| L(s) = 1 | + 2·5-s + 2·13-s + 10·17-s − 25-s − 14·37-s + 20·41-s − 10·53-s + 24·61-s + 4·65-s + 22·73-s + 20·85-s − 26·97-s − 40·101-s + 30·113-s + 22·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.554·13-s + 2.42·17-s − 1/5·25-s − 2.30·37-s + 3.12·41-s − 1.37·53-s + 3.07·61-s + 0.496·65-s + 2.57·73-s + 2.16·85-s − 2.63·97-s − 3.98·101-s + 2.82·113-s + 2·121-s − 1.07·125-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 + 2/13·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.654491335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.654491335\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.66662421035005122413665592620, −9.898988187197046592924298763821, −9.813040406290643222927654857079, −9.629131391752696114180756406965, −8.892777616076392053113395387391, −8.538180891490929448922770681746, −7.86095739511536240670876666551, −7.82855208493149221467800522786, −7.05712736452429743387604982721, −6.69874058099085780973968331031, −6.06599255800950452550590219230, −5.66122178961127731544996289487, −5.37953566580412955310819941787, −4.93054464557888470245271706652, −3.85724090966648733626557431643, −3.80709863824822138295433709877, −2.98570042786697898155467027824, −2.38768127530993581969491773566, −1.58826915175872574550972198792, −0.935267726284866606307417625882,
0.935267726284866606307417625882, 1.58826915175872574550972198792, 2.38768127530993581969491773566, 2.98570042786697898155467027824, 3.80709863824822138295433709877, 3.85724090966648733626557431643, 4.93054464557888470245271706652, 5.37953566580412955310819941787, 5.66122178961127731544996289487, 6.06599255800950452550590219230, 6.69874058099085780973968331031, 7.05712736452429743387604982721, 7.82855208493149221467800522786, 7.86095739511536240670876666551, 8.538180891490929448922770681746, 8.892777616076392053113395387391, 9.629131391752696114180756406965, 9.813040406290643222927654857079, 9.898988187197046592924298763821, 10.66662421035005122413665592620
