| L(s) = 1 | + 3·4-s − 6·5-s − 12·11-s − 4·13-s + 4·16-s − 4·17-s − 18·20-s + 16·23-s + 11·25-s − 14·29-s + 12·31-s − 6·41-s − 36·44-s + 12·47-s + 7·49-s − 12·52-s + 6·53-s + 72·55-s − 8·61-s + 9·64-s + 24·65-s − 12·67-s − 12·68-s − 12·71-s − 30·73-s + 12·79-s − 24·80-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 2.68·5-s − 3.61·11-s − 1.10·13-s + 16-s − 0.970·17-s − 4.02·20-s + 3.33·23-s + 11/5·25-s − 2.59·29-s + 2.15·31-s − 0.937·41-s − 5.42·44-s + 1.75·47-s + 49-s − 1.66·52-s + 0.824·53-s + 9.70·55-s − 1.02·61-s + 9/8·64-s + 2.97·65-s − 1.46·67-s − 1.45·68-s − 1.42·71-s − 3.51·73-s + 1.35·79-s − 2.68·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6185376520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6185376520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 5 | $C_2^2$ | \( ( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 19 | $C_2^3$ | \( 1 + 10 T^{2} - 261 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 115 T^{2} - 804 T^{3} + 5016 T^{4} - 804 p T^{5} + 115 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 86 T^{2} + 4251 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T + 69 T^{2} + 342 T^{3} + 2060 T^{4} + 342 p T^{5} + 69 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^3$ | \( 1 - 65 T^{2} + 2376 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 147 T^{2} - 1188 T^{3} + 9848 T^{4} - 1188 p T^{5} + 147 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T + 5 T^{2} + 450 T^{3} - 3756 T^{4} + 450 p T^{5} + 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 126 T^{2} + 9587 T^{4} - 126 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 8 T + 10 T^{2} - 544 T^{3} - 4709 T^{4} - 544 p T^{5} + 10 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 82 T^{2} + 408 T^{3} - 117 T^{4} + 408 p T^{5} + 82 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 12 T + 139 T^{2} + 1092 T^{3} + 6648 T^{4} + 1092 p T^{5} + 139 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 15 T + 148 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 12 T - 29 T^{2} - 180 T^{3} + 12312 T^{4} - 180 p T^{5} - 29 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 18 T + 301 T^{2} - 3474 T^{3} + 38316 T^{4} - 3474 p T^{5} + 301 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.44320185302878018332866625172, −7.07212731291390002389565129840, −7.00233508631424995119038833384, −6.95075411061233516151617760267, −6.73436033304535145985452725206, −6.19457343705380935279880467336, −5.85981300806408103781013948009, −5.66881625047677998311853729916, −5.42878572671530588298447478557, −5.29182888887573041405004148736, −5.03150211977101162849352580814, −4.58459826465688741655130203172, −4.47337038036019021751706779411, −4.22412996180016135864895004713, −4.17894174282672868760479083984, −3.46330405760525547649910581817, −3.14235839686318368354568700506, −3.13224830595946569089129920730, −2.91183315498519776141480229563, −2.53637715545144850508877551089, −2.27164201374704330691778077303, −2.07322861226253045369013194778, −1.44842099583519400398154705870, −0.60727476220761322079540109663, −0.30147129766992763349773226462,
0.30147129766992763349773226462, 0.60727476220761322079540109663, 1.44842099583519400398154705870, 2.07322861226253045369013194778, 2.27164201374704330691778077303, 2.53637715545144850508877551089, 2.91183315498519776141480229563, 3.13224830595946569089129920730, 3.14235839686318368354568700506, 3.46330405760525547649910581817, 4.17894174282672868760479083984, 4.22412996180016135864895004713, 4.47337038036019021751706779411, 4.58459826465688741655130203172, 5.03150211977101162849352580814, 5.29182888887573041405004148736, 5.42878572671530588298447478557, 5.66881625047677998311853729916, 5.85981300806408103781013948009, 6.19457343705380935279880467336, 6.73436033304535145985452725206, 6.95075411061233516151617760267, 7.00233508631424995119038833384, 7.07212731291390002389565129840, 7.44320185302878018332866625172
