| L(s) = 1 | + 4·2-s + 8·4-s + 8·8-s − 2·16-s − 4·23-s − 20·32-s − 16·46-s + 8·53-s − 32·64-s − 32·92-s + 32·106-s − 8·107-s + 4·113-s + 127-s − 20·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 32·184-s + ⋯ |
| L(s) = 1 | + 4·2-s + 8·4-s + 8·8-s − 2·16-s − 4·23-s − 20·32-s − 16·46-s + 8·53-s − 32·64-s − 32·92-s + 32·106-s − 8·107-s + 4·113-s + 127-s − 20·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 32·184-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.182310687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.182310687\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + T^{4} )^{2} \) |
| good | 2 | \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 13 | \( ( 1 + T^{4} )^{4} \) |
| 17 | \( ( 1 + T^{4} )^{4} \) |
| 19 | \( ( 1 + T^{2} )^{8} \) |
| 23 | \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 31 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 37 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 41 | \( ( 1 + T^{2} )^{8} \) |
| 43 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 - T )^{8}( 1 + T^{2} )^{4} \) |
| 59 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 61 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 67 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 71 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 73 | \( ( 1 + T^{4} )^{4} \) |
| 79 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 83 | \( ( 1 + T^{4} )^{4} \) |
| 89 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 97 | \( ( 1 + T^{4} )^{4} \) |
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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
−4.20109767892290615997803522881, −4.16110148198179180010443735116, −4.12527080358336090050038417874, −4.08300395660795834046983839454, −3.75979719155097544897492757450, −3.60628599443616333668334704571, −3.57696924572056915400755650890, −3.30058867738510469343717204354, −3.26222762422419485066861486374, −3.25229218228598211425576352289, −3.24237934294312350315762593549, −2.79711425493672261724523300334, −2.66662632293459873222333966248, −2.61439408513177880902926591006, −2.41625133245544122439162071957, −2.37974866495687844995343642332, −2.35368353886260169942350334246, −2.22215545205281621639615463225, −1.92369650854773894545859383406, −1.91367649840646536981872745317, −1.71537143237607301999522158749, −1.38043885555316304327893211942, −1.07449077360856215291491042836, −0.790563722286526069708630232352, −0.50012489465889861143531675321,
0.50012489465889861143531675321, 0.790563722286526069708630232352, 1.07449077360856215291491042836, 1.38043885555316304327893211942, 1.71537143237607301999522158749, 1.91367649840646536981872745317, 1.92369650854773894545859383406, 2.22215545205281621639615463225, 2.35368353886260169942350334246, 2.37974866495687844995343642332, 2.41625133245544122439162071957, 2.61439408513177880902926591006, 2.66662632293459873222333966248, 2.79711425493672261724523300334, 3.24237934294312350315762593549, 3.25229218228598211425576352289, 3.26222762422419485066861486374, 3.30058867738510469343717204354, 3.57696924572056915400755650890, 3.60628599443616333668334704571, 3.75979719155097544897492757450, 4.08300395660795834046983839454, 4.12527080358336090050038417874, 4.16110148198179180010443735116, 4.20109767892290615997803522881
Plot not available for L-functions of degree greater than 10.
