| L(s) = 1 | + 4·5-s + 16·19-s + 8·25-s − 8·29-s − 16·31-s − 24·41-s − 2·49-s − 16·59-s + 24·61-s + 24·71-s + 8·79-s − 18·81-s + 40·89-s + 64·95-s + 24·101-s − 8·109-s + 4·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s − 32·145-s + 149-s + 151-s − 64·155-s + 157-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 3.67·19-s + 8/5·25-s − 1.48·29-s − 2.87·31-s − 3.74·41-s − 2/7·49-s − 2.08·59-s + 3.07·61-s + 2.84·71-s + 0.900·79-s − 2·81-s + 4.23·89-s + 6.56·95-s + 2.38·101-s − 0.766·109-s + 4/11·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.65·145-s + 0.0819·149-s + 0.0813·151-s − 5.14·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.851558705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.851558705\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 498 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 998 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 108 T^{2} + 5798 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 128 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 152 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 154 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 24198 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 8838 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.73995901752512594892029298067, −7.68414523634273407074751041424, −7.19258741600170091374277019536, −6.99415327141256090395497325092, −6.90815536370214047351725288857, −6.57452826734288802278571971606, −6.34011920926801764563699016675, −6.02786747564235330744470976082, −5.61574650205583044526235799684, −5.44633409098404133479948268008, −5.40937285065601616465906456078, −5.25152056206810604680905359809, −4.99537457163729420102410260742, −4.65498515269625819021756396253, −4.29419513522135571983399784796, −3.56234497325956004639105698721, −3.48738172840794368963940001970, −3.40736505686345329035316448916, −3.30188167355794241111823405207, −2.62398767237310129532409966925, −2.12770916693379347348022312685, −1.86767772322833534227136792854, −1.79534126096451592561920406277, −1.22009592651033190998187781891, −0.61608851251157899507562863377,
0.61608851251157899507562863377, 1.22009592651033190998187781891, 1.79534126096451592561920406277, 1.86767772322833534227136792854, 2.12770916693379347348022312685, 2.62398767237310129532409966925, 3.30188167355794241111823405207, 3.40736505686345329035316448916, 3.48738172840794368963940001970, 3.56234497325956004639105698721, 4.29419513522135571983399784796, 4.65498515269625819021756396253, 4.99537457163729420102410260742, 5.25152056206810604680905359809, 5.40937285065601616465906456078, 5.44633409098404133479948268008, 5.61574650205583044526235799684, 6.02786747564235330744470976082, 6.34011920926801764563699016675, 6.57452826734288802278571971606, 6.90815536370214047351725288857, 6.99415327141256090395497325092, 7.19258741600170091374277019536, 7.68414523634273407074751041424, 7.73995901752512594892029298067
