| L(s) = 1 | − 4·9-s − 8·47-s + 8·71-s + 10·81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | − 4·9-s − 8·47-s + 8·71-s + 10·81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4058042085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4058042085\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{8} \) |
| 23 | \( 1 + T^{8} \) |
| good | 3 | \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \) |
| 5 | \( 1 + T^{16} \) |
| 7 | \( ( 1 + T^{8} )^{2} \) |
| 11 | \( 1 + T^{16} \) |
| 13 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 17 | \( ( 1 + T^{4} )^{4} \) |
| 19 | \( 1 + T^{16} \) |
| 29 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 31 | \( ( 1 + T^{8} )^{2} \) |
| 37 | \( 1 + T^{16} \) |
| 41 | \( ( 1 + T^{8} )^{2} \) |
| 43 | \( 1 + T^{16} \) |
| 47 | \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \) |
| 53 | \( 1 + T^{16} \) |
| 59 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 61 | \( 1 + T^{16} \) |
| 67 | \( 1 + T^{16} \) |
| 71 | \( ( 1 - T )^{8}( 1 + T^{4} )^{2} \) |
| 73 | \( ( 1 + T^{8} )^{2} \) |
| 79 | \( ( 1 + T^{4} )^{4} \) |
| 83 | \( 1 + T^{16} \) |
| 89 | \( ( 1 + T^{8} )^{2} \) |
| 97 | \( ( 1 + T^{2} )^{8} \) |
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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
−4.27443908926367345796450646942, −3.87651153937540560657556111275, −3.81626626578338268003673427967, −3.75772969322521852975294375082, −3.68110103751905367140933452829, −3.54444734333588368451289385143, −3.54428768663657245820258636813, −3.49294910232623973910955207440, −3.43463789958931329595887662979, −3.00989078808111389277187229653, −2.92855859203135740840330551936, −2.73832121103695529915885000009, −2.68797926960873314046303226804, −2.68534822495083123276756650310, −2.51981893443878802941805110984, −2.41518409875495160665413968061, −2.02955368390360769261372541155, −2.01500556834458439661392516535, −1.79236887342289171329388478872, −1.78986113444135053787637666968, −1.41711752381582801047826787520, −1.38443474047012899719733998476, −0.990964441630214784443484993078, −0.58323883456524963885411174514, −0.45448491946366456762583622661,
0.45448491946366456762583622661, 0.58323883456524963885411174514, 0.990964441630214784443484993078, 1.38443474047012899719733998476, 1.41711752381582801047826787520, 1.78986113444135053787637666968, 1.79236887342289171329388478872, 2.01500556834458439661392516535, 2.02955368390360769261372541155, 2.41518409875495160665413968061, 2.51981893443878802941805110984, 2.68534822495083123276756650310, 2.68797926960873314046303226804, 2.73832121103695529915885000009, 2.92855859203135740840330551936, 3.00989078808111389277187229653, 3.43463789958931329595887662979, 3.49294910232623973910955207440, 3.54428768663657245820258636813, 3.54444734333588368451289385143, 3.68110103751905367140933452829, 3.75772969322521852975294375082, 3.81626626578338268003673427967, 3.87651153937540560657556111275, 4.27443908926367345796450646942
Plot not available for L-functions of degree greater than 10.
