| L(s) = 1 | − 5·2-s + 8·4-s + 5·5-s − 9·7-s + 5·8-s − 25·10-s − 8·11-s − 43·13-s + 45·14-s − 25·16-s + 244·17-s − 118·19-s + 40·20-s + 40·22-s − 213·23-s + 215·26-s − 72·28-s + 224·29-s + 36·31-s + 40·32-s − 1.22e3·34-s − 45·35-s + 412·37-s + 590·38-s + 25·40-s + 413·41-s + 392·43-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 4-s + 0.447·5-s − 0.485·7-s + 0.220·8-s − 0.790·10-s − 0.219·11-s − 0.917·13-s + 0.859·14-s − 0.390·16-s + 3.48·17-s − 1.42·19-s + 0.447·20-s + 0.387·22-s − 1.93·23-s + 1.62·26-s − 0.485·28-s + 1.43·29-s + 0.208·31-s + 0.220·32-s − 6.15·34-s − 0.217·35-s + 1.83·37-s + 2.51·38-s + 0.0988·40-s + 1.57·41-s + 1.39·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9481861685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9481861685\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 5 T + 17 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 9 T - 262 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 8 T - 1267 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 43 T - 348 T^{2} + 43 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 122 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 59 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 213 T + 33202 T^{2} + 213 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 224 T + 25787 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 36 T - 28495 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 206 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 413 T + 101648 T^{2} - 413 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 392 T + 74157 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 311 T - 7102 T^{2} + 311 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 377 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 337 T - 91810 T^{2} - 337 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 40 T - 225381 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 348 T - 179659 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 62 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1214 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 294 T - 406603 T^{2} - 294 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 534 T - 286631 T^{2} - 534 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 810 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 928 T - 51489 T^{2} - 928 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−10.44724554396460045355229778703, −10.27087771101878282954279051556, −10.13336677412703279302436041059, −9.865132684543373014878803904005, −9.166319946083738147286743555203, −9.009453921639664838947535993325, −8.187227456481931034869801924744, −8.037460931841631288222744420249, −7.51499814036303595034648936932, −7.29478940050813931128092727605, −6.22013194273450296198752718570, −5.86477058282750707413137012266, −5.68070380261179145117892025585, −4.56670113856846847007058687048, −4.22441303149087682504392401537, −3.27186059890917882040787797240, −2.65320212878399707592651770514, −1.94667960305857927765196463026, −0.849029250873772318234458793979, −0.64535816991815658587434041234,
0.64535816991815658587434041234, 0.849029250873772318234458793979, 1.94667960305857927765196463026, 2.65320212878399707592651770514, 3.27186059890917882040787797240, 4.22441303149087682504392401537, 4.56670113856846847007058687048, 5.68070380261179145117892025585, 5.86477058282750707413137012266, 6.22013194273450296198752718570, 7.29478940050813931128092727605, 7.51499814036303595034648936932, 8.037460931841631288222744420249, 8.187227456481931034869801924744, 9.009453921639664838947535993325, 9.166319946083738147286743555203, 9.865132684543373014878803904005, 10.13336677412703279302436041059, 10.27087771101878282954279051556, 10.44724554396460045355229778703
