| L(s) = 1 | − 2-s − 5-s + 3·9-s + 10-s − 11-s − 17-s − 3·18-s − 19-s + 22-s − 29-s − 31-s + 34-s − 37-s + 38-s − 43-s − 3·45-s − 47-s + 3·49-s + 55-s + 58-s − 59-s + 62-s + 74-s + 6·81-s + 85-s + 86-s + 3·90-s + ⋯ |
| L(s) = 1 | − 2-s − 5-s + 3·9-s + 10-s − 11-s − 17-s − 3·18-s − 19-s + 22-s − 29-s − 31-s + 34-s − 37-s + 38-s − 43-s − 3·45-s − 47-s + 3·49-s + 55-s + 58-s − 59-s + 62-s + 74-s + 6·81-s + 85-s + 86-s + 3·90-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3442951 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3442951 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1457248293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1457248293\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 151 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 17 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 19 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 31 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 37 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 47 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 59 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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
−11.85438620780584819009386000933, −11.70141259123624616684364305266, −10.99704340753625651184858104778, −10.68997024797752278504909682491, −10.59481161442468845621401137610, −10.19852870201252687896166884705, −9.949700238281880146439100369668, −9.251703683052829018409362241139, −9.243627037633449600472867243059, −9.013192558661001383822127103355, −8.247752176285942870731625179537, −8.116820828412582764542650046626, −7.72172489061707095845989243299, −7.27784716890115809412202436983, −6.95499951952608680352810802672, −6.86724953513348673953115213883, −6.24830716053211365215499522053, −5.48609729130887863451926607972, −5.13978808114584291835441902554, −4.45516567445391320591192585649, −4.20805702542999305543125531760, −3.92589233723069360045659193864, −3.27308661500225015771481383375, −2.17595385646999302389858797878, −1.69473509693998334125644510580,
1.69473509693998334125644510580, 2.17595385646999302389858797878, 3.27308661500225015771481383375, 3.92589233723069360045659193864, 4.20805702542999305543125531760, 4.45516567445391320591192585649, 5.13978808114584291835441902554, 5.48609729130887863451926607972, 6.24830716053211365215499522053, 6.86724953513348673953115213883, 6.95499951952608680352810802672, 7.27784716890115809412202436983, 7.72172489061707095845989243299, 8.116820828412582764542650046626, 8.247752176285942870731625179537, 9.013192558661001383822127103355, 9.243627037633449600472867243059, 9.251703683052829018409362241139, 9.949700238281880146439100369668, 10.19852870201252687896166884705, 10.59481161442468845621401137610, 10.68997024797752278504909682491, 10.99704340753625651184858104778, 11.70141259123624616684364305266, 11.85438620780584819009386000933
