| L(s) = 1 | − 4·9-s + 4·25-s − 16·29-s − 4·49-s + 16·61-s + 10·81-s − 32·89-s + 64·101-s − 48·109-s − 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 4/3·9-s + 4/5·25-s − 2.97·29-s − 4/7·49-s + 2.04·61-s + 10/9·81-s − 3.39·89-s + 6.36·101-s − 4.59·109-s − 5.09·121-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 + 5.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.668224853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.668224853\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{4} \) |
| 5 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( ( 1 + T^{2} )^{4} \) |
| good | 11 | \( ( 1 + 28 T^{2} + 390 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 36 T^{2} + 614 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 60 T^{2} + 1574 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 31 | \( ( 1 + 76 T^{2} + 2934 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 84 T^{2} + 3734 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - p T^{2} )^{8} \) |
| 47 | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 196 T^{2} + 15174 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 172 T^{2} + 13590 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 76 T^{2} + 726 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 244 T^{2} + 25110 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 204 T^{2} + 21110 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 84 T^{2} + 19382 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.44311883381474095354139773619, −3.30284003105366440590343900876, −3.28587214511411595868204712635, −3.26801664019519118691289433448, −3.16600677189481060480163962045, −3.04695611715916639638481142778, −2.99282154084576741172713582884, −2.64980598558784697004833727254, −2.53599036361039427885615032155, −2.45874140469450225203725537542, −2.37254822829258166466936879713, −2.27429213145082889465109589737, −2.11949240402335147149954501138, −2.05374272868452927162341984246, −1.90645958610532300765859408643, −1.78682784156058713756579130569, −1.43443034718655029285554697649, −1.35873398573997781449202505694, −1.31575797634600168357277384193, −1.05897560174545487269078743296, −0.939687307857901478884557229923, −0.906840384379141875666007663580, −0.28093008700947020329003300966, −0.27432044951849413160776719350, −0.23448431017017511267399502160,
0.23448431017017511267399502160, 0.27432044951849413160776719350, 0.28093008700947020329003300966, 0.906840384379141875666007663580, 0.939687307857901478884557229923, 1.05897560174545487269078743296, 1.31575797634600168357277384193, 1.35873398573997781449202505694, 1.43443034718655029285554697649, 1.78682784156058713756579130569, 1.90645958610532300765859408643, 2.05374272868452927162341984246, 2.11949240402335147149954501138, 2.27429213145082889465109589737, 2.37254822829258166466936879713, 2.45874140469450225203725537542, 2.53599036361039427885615032155, 2.64980598558784697004833727254, 2.99282154084576741172713582884, 3.04695611715916639638481142778, 3.16600677189481060480163962045, 3.26801664019519118691289433448, 3.28587214511411595868204712635, 3.30284003105366440590343900876, 3.44311883381474095354139773619
Plot not available for L-functions of degree greater than 10.
