| L(s) = 1 | − 2·3-s + 4-s + 4·5-s + 2·9-s − 2·11-s − 2·12-s − 2·13-s − 8·15-s − 3·16-s + 2·17-s − 4·19-s + 4·20-s − 4·23-s + 2·25-s − 6·27-s + 8·29-s + 10·31-s + 4·33-s + 2·36-s − 8·37-s + 4·39-s + 18·41-s + 16·43-s − 2·44-s + 8·45-s − 10·47-s + 6·48-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1.78·5-s + 2/3·9-s − 0.603·11-s − 0.577·12-s − 0.554·13-s − 2.06·15-s − 3/4·16-s + 0.485·17-s − 0.917·19-s + 0.894·20-s − 0.834·23-s + 2/5·25-s − 1.15·27-s + 1.48·29-s + 1.79·31-s + 0.696·33-s + 1/3·36-s − 1.31·37-s + 0.640·39-s + 2.81·41-s + 2.43·43-s − 0.301·44-s + 1.19·45-s − 1.45·47-s + 0.866·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 290521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290521 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.606863429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.606863429\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 18 T + 158 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 142 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 150 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.96537080075096561677003009513, −10.62488177703540135650348344723, −10.03522080405151530595692497139, −9.987651779959547751517389905091, −9.472926087534246252774508044574, −9.063174999709150314402136304462, −8.147663586618887641096158432684, −8.075258193869289737435894639904, −7.18424524510303540791015816359, −6.85927092204615940985358372739, −6.26454448036419840097745758653, −5.94113032611647041502513696057, −5.69073997891943019562563874218, −5.19122582593861883092161953530, −4.44982373744121376579929552492, −4.14186486690873075288704673450, −2.87901426576957832639051345580, −2.26615973375245716475657042143, −2.02151533514478628667616299223, −0.77902122701161020743563112113,
0.77902122701161020743563112113, 2.02151533514478628667616299223, 2.26615973375245716475657042143, 2.87901426576957832639051345580, 4.14186486690873075288704673450, 4.44982373744121376579929552492, 5.19122582593861883092161953530, 5.69073997891943019562563874218, 5.94113032611647041502513696057, 6.26454448036419840097745758653, 6.85927092204615940985358372739, 7.18424524510303540791015816359, 8.075258193869289737435894639904, 8.147663586618887641096158432684, 9.063174999709150314402136304462, 9.472926087534246252774508044574, 9.987651779959547751517389905091, 10.03522080405151530595692497139, 10.62488177703540135650348344723, 10.96537080075096561677003009513
