| L(s) = 1 | − 3·2-s + 6·4-s − 5-s − 10·8-s + 3·9-s + 3·10-s − 13-s + 15·16-s + 17-s − 9·18-s − 6·20-s + 3·26-s − 21·32-s − 3·34-s + 18·36-s + 37-s + 10·40-s + 41-s − 3·45-s + 3·49-s − 6·52-s − 53-s + 61-s + 28·64-s + 65-s + 6·68-s − 30·72-s + ⋯ |
| L(s) = 1 | − 3·2-s + 6·4-s − 5-s − 10·8-s + 3·9-s + 3·10-s − 13-s + 15·16-s + 17-s − 9·18-s − 6·20-s + 3·26-s − 21·32-s − 3·34-s + 18·36-s + 37-s + 10·40-s + 41-s − 3·45-s + 3·49-s − 6·52-s − 53-s + 61-s + 28·64-s + 65-s + 6·68-s − 30·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5000106439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5000106439\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 29 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 5 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 17 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 41 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 61 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 97 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.903413278117031458969881964708, −7.47709001309989253339558478685, −7.45749054458501271312042975999, −7.28972062358337463590929590647, −7.10100573146195452117070645446, −6.78578388401659292837254197821, −6.70897484567356386917193138464, −6.09753885077374383339514090944, −5.98559493231980408096428427340, −5.94584046402975017574545687181, −5.19865544382881879617341048410, −5.14352511238117401696326059677, −4.72500517201083121061096553805, −4.17906567621252847752725643778, −4.12748198461772658322868111250, −3.78078859792851565068141242302, −3.36421363108133430410468303240, −3.21068963080272709495590806761, −2.71026866566324459202221696453, −2.19015294019073983310212102479, −2.11863338502720460257551192956, −1.89686527629691327744203762069, −1.11816132954976476856221113807, −1.04586633872361525407601006603, −0.71777170660935102793492447871,
0.71777170660935102793492447871, 1.04586633872361525407601006603, 1.11816132954976476856221113807, 1.89686527629691327744203762069, 2.11863338502720460257551192956, 2.19015294019073983310212102479, 2.71026866566324459202221696453, 3.21068963080272709495590806761, 3.36421363108133430410468303240, 3.78078859792851565068141242302, 4.12748198461772658322868111250, 4.17906567621252847752725643778, 4.72500517201083121061096553805, 5.14352511238117401696326059677, 5.19865544382881879617341048410, 5.94584046402975017574545687181, 5.98559493231980408096428427340, 6.09753885077374383339514090944, 6.70897484567356386917193138464, 6.78578388401659292837254197821, 7.10100573146195452117070645446, 7.28972062358337463590929590647, 7.45749054458501271312042975999, 7.47709001309989253339558478685, 7.903413278117031458969881964708
