| 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
| \(L(1)\) |
\(\approx\) |
\(1.224379880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.224379880\) |
| \(L(\frac{3}{2})\) |
|
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
−6.17421436061508557572265711817, −6.15550643422303230488989934119, −6.08787232981047724639073416754, −5.84415165181649602377775222850, −5.76467461938641941800474404669, −5.17000955006923245417223006111, −5.07595883104042653640973737903, −4.83674971239023945928439012182, −4.54151946418519482345212078162, −4.38320070169824173031072658436, −4.08787944716623970609163785502, −4.05692503230653688573751212926, −3.80644215437550200190973260076, −3.75497182336446420327204412285, −3.59470178482629253270225822431, −3.08439136072330616554662316292, −3.07963901199321753829676663192, −2.45059733443672442533801330584, −2.43709650691966385980552313137, −2.11233070842836190345554822279, −1.85139346523114944866609755625, −1.31052383753195925082816742878, −0.72576440225218819714584265843, −0.54037091130049667997643711500, −0.39731439156352676821313039217,
0.39731439156352676821313039217, 0.54037091130049667997643711500, 0.72576440225218819714584265843, 1.31052383753195925082816742878, 1.85139346523114944866609755625, 2.11233070842836190345554822279, 2.43709650691966385980552313137, 2.45059733443672442533801330584, 3.07963901199321753829676663192, 3.08439136072330616554662316292, 3.59470178482629253270225822431, 3.75497182336446420327204412285, 3.80644215437550200190973260076, 4.05692503230653688573751212926, 4.08787944716623970609163785502, 4.38320070169824173031072658436, 4.54151946418519482345212078162, 4.83674971239023945928439012182, 5.07595883104042653640973737903, 5.17000955006923245417223006111, 5.76467461938641941800474404669, 5.84415165181649602377775222850, 6.08787232981047724639073416754, 6.15550643422303230488989934119, 6.17421436061508557572265711817
