| L(s) = 1 | + 8·5-s − 8·13-s + 24·25-s + 24·29-s − 8·37-s + 16·41-s − 12·49-s + 24·53-s − 24·61-s − 64·65-s + 8·73-s − 8·89-s + 32·97-s + 40·101-s + 24·109-s + 8·113-s − 4·121-s + 8·125-s + 127-s + 131-s + 137-s + 139-s + 192·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 2.21·13-s + 24/5·25-s + 4.45·29-s − 1.31·37-s + 2.49·41-s − 1.71·49-s + 3.29·53-s − 3.07·61-s − 7.93·65-s + 0.936·73-s − 0.847·89-s + 3.24·97-s + 3.98·101-s + 2.29·109-s + 0.752·113-s − 0.363·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 15.9·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.085968363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.085968363\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2:C_4$ | \( 1 + 12 T^{2} + 102 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 + 4 T^{2} + 238 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 36 T^{2} + 654 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 12 T^{2} + 1062 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 60 T^{2} + 2310 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 43 | $C_2^2:C_4$ | \( 1 + 164 T^{2} + 10414 T^{4} + 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 124 T^{2} + 7750 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 140 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 228 T^{2} + 19950 T^{4} + 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 108 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 164 T^{2} + 13390 T^{4} + 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 76 T^{2} + 9958 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 60 T^{2} + 5190 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 132 T^{2} + 10446 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 174 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 250 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.51245643345618831060533296474, −5.16051772802319597396429900492, −4.99184284003444374814508651688, −4.94292750858969238332474964933, −4.89431044042490087014278762260, −4.43421999045745900996774178152, −4.39654270232539166449484612738, −4.32678298577289670732472416706, −4.22207167097453826938374598142, −3.44978020427279514449331512107, −3.39552769413043676028813203711, −3.28508627262278690585487828849, −3.22513146189023912108186503466, −2.70238174324831103660305658054, −2.52602173933100596976584331262, −2.47452323554506054653881933390, −2.34375222322220878823217840625, −2.08369959087877303281486562872, −1.98851783320433540659273266238, −1.64935330006008947707760524552, −1.61079682983465789731624670871, −0.971663631570427787010944184089, −0.920265768119898167165551029765, −0.806798817382386188210046916022, −0.24062285561784008402405755151,
0.24062285561784008402405755151, 0.806798817382386188210046916022, 0.920265768119898167165551029765, 0.971663631570427787010944184089, 1.61079682983465789731624670871, 1.64935330006008947707760524552, 1.98851783320433540659273266238, 2.08369959087877303281486562872, 2.34375222322220878823217840625, 2.47452323554506054653881933390, 2.52602173933100596976584331262, 2.70238174324831103660305658054, 3.22513146189023912108186503466, 3.28508627262278690585487828849, 3.39552769413043676028813203711, 3.44978020427279514449331512107, 4.22207167097453826938374598142, 4.32678298577289670732472416706, 4.39654270232539166449484612738, 4.43421999045745900996774178152, 4.89431044042490087014278762260, 4.94292750858969238332474964933, 4.99184284003444374814508651688, 5.16051772802319597396429900492, 5.51245643345618831060533296474
