| L(s) = 1 | − 4-s − 9-s + 16-s + 8·19-s + 12·29-s − 8·31-s + 36-s + 12·41-s − 49-s + 24·59-s + 4·61-s − 64-s − 8·76-s + 32·79-s + 81-s − 12·89-s − 12·101-s − 28·109-s − 12·116-s − 22·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 1/4·16-s + 1.83·19-s + 2.22·29-s − 1.43·31-s + 1/6·36-s + 1.87·41-s − 1/7·49-s + 3.12·59-s + 0.512·61-s − 1/8·64-s − 0.917·76-s + 3.60·79-s + 1/9·81-s − 1.27·89-s − 1.19·101-s − 2.68·109-s − 1.11·116-s − 2·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.921893716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.921893716\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.965647564677271618009044951800, −9.638384616400734223323281257554, −9.408060724752268989045128491625, −8.886528314111039536935450220269, −8.497059965004205102174048898332, −8.066584671532649632407287729800, −7.68571631476451150590522686968, −7.27920592613878604214021570500, −6.64548407826923498055455695336, −6.48574573038029265815665440974, −5.52525918119777978898168474243, −5.52301483738733649453254187189, −5.02875731541522540415598143449, −4.44747930833595349098186192469, −3.79560114778102437308091701121, −3.55377399215056524140481136791, −2.69846260282764285658579818210, −2.43629996176026191335162435104, −1.28835933150216580136909444409, −0.71568837696936243949026403270,
0.71568837696936243949026403270, 1.28835933150216580136909444409, 2.43629996176026191335162435104, 2.69846260282764285658579818210, 3.55377399215056524140481136791, 3.79560114778102437308091701121, 4.44747930833595349098186192469, 5.02875731541522540415598143449, 5.52301483738733649453254187189, 5.52525918119777978898168474243, 6.48574573038029265815665440974, 6.64548407826923498055455695336, 7.27920592613878604214021570500, 7.68571631476451150590522686968, 8.066584671532649632407287729800, 8.497059965004205102174048898332, 8.886528314111039536935450220269, 9.408060724752268989045128491625, 9.638384616400734223323281257554, 9.965647564677271618009044951800
