| L(s) = 1 | + 8·2-s − 18·3-s + 48·4-s − 5·5-s − 144·6-s − 105·7-s + 256·8-s + 243·9-s − 40·10-s + 451·11-s − 864·12-s + 1.09e3·13-s − 840·14-s + 90·15-s + 1.28e3·16-s + 3.05e3·17-s + 1.94e3·18-s − 722·19-s − 240·20-s + 1.89e3·21-s + 3.60e3·22-s − 386·23-s − 4.60e3·24-s − 739·25-s + 8.76e3·26-s − 2.91e3·27-s − 5.04e3·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.0894·5-s − 1.63·6-s − 0.809·7-s + 1.41·8-s + 9-s − 0.126·10-s + 1.12·11-s − 1.73·12-s + 1.79·13-s − 1.14·14-s + 0.103·15-s + 5/4·16-s + 2.56·17-s + 1.41·18-s − 0.458·19-s − 0.134·20-s + 0.935·21-s + 1.58·22-s − 0.152·23-s − 1.63·24-s − 0.236·25-s + 2.54·26-s − 0.769·27-s − 1.21·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.009351574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.009351574\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + p T + 764 T^{2} + p^{6} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 15 p T + 30878 T^{2} + 15 p^{6} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 41 p T + 103832 T^{2} - 41 p^{6} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1096 T + 955014 T^{2} - 1096 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3057 T + 5126596 T^{2} - 3057 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 386 T + 10251686 T^{2} + 386 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3446 T - 8756542 T^{2} - 3446 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 15362 T + 109907022 T^{2} - 15362 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5174 T + 144831258 T^{2} - 5174 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2128 T + 182227922 T^{2} + 2128 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 157 T + 290019198 T^{2} + 157 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1343 T + 337207484 T^{2} + 1343 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5326 T + 843284834 T^{2} - 5326 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5796 T + 1438159126 T^{2} + 5796 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1009 T + 1401428040 T^{2} - 1009 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 45720 T + 3132844790 T^{2} - 45720 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 50876 T + 3249458222 T^{2} + 50876 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 95907 T + 5499778592 T^{2} - 95907 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5132 T + 4209762078 T^{2} + 5132 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 61662 T + 8353903726 T^{2} + 61662 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 726 T + 11149774738 T^{2} + 726 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 75776 T + 7158092862 T^{2} - 75776 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−12.77195654328121871630227782082, −12.35688782143548740065521299961, −11.83313149620730011877297116592, −11.76610364004699031117057310242, −10.98498887350673506232658669223, −10.50853135392703653540824092679, −9.869594945363362683942622225311, −9.551563048208542500623583737343, −8.176172298262866490363332207696, −8.096331333881740138514254269342, −6.75590990315537048149233633719, −6.64203591289298639773972733858, −5.98749407144771624271205492221, −5.71721899372470958340748983309, −4.82123052689538584040824169157, −4.09821604949240528335569919705, −3.56868888781204854315592755962, −2.90698189031179807550198590993, −1.32229183085121059057301767231, −0.941523836456965097735852620212,
0.941523836456965097735852620212, 1.32229183085121059057301767231, 2.90698189031179807550198590993, 3.56868888781204854315592755962, 4.09821604949240528335569919705, 4.82123052689538584040824169157, 5.71721899372470958340748983309, 5.98749407144771624271205492221, 6.64203591289298639773972733858, 6.75590990315537048149233633719, 8.096331333881740138514254269342, 8.176172298262866490363332207696, 9.551563048208542500623583737343, 9.869594945363362683942622225311, 10.50853135392703653540824092679, 10.98498887350673506232658669223, 11.76610364004699031117057310242, 11.83313149620730011877297116592, 12.35688782143548740065521299961, 12.77195654328121871630227782082
