| L(s) = 1 | + 4·19-s − 4·31-s + 2·49-s − 4·79-s − 4·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 4·19-s − 4·31-s + 2·49-s − 4·79-s − 4·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7333190568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7333190568\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T^{4} + T^{8} \) |
| 7 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| good | 11 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 13 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 17 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 19 | \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \) |
| 23 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 29 | \( ( 1 + T^{4} )^{4} \) |
| 31 | \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \) |
| 37 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 43 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 47 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 53 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 59 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 61 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 67 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 73 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \) |
| 83 | \( ( 1 + T^{2} )^{8} \) |
| 89 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 97 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
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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
−4.32670325324356331419844706423, −4.04988577145120062114290003017, −3.99282451379923996282847357908, −3.94140478257495277929620362253, −3.92437692347078370072152167570, −3.67992659247001805644923405695, −3.58943625747657241498378375885, −3.53826080062410070942245004976, −3.48100328327064860352772278580, −3.24497150741637261208656323365, −2.99253390799909931709292648865, −2.81411162461294760922285626071, −2.70738857401396053801765080566, −2.67052586483773711914503354483, −2.62346757628067691120612645287, −2.47309353936923648400672123748, −2.31027967555264573545382235650, −1.85144213327307794212628730623, −1.79428896087623349626107626926, −1.66226024200484674677880923368, −1.40518854163695122598186741464, −1.30642787499293440883789710084, −1.15012761045758731498014932025, −1.14334442285070001189928614358, −0.45689532563816288200387315070,
0.45689532563816288200387315070, 1.14334442285070001189928614358, 1.15012761045758731498014932025, 1.30642787499293440883789710084, 1.40518854163695122598186741464, 1.66226024200484674677880923368, 1.79428896087623349626107626926, 1.85144213327307794212628730623, 2.31027967555264573545382235650, 2.47309353936923648400672123748, 2.62346757628067691120612645287, 2.67052586483773711914503354483, 2.70738857401396053801765080566, 2.81411162461294760922285626071, 2.99253390799909931709292648865, 3.24497150741637261208656323365, 3.48100328327064860352772278580, 3.53826080062410070942245004976, 3.58943625747657241498378375885, 3.67992659247001805644923405695, 3.92437692347078370072152167570, 3.94140478257495277929620362253, 3.99282451379923996282847357908, 4.04988577145120062114290003017, 4.32670325324356331419844706423
Plot not available for L-functions of degree greater than 10.
