| L(s) = 1 | − 6·9-s − 6·11-s − 2·19-s + 2·29-s − 2·31-s − 30·41-s + 2·59-s − 24·61-s + 12·71-s + 14·79-s + 9·81-s − 28·89-s + 36·99-s − 24·101-s − 16·109-s + 7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 12·171-s + ⋯ |
| L(s) = 1 | − 2·9-s − 1.80·11-s − 0.458·19-s + 0.371·29-s − 0.359·31-s − 4.68·41-s + 0.260·59-s − 3.07·61-s + 1.42·71-s + 1.57·79-s + 81-s − 2.96·89-s + 3.61·99-s − 2.38·101-s − 1.53·109-s + 7/11·121-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 − 0.615·169-s + 0.917·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 126 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 45 T^{2} + 956 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - T + 44 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 17 T^{2} + 1656 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 15 T + 124 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 125 T^{2} + 7248 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 3 p T^{2} + 9032 T^{4} + 3 p^{3} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 125 T^{2} + 9396 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - T + 104 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 62 T^{2} + 1731 T^{4} + 62 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 245 T^{2} + 25308 T^{4} + 245 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 7 T + 156 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 245 T^{2} + 28656 T^{4} + 245 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.37048779003466986050488837543, −5.86170155937473186147973948889, −5.66581161682903938842761723034, −5.61742069232045284834503696962, −5.58523860571461149420469094059, −5.05210379333635105301566586495, −4.99810927295076349560446139709, −4.98944973476097996149396645089, −4.98610002652881137021543725628, −4.58421084460227091724048603117, −4.15171451778313039751111306220, −3.95298431394978937391033101649, −3.91880603855696050327919087215, −3.55993157071463785888079402183, −3.31565640203143152830727708660, −3.17307707750051355538731660997, −3.03505922042885497678002408675, −2.62814911109017780709419780341, −2.56743690826228254211114167308, −2.41939089101665048479699773959, −2.24380582052460664552484634844, −1.66690449296031921304705120978, −1.60595642976590535776053201926, −1.21905613991825137378597035857, −1.07342434896189540411707306242, 0, 0, 0, 0,
1.07342434896189540411707306242, 1.21905613991825137378597035857, 1.60595642976590535776053201926, 1.66690449296031921304705120978, 2.24380582052460664552484634844, 2.41939089101665048479699773959, 2.56743690826228254211114167308, 2.62814911109017780709419780341, 3.03505922042885497678002408675, 3.17307707750051355538731660997, 3.31565640203143152830727708660, 3.55993157071463785888079402183, 3.91880603855696050327919087215, 3.95298431394978937391033101649, 4.15171451778313039751111306220, 4.58421084460227091724048603117, 4.98610002652881137021543725628, 4.98944973476097996149396645089, 4.99810927295076349560446139709, 5.05210379333635105301566586495, 5.58523860571461149420469094059, 5.61742069232045284834503696962, 5.66581161682903938842761723034, 5.86170155937473186147973948889, 6.37048779003466986050488837543
