| L(s) = 1 | + 4·3-s + 10·9-s + 32·27-s + 12·41-s − 40·43-s + 28·49-s + 28·67-s + 89·81-s + 36·83-s + 36·89-s + 12·107-s + 14·121-s + 48·123-s + 127-s − 160·129-s + 131-s + 137-s + 139-s + 112·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 10/3·9-s + 6.15·27-s + 1.87·41-s − 6.09·43-s + 4·49-s + 3.42·67-s + 89/9·81-s + 3.95·83-s + 3.81·89-s + 1.16·107-s + 1.27·121-s + 4.32·123-s + 0.0887·127-s − 14.0·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 9.23·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.49321915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.49321915\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^3$ | \( 1 - 14 T^{2} + 75 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 142 T^{2} + 14835 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.70969627662154486021029465365, −6.69676303741450502161001887161, −6.30997886198093302931913416904, −6.16963219705657633235645357219, −6.06018293533426589488756160905, −5.43877429435754413706936530596, −5.31259092739321913687551925083, −5.10764561584770146250117108047, −4.91550238924228285810777705781, −4.62885209241547409491426266975, −4.50242631012420504901707838228, −4.29515526130043392648543580820, −3.81457345230701978473962365937, −3.49025926622925289617435377717, −3.40630118259155737007650150794, −3.40580006320534725701013349181, −3.31895209490381357167892992056, −2.45303313972012190659072963380, −2.39882166790071681086385799714, −2.35999444913599819707844510303, −2.20292753354796729517817336163, −1.63303072257326074312814177864, −1.30763718413459271204052422426, −0.821500104886940392774969281301, −0.66800668014292607595188688813,
0.66800668014292607595188688813, 0.821500104886940392774969281301, 1.30763718413459271204052422426, 1.63303072257326074312814177864, 2.20292753354796729517817336163, 2.35999444913599819707844510303, 2.39882166790071681086385799714, 2.45303313972012190659072963380, 3.31895209490381357167892992056, 3.40580006320534725701013349181, 3.40630118259155737007650150794, 3.49025926622925289617435377717, 3.81457345230701978473962365937, 4.29515526130043392648543580820, 4.50242631012420504901707838228, 4.62885209241547409491426266975, 4.91550238924228285810777705781, 5.10764561584770146250117108047, 5.31259092739321913687551925083, 5.43877429435754413706936530596, 6.06018293533426589488756160905, 6.16963219705657633235645357219, 6.30997886198093302931913416904, 6.69676303741450502161001887161, 6.70969627662154486021029465365
