| L(s) = 1 | + 12·11-s − 2·19-s + 12·29-s + 6·31-s − 8·41-s − 11·49-s − 12·59-s + 6·61-s + 24·71-s + 16·79-s − 32·89-s + 16·101-s + 14·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 3.61·11-s − 0.458·19-s + 2.22·29-s + 1.07·31-s − 1.24·41-s − 1.57·49-s − 1.56·59-s + 0.768·61-s + 2.84·71-s + 1.80·79-s − 3.39·89-s + 1.59·101-s + 1.34·109-s + 7.81·121-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 + 1.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.513443055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.513443055\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.407202015703902848825397984217, −9.169099916481333577042128737426, −8.595368586648690634672496436619, −8.382962759418449624942456372720, −8.156007475410420899399204764692, −7.40460976196614024470719546334, −6.79647414943752038285844377992, −6.69448113285226935091825975717, −6.31445540855114474485551391928, −6.23780407860912771200427177611, −5.43584808474449039997366784673, −4.72348164855584968301861870293, −4.58559624166148458569392310252, −4.09474211125503989548263842210, −3.51991588285518470925247481426, −3.34974303329519965935740777753, −2.53658007805095236408202283035, −1.80081316668870256100744924672, −1.30666411591999157523324950156, −0.804277555809628234372991598527,
0.804277555809628234372991598527, 1.30666411591999157523324950156, 1.80081316668870256100744924672, 2.53658007805095236408202283035, 3.34974303329519965935740777753, 3.51991588285518470925247481426, 4.09474211125503989548263842210, 4.58559624166148458569392310252, 4.72348164855584968301861870293, 5.43584808474449039997366784673, 6.23780407860912771200427177611, 6.31445540855114474485551391928, 6.69448113285226935091825975717, 6.79647414943752038285844377992, 7.40460976196614024470719546334, 8.156007475410420899399204764692, 8.382962759418449624942456372720, 8.595368586648690634672496436619, 9.169099916481333577042128737426, 9.407202015703902848825397984217
