| L(s) = 1 | − 4·9-s + 3·11-s − 13-s + 2·17-s + 3·19-s − 2·23-s + 3·27-s − 29-s + 3·31-s + 37-s − 9·41-s + 43-s + 13·47-s − 16·49-s + 9·53-s + 10·59-s − 21·61-s + 13·67-s − 71-s + 17·73-s + 20·79-s + 4·81-s − 83-s + 4·89-s + 15·97-s − 12·99-s − 16·101-s + ⋯ |
| L(s) = 1 | − 4/3·9-s + 0.904·11-s − 0.277·13-s + 0.485·17-s + 0.688·19-s − 0.417·23-s + 0.577·27-s − 0.185·29-s + 0.538·31-s + 0.164·37-s − 1.40·41-s + 0.152·43-s + 1.89·47-s − 2.28·49-s + 1.23·53-s + 1.30·59-s − 2.68·61-s + 1.58·67-s − 0.118·71-s + 1.98·73-s + 2.25·79-s + 4/9·81-s − 0.109·83-s + 0.423·89-s + 1.52·97-s − 1.20·99-s − 1.59·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.138147308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.138147308\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 4 T^{2} - p T^{3} + 4 p T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 16 T^{2} + 3 T^{3} + 16 p T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 3 T + 31 T^{2} - 65 T^{3} + 31 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + T + 35 T^{2} + 23 T^{3} + 35 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 32 T^{2} - 23 T^{3} + 32 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 62 T^{2} + 87 T^{3} + 62 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + T + 9 T^{2} + 107 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 57 T^{2} - 105 T^{3} + 57 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - T + 87 T^{2} - 29 T^{3} + 87 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 9 T + 103 T^{2} + 563 T^{3} + 103 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - T + 88 T^{2} + 27 T^{3} + 88 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 13 T + 171 T^{2} - 1213 T^{3} + 171 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 166 T^{2} - 945 T^{3} + 166 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 10 T + 193 T^{2} - 1172 T^{3} + 193 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 21 T + 265 T^{2} + 37 p T^{3} + 265 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 13 T + 237 T^{2} - 1767 T^{3} + 237 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + T + 132 T^{2} - 139 T^{3} + 132 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 17 T + 177 T^{2} - 1407 T^{3} + 177 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 20 T + 292 T^{2} - 2859 T^{3} + 292 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + T + 131 T^{2} + 57 T^{3} + 131 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 4 T + 148 T^{2} - 1067 T^{3} + 148 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 15 T + 327 T^{2} - 2797 T^{3} + 327 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−7.04359078618064592789397688194, −6.59640468724440912746467051208, −6.43251347559225452320834435285, −6.40228643086464313250830305434, −5.95612796513108685300930136310, −5.84284584137753179560479485599, −5.61453006491792719253143023063, −5.22247068409608625178184599529, −5.09743560431841196345257280817, −4.93995352759426478769253455069, −4.56211357481779345236908163252, −4.40235447793232113066141594491, −3.97764712925113173969753415517, −3.62180664630984072952840358602, −3.54846785922016636985988336956, −3.41477090455186116209321924847, −2.88049943386810071603520145493, −2.67546180104255472066642430831, −2.62801746698960412056659645496, −1.96158404338984469987799207071, −1.78615308264847434476178639419, −1.64343460817352830999490658446, −0.897031711372517480023002984809, −0.58162228428375042775719492280, −0.55778423598784751467143217399,
0.55778423598784751467143217399, 0.58162228428375042775719492280, 0.897031711372517480023002984809, 1.64343460817352830999490658446, 1.78615308264847434476178639419, 1.96158404338984469987799207071, 2.62801746698960412056659645496, 2.67546180104255472066642430831, 2.88049943386810071603520145493, 3.41477090455186116209321924847, 3.54846785922016636985988336956, 3.62180664630984072952840358602, 3.97764712925113173969753415517, 4.40235447793232113066141594491, 4.56211357481779345236908163252, 4.93995352759426478769253455069, 5.09743560431841196345257280817, 5.22247068409608625178184599529, 5.61453006491792719253143023063, 5.84284584137753179560479485599, 5.95612796513108685300930136310, 6.40228643086464313250830305434, 6.43251347559225452320834435285, 6.59640468724440912746467051208, 7.04359078618064592789397688194
