| L(s) = 1 | − 12·11-s − 4·19-s + 12·29-s − 20·31-s + 12·41-s − 49-s + 28·61-s − 12·71-s + 32·79-s + 12·89-s − 12·101-s − 4·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 3.61·11-s − 0.917·19-s + 2.22·29-s − 3.59·31-s + 1.87·41-s − 1/7·49-s + 3.58·61-s − 1.42·71-s + 3.60·79-s + 1.27·89-s − 1.19·101-s − 0.383·109-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 + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.592748039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.592748039\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−8.037672269532777277215710453566, −7.974993749143955514348494516331, −7.56293888691250174193412738247, −7.27864900614843589322956216799, −6.73539172803285177306077442875, −6.68472230711466095013779907733, −5.75694998868896289625131497731, −5.71640828628891669085236793668, −5.45648175859613797717418998714, −5.06809646210375517438779108257, −4.57962017951305505505833440253, −4.46001566948391859295971699818, −3.66542247193884457469287440118, −3.45085974324065457683500676928, −2.80724755414600459599811911102, −2.58530176153702754606729741851, −2.01713772566379161559439994364, −2.00099877705111668925064305878, −0.68922434238440671713520378046, −0.45586460558548579626281711890,
0.45586460558548579626281711890, 0.68922434238440671713520378046, 2.00099877705111668925064305878, 2.01713772566379161559439994364, 2.58530176153702754606729741851, 2.80724755414600459599811911102, 3.45085974324065457683500676928, 3.66542247193884457469287440118, 4.46001566948391859295971699818, 4.57962017951305505505833440253, 5.06809646210375517438779108257, 5.45648175859613797717418998714, 5.71640828628891669085236793668, 5.75694998868896289625131497731, 6.68472230711466095013779907733, 6.73539172803285177306077442875, 7.27864900614843589322956216799, 7.56293888691250174193412738247, 7.974993749143955514348494516331, 8.037672269532777277215710453566
