| L(s) = 1 | + 2·3-s + 3·9-s − 4·11-s − 8·17-s + 8·19-s + 6·25-s + 4·27-s − 8·33-s + 8·43-s + 49-s − 16·51-s + 16·57-s + 16·59-s + 16·67-s − 4·73-s + 12·75-s + 5·81-s + 8·83-s − 4·97-s − 12·99-s − 36·107-s + 20·113-s − 10·121-s + 127-s + 16·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 1.20·11-s − 1.94·17-s + 1.83·19-s + 6/5·25-s + 0.769·27-s − 1.39·33-s + 1.21·43-s + 1/7·49-s − 2.24·51-s + 2.11·57-s + 2.08·59-s + 1.95·67-s − 0.468·73-s + 1.38·75-s + 5/9·81-s + 0.878·83-s − 0.406·97-s − 1.20·99-s − 3.48·107-s + 1.88·113-s − 0.909·121-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.703498957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.703498957\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.617092129725433919332782114740, −8.106251167080634675502825469413, −7.76556317647643738947138137602, −7.30207867772318096349635238411, −6.76281206095777722703467003614, −6.59023808703041071787627483496, −5.54203834680246413327649010589, −5.30046762765077591142520164517, −4.73251701657473593050786657983, −4.12575993015631772148699876717, −3.64261842522368699265247670760, −2.75718031114594033742569839966, −2.68731442866574654307984003997, −1.92538980972810959214451574073, −0.837969571004769044543745728003,
0.837969571004769044543745728003, 1.92538980972810959214451574073, 2.68731442866574654307984003997, 2.75718031114594033742569839966, 3.64261842522368699265247670760, 4.12575993015631772148699876717, 4.73251701657473593050786657983, 5.30046762765077591142520164517, 5.54203834680246413327649010589, 6.59023808703041071787627483496, 6.76281206095777722703467003614, 7.30207867772318096349635238411, 7.76556317647643738947138137602, 8.106251167080634675502825469413, 8.617092129725433919332782114740
