| L(s) = 1 | − 4·3-s + 6·9-s + 8·19-s − 4·25-s + 4·27-s + 40·43-s − 2·49-s − 32·57-s − 24·67-s + 40·73-s + 16·75-s − 37·81-s − 8·97-s − 20·121-s + 127-s − 160·129-s + 131-s + 137-s + 139-s + 8·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 48·171-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 2·9-s + 1.83·19-s − 4/5·25-s + 0.769·27-s + 6.09·43-s − 2/7·49-s − 4.23·57-s − 2.93·67-s + 4.68·73-s + 1.84·75-s − 4.11·81-s − 0.812·97-s − 1.81·121-s + 0.0887·127-s − 14.0·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.659·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.53·169-s + 3.67·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.093860400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.093860400\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
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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
−7.63835856636368030420013946111, −7.34831507248671814067076624446, −6.96540789622041837596297961291, −6.78391772049044462169822859165, −6.52732043184584880743036062622, −6.29649974716782611256341768716, −6.10392986975012395256590188592, −5.83165540324872493710637869314, −5.67604898182184326782463530262, −5.40813517959493494027930335587, −5.30958829214338860243821758092, −5.03221272842956234143857253410, −4.82797027588718916675367524570, −4.29793808407072369004329844798, −4.23041151525160802007333464727, −3.88280834827441278101897894534, −3.81009593026033240300305234475, −3.04501657203549755895224361767, −2.81487258407880701286369656536, −2.80831787735347733612055588396, −2.20849493224236845721345450712, −1.76649967739551748134831501980, −1.15074566330991639486068196642, −0.825030282542565167897527962824, −0.51899824804953233780099688543,
0.51899824804953233780099688543, 0.825030282542565167897527962824, 1.15074566330991639486068196642, 1.76649967739551748134831501980, 2.20849493224236845721345450712, 2.80831787735347733612055588396, 2.81487258407880701286369656536, 3.04501657203549755895224361767, 3.81009593026033240300305234475, 3.88280834827441278101897894534, 4.23041151525160802007333464727, 4.29793808407072369004329844798, 4.82797027588718916675367524570, 5.03221272842956234143857253410, 5.30958829214338860243821758092, 5.40813517959493494027930335587, 5.67604898182184326782463530262, 5.83165540324872493710637869314, 6.10392986975012395256590188592, 6.29649974716782611256341768716, 6.52732043184584880743036062622, 6.78391772049044462169822859165, 6.96540789622041837596297961291, 7.34831507248671814067076624446, 7.63835856636368030420013946111
