| L(s) = 1 | + 4·4-s + 4·7-s + 8·9-s − 8·13-s + 7·16-s + 12·23-s + 16·28-s + 32·36-s − 12·49-s − 32·52-s + 24·59-s + 32·63-s + 8·64-s + 4·67-s + 24·71-s + 30·81-s + 12·83-s − 32·91-s + 48·92-s − 20·103-s + 12·107-s + 8·109-s + 28·112-s − 64·117-s + 16·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.51·7-s + 8/3·9-s − 2.21·13-s + 7/4·16-s + 2.50·23-s + 3.02·28-s + 16/3·36-s − 1.71·49-s − 4.43·52-s + 3.12·59-s + 4.03·63-s + 64-s + 0.488·67-s + 2.84·71-s + 10/3·81-s + 1.31·83-s − 3.35·91-s + 5.00·92-s − 1.97·103-s + 1.16·107-s + 0.766·109-s + 2.64·112-s − 5.91·117-s + 1.45·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.335709003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.335709003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p^{2} T^{2} + 9 T^{4} - p^{4} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 114 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 450 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 3606 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 78 p T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 61 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 6054 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 2 T - 12 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 40 T^{2} - 4494 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 64 T^{2} - 2046 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 292 T^{2} + 36390 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.43435923643187234190269629361, −7.37693381649319866199212225577, −7.05320362078278367621600406301, −6.80112126385499493772641153530, −6.63645301338411633050681161612, −6.53983115795204181573989250127, −6.34069062356632427407923196621, −5.90069009602319566385101773986, −5.34460261526050337699763283611, −5.23599576885116948224245086921, −4.98426893841827592891608720042, −4.86412691002601595408511865744, −4.86285008496600651860254862614, −4.35601962695981938494462000411, −3.93451538623523367576980498957, −3.88483282143954472080100629768, −3.48148150578803842324074147406, −3.01731365351622244333667048381, −2.73639366867112909095620207220, −2.41315717598866500278491992387, −2.04727898389295466763033248217, −2.04185263399383707319209824436, −1.53000131070270396062639290106, −1.20055445441737969979034020740, −0.817685260961591954089541976270,
0.817685260961591954089541976270, 1.20055445441737969979034020740, 1.53000131070270396062639290106, 2.04185263399383707319209824436, 2.04727898389295466763033248217, 2.41315717598866500278491992387, 2.73639366867112909095620207220, 3.01731365351622244333667048381, 3.48148150578803842324074147406, 3.88483282143954472080100629768, 3.93451538623523367576980498957, 4.35601962695981938494462000411, 4.86285008496600651860254862614, 4.86412691002601595408511865744, 4.98426893841827592891608720042, 5.23599576885116948224245086921, 5.34460261526050337699763283611, 5.90069009602319566385101773986, 6.34069062356632427407923196621, 6.53983115795204181573989250127, 6.63645301338411633050681161612, 6.80112126385499493772641153530, 7.05320362078278367621600406301, 7.37693381649319866199212225577, 7.43435923643187234190269629361
