L(s) = 1 | − 8·5-s + 16·23-s + 24·25-s + 8·29-s − 16·43-s − 16·47-s + 4·49-s − 8·53-s − 48·67-s + 16·71-s − 16·73-s + 32·97-s + 8·101-s − 128·115-s + 28·121-s − 8·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-s + ⋯ |
L(s) = 1 | − 3.57·5-s + 3.33·23-s + 24/5·25-s + 1.48·29-s − 2.43·43-s − 2.33·47-s + 4/7·49-s − 1.09·53-s − 5.86·67-s + 1.89·71-s − 1.87·73-s + 3.24·97-s + 0.796·101-s − 11.9·115-s + 2.54·121-s − 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.76·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3347200439\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3347200439\) |
\(L(\frac{3}{2})\) |
|
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 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 706 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 1094 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3670 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7330 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 44 T^{2} - 746 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 14326 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 79 | $C_4\times C_2$ | \( 1 - 164 T^{2} + 18054 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 20566 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 192 T^{2} + 20450 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 226 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.41603924783090882047493800421, −6.23085779363202262461065026688, −6.14416116544354416133327073911, −5.82015209025169242814012194775, −5.50713345873203090246182594790, −5.10141878352736564304041784137, −5.01980469276290794111066793055, −4.74800779763009279783004727594, −4.62399325594692857105404542407, −4.59788664525257743395550151961, −4.38989382688236103077311307779, −4.01357138477230763619734884290, −3.63113259690906198836116673394, −3.56164415539871279270975546360, −3.47744224066128365741020750061, −3.08080619841835935010827420982, −2.97626125278471361670244388285, −2.96513761039295894798962152263, −2.43462722508434103923821731310, −1.99755586204245242815747637772, −1.61906704823410697971713453222, −1.19103445477110279949797461047, −1.16800472445412204668006113520, −0.45237319736271570199439447172, −0.18672270352122442374392895860,
0.18672270352122442374392895860, 0.45237319736271570199439447172, 1.16800472445412204668006113520, 1.19103445477110279949797461047, 1.61906704823410697971713453222, 1.99755586204245242815747637772, 2.43462722508434103923821731310, 2.96513761039295894798962152263, 2.97626125278471361670244388285, 3.08080619841835935010827420982, 3.47744224066128365741020750061, 3.56164415539871279270975546360, 3.63113259690906198836116673394, 4.01357138477230763619734884290, 4.38989382688236103077311307779, 4.59788664525257743395550151961, 4.62399325594692857105404542407, 4.74800779763009279783004727594, 5.01980469276290794111066793055, 5.10141878352736564304041784137, 5.50713345873203090246182594790, 5.82015209025169242814012194775, 6.14416116544354416133327073911, 6.23085779363202262461065026688, 6.41603924783090882047493800421