| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s − 2·7-s + 4·8-s + 2·9-s − 4·10-s + 6·13-s − 4·14-s + 5·16-s + 4·18-s − 6·20-s − 25-s + 12·26-s − 6·28-s + 12·29-s + 6·32-s + 4·35-s + 6·36-s + 12·37-s − 8·40-s − 4·45-s + 3·49-s − 2·50-s + 18·52-s − 8·56-s + 24·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s − 0.755·7-s + 1.41·8-s + 2/3·9-s − 1.26·10-s + 1.66·13-s − 1.06·14-s + 5/4·16-s + 0.942·18-s − 1.34·20-s − 1/5·25-s + 2.35·26-s − 1.13·28-s + 2.22·29-s + 1.06·32-s + 0.676·35-s + 36-s + 1.97·37-s − 1.26·40-s − 0.596·45-s + 3/7·49-s − 0.282·50-s + 2.49·52-s − 1.06·56-s + 3.15·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.812113476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.812113476\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−10.32129560548360186822717810343, −10.13409162090470391946822969131, −9.572286512829513066235339645118, −8.916796765657529391865762344095, −8.659405714273056353805415891272, −7.926407037489289940509224525594, −7.77664105918032494423833267340, −7.26461399005716480830641530470, −6.58307179723680222256169822772, −6.49316190282479888105567661950, −5.83139711372591017460846882606, −5.78041661581863159137831522441, −4.63528533071231980593776403887, −4.50272825299940493342893435526, −4.17280262507474369555525950347, −3.43072499905650884314082867346, −3.19436875295639557327684188603, −2.63492315635532807709080821589, −1.64770219879929603046448867580, −0.893239535709086100249262301523,
0.893239535709086100249262301523, 1.64770219879929603046448867580, 2.63492315635532807709080821589, 3.19436875295639557327684188603, 3.43072499905650884314082867346, 4.17280262507474369555525950347, 4.50272825299940493342893435526, 4.63528533071231980593776403887, 5.78041661581863159137831522441, 5.83139711372591017460846882606, 6.49316190282479888105567661950, 6.58307179723680222256169822772, 7.26461399005716480830641530470, 7.77664105918032494423833267340, 7.926407037489289940509224525594, 8.659405714273056353805415891272, 8.916796765657529391865762344095, 9.572286512829513066235339645118, 10.13409162090470391946822969131, 10.32129560548360186822717810343
