| L(s) = 1 | − 8·4-s + 40·16-s − 10·25-s − 2·49-s − 28·61-s − 160·64-s + 44·79-s + 80·100-s + 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 16·196-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 4·4-s + 10·16-s − 2·25-s − 2/7·49-s − 3.58·61-s − 20·64-s + 4.95·79-s + 8·100-s + 3.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 8/7·196-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4409726671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4409726671\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 61 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 103 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 173 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 181 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.87519005697604201429292484777, −7.81341848863773237605860557535, −7.51166875015286550108672545056, −7.15971788054442718420816909103, −6.66370143666822466027739805187, −6.26370395043105482593932232570, −6.12852378394196762989542180482, −6.11926811965491154687413270640, −5.48767134600582876150675813109, −5.35786814989100892331454260842, −5.34921946222101366361750820004, −4.85373335881621089496087951450, −4.82696543800890614048691086724, −4.32471463026495286290800673739, −4.27441167677038615005817656131, −4.12964500371094541039026739454, −3.58807539949549318511065618997, −3.52111941443955530834076750881, −3.30771624253727757590719213019, −2.91522951295656554951821953702, −2.30417404089142620941606930762, −1.67829559750508934113047189298, −1.43178131077830570229078081033, −0.68224825621525657989964919455, −0.36881477182371913175736337206,
0.36881477182371913175736337206, 0.68224825621525657989964919455, 1.43178131077830570229078081033, 1.67829559750508934113047189298, 2.30417404089142620941606930762, 2.91522951295656554951821953702, 3.30771624253727757590719213019, 3.52111941443955530834076750881, 3.58807539949549318511065618997, 4.12964500371094541039026739454, 4.27441167677038615005817656131, 4.32471463026495286290800673739, 4.82696543800890614048691086724, 4.85373335881621089496087951450, 5.34921946222101366361750820004, 5.35786814989100892331454260842, 5.48767134600582876150675813109, 6.11926811965491154687413270640, 6.12852378394196762989542180482, 6.26370395043105482593932232570, 6.66370143666822466027739805187, 7.15971788054442718420816909103, 7.51166875015286550108672545056, 7.81341848863773237605860557535, 7.87519005697604201429292484777
