| L(s) = 1 | − 2·4-s + 5·9-s + 8·11-s + 3·16-s − 10·19-s + 20·29-s + 8·31-s − 10·36-s − 12·41-s − 16·44-s + 10·49-s − 22·61-s − 4·64-s − 12·71-s + 20·76-s + 20·79-s + 12·81-s + 40·99-s − 2·101-s + 60·109-s − 40·116-s + 6·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 4-s + 5/3·9-s + 2.41·11-s + 3/4·16-s − 2.29·19-s + 3.71·29-s + 1.43·31-s − 5/3·36-s − 1.87·41-s − 2.41·44-s + 10/7·49-s − 2.81·61-s − 1/2·64-s − 1.42·71-s + 2.29·76-s + 2.25·79-s + 4/3·81-s + 4.02·99-s − 0.199·101-s + 5.74·109-s − 3.71·116-s + 6/11·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.201358089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.201358089\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 13 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 5 T^{2} - 327 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 50 T^{2} + 1363 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 10 T + 63 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 61 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 2833 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 130 T^{2} + 7603 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 10 T^{2} + 1563 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 5238 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 130 T^{2} + 11923 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 200 T^{2} + 20478 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 10 T + 178 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 130 T^{2} + 11523 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 185 T^{2} + 27273 T^{4} - 185 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.96627497438097932200129665552, −6.48742570445656358083308216480, −6.41669150895263320496307641563, −6.35547004316027443783689250087, −6.31484923910832774232050583647, −6.06832231843345329201761513885, −5.54270491105238344165095561711, −5.34499805655237275091365696521, −4.84112942939226211244088273030, −4.73586548805721697813321347915, −4.67790530456344968264604590700, −4.47791658744744135279791929013, −4.15269854597057949304626604280, −4.10728643025840240001546436719, −3.80453512897240379575696678717, −3.55823356814939750772689012486, −3.16091563515137461281044982963, −2.94502519236222487858372176874, −2.66779935051917070893247625101, −2.09094740760426338672724572724, −1.95776957835675333042976725576, −1.42045734999397464307731722005, −1.34650779124563106976574123055, −0.859212574906169052191574286132, −0.52906112539768869495374409748,
0.52906112539768869495374409748, 0.859212574906169052191574286132, 1.34650779124563106976574123055, 1.42045734999397464307731722005, 1.95776957835675333042976725576, 2.09094740760426338672724572724, 2.66779935051917070893247625101, 2.94502519236222487858372176874, 3.16091563515137461281044982963, 3.55823356814939750772689012486, 3.80453512897240379575696678717, 4.10728643025840240001546436719, 4.15269854597057949304626604280, 4.47791658744744135279791929013, 4.67790530456344968264604590700, 4.73586548805721697813321347915, 4.84112942939226211244088273030, 5.34499805655237275091365696521, 5.54270491105238344165095561711, 6.06832231843345329201761513885, 6.31484923910832774232050583647, 6.35547004316027443783689250087, 6.41669150895263320496307641563, 6.48742570445656358083308216480, 6.96627497438097932200129665552
