| L(s) = 1 | + 8·5-s + 12·13-s − 24·17-s + 32·25-s + 16·29-s + 12·37-s − 8·49-s + 16·53-s + 12·61-s + 96·65-s − 81-s − 192·85-s − 16·97-s + 32·101-s − 28·109-s + 24·113-s + 104·125-s + 127-s + 131-s + 137-s + 139-s + 128·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 3.57·5-s + 3.32·13-s − 5.82·17-s + 32/5·25-s + 2.97·29-s + 1.97·37-s − 8/7·49-s + 2.19·53-s + 1.53·61-s + 11.9·65-s − 1/9·81-s − 20.8·85-s − 1.62·97-s + 3.18·101-s − 2.68·109-s + 2.25·113-s + 9.30·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.6·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.419198324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.419198324\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 206 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 1198 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 4946 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 140 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 5678 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
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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
−7.17541090524028005340860416239, −6.90766429471259305559175216361, −6.60530406817808809405330585777, −6.59514634713179358768277490491, −6.55916352975344041183676534129, −6.31227264158126278770763844670, −6.11445119591430064242779320287, −5.84178694129719924994683965658, −5.72070719485549474035647467450, −5.41853600021421866967305718659, −5.13995183399375656247404544564, −4.64452540296242581733366135328, −4.55333170601221371658323845706, −4.28657100691365784008816074475, −4.27793153380316927668091195971, −3.77575529505352704074050354129, −3.37833124370845631094842673982, −2.86337915081147267109893599540, −2.56869848048385159799366707749, −2.56414720835923351115198045294, −1.97386060038264469952212255906, −1.90312982309361220561424011402, −1.79390694285162107155781629995, −0.948955856369256371994491761986, −0.851831776320703403346591185809,
0.851831776320703403346591185809, 0.948955856369256371994491761986, 1.79390694285162107155781629995, 1.90312982309361220561424011402, 1.97386060038264469952212255906, 2.56414720835923351115198045294, 2.56869848048385159799366707749, 2.86337915081147267109893599540, 3.37833124370845631094842673982, 3.77575529505352704074050354129, 4.27793153380316927668091195971, 4.28657100691365784008816074475, 4.55333170601221371658323845706, 4.64452540296242581733366135328, 5.13995183399375656247404544564, 5.41853600021421866967305718659, 5.72070719485549474035647467450, 5.84178694129719924994683965658, 6.11445119591430064242779320287, 6.31227264158126278770763844670, 6.55916352975344041183676534129, 6.59514634713179358768277490491, 6.60530406817808809405330585777, 6.90766429471259305559175216361, 7.17541090524028005340860416239
