| L(s) = 1 | − 2·9-s + 8·11-s + 8·19-s + 10·25-s + 16·29-s − 16·31-s − 32·41-s − 2·49-s − 32·61-s − 32·71-s + 3·81-s + 16·89-s − 16·99-s − 24·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s − 16·171-s + 173-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 2.41·11-s + 1.83·19-s + 2·25-s + 2.97·29-s − 2.87·31-s − 4.99·41-s − 2/7·49-s − 4.09·61-s − 3.79·71-s + 1/3·81-s + 1.69·89-s − 1.60·99-s − 2.29·109-s − 0.363·121-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 + 0.923·169-s − 1.22·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.653024417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.653024417\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4918 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_4$ | \( ( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7222 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 8998 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_4$ | \( ( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 12182 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 16 T + 186 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 36 T^{2} - 538 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 13558 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 8 T + 14 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 25798 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.67427514917933090639374342942, −6.62754550826837263948662540680, −6.44648988037086414665311803493, −6.02589171913680573336794383658, −5.69562896916051872321719933051, −5.60666589712115886154968881518, −5.40008369206042728836092980195, −5.21822241519116633952655098494, −4.84800183149000539712994214537, −4.59429385487275223150575003965, −4.59066596833036635818773195792, −4.36479531582599764469203548637, −3.96408555542793988421582208168, −3.66964663581635087881691048687, −3.32299895968630026163813232651, −3.14043716612225681638802522076, −3.11286114229709259824528097070, −3.07479837211408391179632462678, −2.55117428671047054305860631940, −1.98393875322555708005438014281, −1.67740086912831011580957148508, −1.48169200962818908037988918695, −1.31107122907419097395558461800, −0.950268170574869678856090109351, −0.23296643099176709284440640549,
0.23296643099176709284440640549, 0.950268170574869678856090109351, 1.31107122907419097395558461800, 1.48169200962818908037988918695, 1.67740086912831011580957148508, 1.98393875322555708005438014281, 2.55117428671047054305860631940, 3.07479837211408391179632462678, 3.11286114229709259824528097070, 3.14043716612225681638802522076, 3.32299895968630026163813232651, 3.66964663581635087881691048687, 3.96408555542793988421582208168, 4.36479531582599764469203548637, 4.59066596833036635818773195792, 4.59429385487275223150575003965, 4.84800183149000539712994214537, 5.21822241519116633952655098494, 5.40008369206042728836092980195, 5.60666589712115886154968881518, 5.69562896916051872321719933051, 6.02589171913680573336794383658, 6.44648988037086414665311803493, 6.62754550826837263948662540680, 6.67427514917933090639374342942
