| L(s) = 1 | + 2·4-s + 8·7-s + 2·9-s − 3·16-s + 16·28-s + 4·36-s − 32·37-s + 16·43-s + 12·49-s + 16·63-s − 6·64-s + 40·67-s − 32·79-s + 6·81-s + 40·109-s − 24·112-s + 32·121-s + 127-s + 131-s + 137-s + 139-s − 6·144-s − 64·148-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 4-s + 3.02·7-s + 2/3·9-s − 3/4·16-s + 3.02·28-s + 2/3·36-s − 5.26·37-s + 2.43·43-s + 12/7·49-s + 2.01·63-s − 3/4·64-s + 4.88·67-s − 3.60·79-s + 2/3·81-s + 3.83·109-s − 2.26·112-s + 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/2·144-s − 5.26·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.063287579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.063287579\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 2 T^{2} - 2 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| good | 2 | \( ( 1 - T^{2} + 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 16 T^{2} + 285 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 26 T^{2} + 558 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 10 T^{2} + 558 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 64 T^{2} + 1893 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 88 T^{2} + 3429 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 58 T^{2} + 2574 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 38 T^{2} + 2022 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 4 T + 69 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 146 T^{2} + 9558 T^{4} + 146 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 16 T^{2} - 1122 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 68 T^{2} + 7362 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 178 T^{2} + 15174 T^{4} - 178 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 256 T^{2} + 26445 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 8 T^{2} - 1422 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 8 T + 153 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 38 T^{2} + 4878 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 188 T^{2} + 23922 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 250 T^{2} + 29718 T^{4} - 250 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.78048416539688629214989722120, −4.59770304714564471451900906598, −4.47517564871790222332110039292, −4.31508523882146213789451170327, −4.24984747350407319077035269594, −4.23979875019359058974697992797, −4.00841152443874630923297015079, −3.80761358303577913714104596630, −3.54205074566745684303065796964, −3.31152443477967624392690482546, −3.28906316037601837218100511244, −3.25644812966638130560655026771, −3.15803913504395878615806680332, −2.64456544478321384894890773802, −2.48218510391075962782045019821, −2.42912930719197130887444405818, −2.28437785562087568380613836884, −1.82551837125084223685609401573, −1.78151578642576374639183407985, −1.76347781610699772212675850732, −1.70946924004765435134188156464, −1.60912719031927549195505142310, −0.896725275781526178357933044378, −0.878752499354625664433753105713, −0.43140032810367497566720297571,
0.43140032810367497566720297571, 0.878752499354625664433753105713, 0.896725275781526178357933044378, 1.60912719031927549195505142310, 1.70946924004765435134188156464, 1.76347781610699772212675850732, 1.78151578642576374639183407985, 1.82551837125084223685609401573, 2.28437785562087568380613836884, 2.42912930719197130887444405818, 2.48218510391075962782045019821, 2.64456544478321384894890773802, 3.15803913504395878615806680332, 3.25644812966638130560655026771, 3.28906316037601837218100511244, 3.31152443477967624392690482546, 3.54205074566745684303065796964, 3.80761358303577913714104596630, 4.00841152443874630923297015079, 4.23979875019359058974697992797, 4.24984747350407319077035269594, 4.31508523882146213789451170327, 4.47517564871790222332110039292, 4.59770304714564471451900906598, 4.78048416539688629214989722120
Plot not available for L-functions of degree greater than 10.
