| L(s) = 1 | + 12·3-s − 54·5-s + 243·9-s − 540·11-s − 836·13-s − 648·15-s − 594·17-s − 836·19-s + 4.10e3·23-s + 3.12e3·25-s + 7.02e3·27-s − 1.18e3·29-s − 4.25e3·31-s − 6.48e3·33-s + 298·37-s − 1.00e4·39-s + 3.44e4·41-s − 2.42e4·43-s − 1.31e4·45-s + 1.29e3·47-s − 7.12e3·51-s − 1.94e4·53-s + 2.91e4·55-s − 1.00e4·57-s + 7.66e3·59-s + 3.47e4·61-s + 4.51e4·65-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 0.965·5-s + 9-s − 1.34·11-s − 1.37·13-s − 0.743·15-s − 0.498·17-s − 0.531·19-s + 1.61·23-s + 25-s + 1.85·27-s − 0.262·29-s − 0.795·31-s − 1.03·33-s + 0.0357·37-s − 1.05·39-s + 3.20·41-s − 1.99·43-s − 0.965·45-s + 0.0855·47-s − 0.383·51-s − 0.953·53-s + 1.29·55-s − 0.408·57-s + 0.286·59-s + 1.19·61-s + 1.32·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.336287809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.336287809\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 4 p T - 11 p^{2} T^{2} - 4 p^{6} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 54 T - 209 T^{2} + 54 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 540 T + 130549 T^{2} + 540 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 418 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 594 T - 1067021 T^{2} + 594 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 44 p T - 4923 p^{2} T^{2} + 44 p^{6} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4104 T + 10406473 T^{2} - 4104 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 594 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 4256 T - 10515615 T^{2} + 4256 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 298 T - 69255153 T^{2} - 298 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 17226 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12100 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 1296 T - 227665391 T^{2} - 1296 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 19494 T - 38179457 T^{2} + 19494 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7668 T - 656126075 T^{2} - 7668 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 34738 T + 362132343 T^{2} - 34738 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 21812 T - 874361763 T^{2} + 21812 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 46872 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 67562 T + 2491552251 T^{2} + 67562 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 76912 T + 2838399345 T^{2} - 76912 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 67716 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 29754 T - 4698758933 T^{2} + 29754 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 122398 T + p^{5} T^{2} )^{2} \) |
| 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
−12.11799584581073320235122214514, −11.15048160276007465191240255165, −10.84997564948463840356854326260, −10.55336826916028740820779180017, −9.708907476793500710406030791969, −9.487030230800900125724932781206, −8.640381298613738334087411837506, −8.474678544708176535168099725471, −7.65677623084390485558386702642, −7.37863734862598837604263295449, −7.04319823267137191253757345171, −6.28118078188709559537665221042, −5.24065990559463151874655812455, −4.80349879031126003542564686476, −4.37176566652626612478305956345, −3.55435132408510697222573774724, −2.67692437645262519378295633483, −2.56291338358098939148822905029, −1.34348112205492016122095831886, −0.34674561365195753118595261140,
0.34674561365195753118595261140, 1.34348112205492016122095831886, 2.56291338358098939148822905029, 2.67692437645262519378295633483, 3.55435132408510697222573774724, 4.37176566652626612478305956345, 4.80349879031126003542564686476, 5.24065990559463151874655812455, 6.28118078188709559537665221042, 7.04319823267137191253757345171, 7.37863734862598837604263295449, 7.65677623084390485558386702642, 8.474678544708176535168099725471, 8.640381298613738334087411837506, 9.487030230800900125724932781206, 9.708907476793500710406030791969, 10.55336826916028740820779180017, 10.84997564948463840356854326260, 11.15048160276007465191240255165, 12.11799584581073320235122214514
