| L(s) = 1 | − 2·9-s − 24·17-s + 20·25-s + 24·41-s − 24·49-s + 3·81-s − 24·89-s − 48·97-s − 24·113-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·153-s + 157-s + 163-s + 167-s + 48·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 2/3·9-s − 5.82·17-s + 4·25-s + 3.74·41-s − 3.42·49-s + 1/3·81-s − 2.54·89-s − 4.87·97-s − 2.25·113-s + 4·121-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 + 3.88·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.69·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.526407971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.526407971\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 60 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 72 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 156 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.32795152709152376496503148672, −5.98975169266293578725729303150, −5.79024754387510605379696188808, −5.62526961381169072793773373830, −5.31395498027439616614026610426, −5.07300400569847056263394025325, −4.86464752291098193374799709576, −4.60123258485999828188728030835, −4.47685898806043931163223850136, −4.41503864359937703931414464297, −4.25882852792449715463878995562, −4.07936923970635971950703734797, −3.74104443628685192992003923582, −3.24001953078518237532610773919, −3.09061510428590603142983242932, −2.73667264220938017123003260660, −2.72438191421909186405267718005, −2.70944622361400824579772991400, −2.16158226783656086960862977224, −1.97040003953426461633677681274, −1.84107720991729302405582399393, −1.32588051780252714806230688703, −1.02287981619619312505588935403, −0.48101021454616031929391924967, −0.28339864770998730113055690983,
0.28339864770998730113055690983, 0.48101021454616031929391924967, 1.02287981619619312505588935403, 1.32588051780252714806230688703, 1.84107720991729302405582399393, 1.97040003953426461633677681274, 2.16158226783656086960862977224, 2.70944622361400824579772991400, 2.72438191421909186405267718005, 2.73667264220938017123003260660, 3.09061510428590603142983242932, 3.24001953078518237532610773919, 3.74104443628685192992003923582, 4.07936923970635971950703734797, 4.25882852792449715463878995562, 4.41503864359937703931414464297, 4.47685898806043931163223850136, 4.60123258485999828188728030835, 4.86464752291098193374799709576, 5.07300400569847056263394025325, 5.31395498027439616614026610426, 5.62526961381169072793773373830, 5.79024754387510605379696188808, 5.98975169266293578725729303150, 6.32795152709152376496503148672
