} + \)\(10\!\cdots\!40\)\( T^{6} + \)\(68\!\cdots\!47\)\( p^{12} T^{8} + 231976424442 p^{24} T^{10} + p^{36} T^{12} \) |
| 97 | \( ( 1 + 946251 T + 1500955250454 T^{2} + 756915606638423071 T^{3} + 1500955250454 p^{6} T^{4} + 946251 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
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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
−4.89490104348511593948492962271, −4.83860257483138635159198607283, −4.80129328068624354223898018691, −4.63308081623804806015887592384, −4.51478880330832460094354635041, −3.94240689488957828944319769716, −3.71785382414059770791506475873, −3.66201189460270109152466920632, −3.61119585117219099918000446050, −3.56418587459662873460004784273, −2.86849250204852570362858709719, −2.69993694329356881629341770769, −2.63676836315979731414926990770, −2.63427408097757336028390683965, −2.56173746386547187832463303493, −2.11068866376837603163363514070, −1.53664022178059830668288278891, −1.53242535930378688447367885146, −1.47935924960178305538974399471, −1.41748257933384253190982342965, −1.20066846072107090776391186506, −1.08132839598015586707194651515, −0.22892656981290655959560015184, −0.22421775623090976032330158481, −0.17891675658041108386525715538,
0.17891675658041108386525715538, 0.22421775623090976032330158481, 0.22892656981290655959560015184, 1.08132839598015586707194651515, 1.20066846072107090776391186506, 1.41748257933384253190982342965, 1.47935924960178305538974399471, 1.53242535930378688447367885146, 1.53664022178059830668288278891, 2.11068866376837603163363514070, 2.56173746386547187832463303493, 2.63427408097757336028390683965, 2.63676836315979731414926990770, 2.69993694329356881629341770769, 2.86849250204852570362858709719, 3.56418587459662873460004784273, 3.61119585117219099918000446050, 3.66201189460270109152466920632, 3.71785382414059770791506475873, 3.94240689488957828944319769716, 4.51478880330832460094354635041, 4.63308081623804806015887592384, 4.80129328068624354223898018691, 4.83860257483138635159198607283, 4.89490104348511593948492962271
Plot not available for L-functions of degree greater than 10.
