| L(s) = 1 | − 9-s − 8·11-s + 16·19-s + 12·29-s − 12·41-s + 14·49-s + 24·59-s + 28·61-s − 16·79-s + 81-s − 4·89-s + 8·99-s − 28·101-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s − 16·171-s + 173-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 2.41·11-s + 3.67·19-s + 2.22·29-s − 1.87·41-s + 2·49-s + 3.12·59-s + 3.58·61-s − 1.80·79-s + 1/9·81-s − 0.423·89-s + 0.804·99-s − 2.78·101-s − 2.68·109-s + 2.36·121-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 + 1.69·169-s − 1.22·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.432521223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.432521223\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.201145566901437476918609800502, −8.529389122583693293741640865042, −8.270377436592026487822030625073, −8.190393920947441415700813720900, −7.66662785429589770897642491505, −7.15806364265476234137745060548, −6.90459188054174155988711236385, −6.75148687208535328069321928687, −5.58966397014236965753719663333, −5.55339513595594029290523741562, −5.32840170322913764121323210682, −5.12042229147671765131384252835, −4.36816029174412791771296539175, −3.89059680202904436367062816236, −3.20338781620179052333488351100, −2.87783097938211242420526184109, −2.68003945870910668634102494478, −1.99851166935410544626595899634, −1.00217395972657893711427271711, −0.65018523303646176530392063763,
0.65018523303646176530392063763, 1.00217395972657893711427271711, 1.99851166935410544626595899634, 2.68003945870910668634102494478, 2.87783097938211242420526184109, 3.20338781620179052333488351100, 3.89059680202904436367062816236, 4.36816029174412791771296539175, 5.12042229147671765131384252835, 5.32840170322913764121323210682, 5.55339513595594029290523741562, 5.58966397014236965753719663333, 6.75148687208535328069321928687, 6.90459188054174155988711236385, 7.15806364265476234137745060548, 7.66662785429589770897642491505, 8.190393920947441415700813720900, 8.270377436592026487822030625073, 8.529389122583693293741640865042, 9.201145566901437476918609800502
