| L(s) = 1 | − 4·25-s + 16·37-s − 24·47-s + 14·49-s − 12·59-s + 48·83-s − 64·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 4/5·25-s + 2.63·37-s − 3.50·47-s + 2·49-s − 1.56·59-s + 5.26·83-s − 6.13·109-s + 4·121-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 + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.911456472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.911456472\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 121 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 151 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−5.98981763710270761208689160428, −5.89345151799220228361544062480, −5.88057113563610627400037320937, −5.87370656747971866759144125921, −5.22563979237517673582218421057, −5.21518871867730154009282005862, −4.81265479800648420960377046923, −4.76335103094274571025763720278, −4.63215516681622433331001492695, −4.38317100606364102124423469455, −4.12535876584937275948195842457, −3.88018829935572687095994075020, −3.54581441697250822623071516613, −3.43075461060917871380850270448, −3.33700483409131087015183657450, −2.99692983859679776130038916266, −2.68168200761265926274422247207, −2.36889557002579166584904714467, −2.21486702787989740790102378284, −2.10826812791739662003575078402, −1.53900610636398936446682359892, −1.44282694542579214612777598933, −1.02634958463653788423674866876, −0.64137665604591634232310830250, −0.30996924983463594574558091552,
0.30996924983463594574558091552, 0.64137665604591634232310830250, 1.02634958463653788423674866876, 1.44282694542579214612777598933, 1.53900610636398936446682359892, 2.10826812791739662003575078402, 2.21486702787989740790102378284, 2.36889557002579166584904714467, 2.68168200761265926274422247207, 2.99692983859679776130038916266, 3.33700483409131087015183657450, 3.43075461060917871380850270448, 3.54581441697250822623071516613, 3.88018829935572687095994075020, 4.12535876584937275948195842457, 4.38317100606364102124423469455, 4.63215516681622433331001492695, 4.76335103094274571025763720278, 4.81265479800648420960377046923, 5.21518871867730154009282005862, 5.22563979237517673582218421057, 5.87370656747971866759144125921, 5.88057113563610627400037320937, 5.89345151799220228361544062480, 5.98981763710270761208689160428
