| L(s) = 1 | + 2·2-s − 4·3-s − 4·5-s − 8·6-s − 2·8-s + 10·9-s − 8·10-s + 8·11-s − 6·13-s + 16·15-s − 16-s − 2·17-s + 20·18-s + 2·19-s + 16·22-s + 8·23-s + 8·24-s + 10·25-s − 12·26-s − 20·27-s + 20·29-s + 32·30-s − 4·31-s − 32·33-s − 4·34-s + 4·38-s + 24·39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 2.30·3-s − 1.78·5-s − 3.26·6-s − 0.707·8-s + 10/3·9-s − 2.52·10-s + 2.41·11-s − 1.66·13-s + 4.13·15-s − 1/4·16-s − 0.485·17-s + 4.71·18-s + 0.458·19-s + 3.41·22-s + 1.66·23-s + 1.63·24-s + 2·25-s − 2.35·26-s − 3.84·27-s + 3.71·29-s + 5.84·30-s − 0.718·31-s − 5.57·33-s − 0.685·34-s + 0.648·38-s + 3.84·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 47^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 47^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.745478974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.745478974\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 47 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - p T + p^{2} T^{2} - 3 p T^{3} + 9 T^{4} - 3 p^{2} T^{5} + p^{4} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 2 p T^{2} - 16 T^{3} + 103 T^{4} - 16 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 8 T + 56 T^{2} - 256 T^{3} + 978 T^{4} - 256 p T^{5} + 56 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 6 T + 60 T^{2} + 232 T^{3} + 1219 T^{4} + 232 p T^{5} + 60 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 2 T + 28 T^{2} - 40 T^{3} + 251 T^{4} - 40 p T^{5} + 28 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 2 T + 36 T^{2} - 104 T^{3} + 687 T^{4} - 104 p T^{5} + 36 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 8 T + 62 T^{2} - 336 T^{3} + 1879 T^{4} - 336 p T^{5} + 62 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 4 T + 44 T^{2} + 236 T^{3} + 1370 T^{4} + 236 p T^{5} + 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 96 T^{2} + 8 T^{3} + 4930 T^{4} + 8 p T^{5} + 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 20 T + 302 T^{2} - 2832 T^{3} + 21691 T^{4} - 2832 p T^{5} + 302 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 8 T + 76 T^{2} + 328 T^{3} + 3414 T^{4} + 328 p T^{5} + 76 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 10 T + 196 T^{2} - 1392 T^{3} + 15451 T^{4} - 1392 p T^{5} + 196 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 10 T + 244 T^{2} - 1672 T^{3} + 21815 T^{4} - 1672 p T^{5} + 244 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 20 T + 210 T^{2} + 1296 T^{3} + 8331 T^{4} + 1296 p T^{5} + 210 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 180 T^{2} + 224 T^{3} + 15386 T^{4} + 224 p T^{5} + 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 18 T + 188 T^{2} - 1144 T^{3} + 8583 T^{4} - 1144 p T^{5} + 188 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 4 T + 28 T^{2} + 308 T^{3} + 10970 T^{4} + 308 p T^{5} + 28 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 20 T + 312 T^{2} - 3716 T^{3} + 39426 T^{4} - 3716 p T^{5} + 312 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 28 T + 476 T^{2} + 5388 T^{3} + 53158 T^{4} + 5388 p T^{5} + 476 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 68 T^{2} + 368 T^{3} + 4906 T^{4} + 368 p T^{5} + 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^3:S_4$ | \( 1 + 20 T + 404 T^{2} + 5164 T^{3} + 58358 T^{4} + 5164 p T^{5} + 404 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−7.32515830043046227134114113916, −7.04238113089590116829006909756, −6.91436885347375842537284156254, −6.79299653239344968462066310351, −6.49098677832349941816677725236, −6.43155063641086419350441862929, −6.04400112278807799385136038317, −5.85056887388041110113384980556, −5.45576548110866131389305971815, −5.08176404446748032357770544102, −4.96656706775210278815858309841, −4.93081218936936467509974121311, −4.67891716524379161904037165412, −4.21947527178244387581857858706, −4.19524433718368433358398962132, −4.05923055466294356652616400216, −4.01124921930819702221244830145, −3.21324240661074982387176291190, −3.08653713917297007128156976107, −2.90428787350610435929061426855, −2.30999976352384243564119616520, −1.65682659682459983161283038721, −1.20480822413186362905384904556, −0.65866912698569379408310025305, −0.64389631053701918071447175730,
0.64389631053701918071447175730, 0.65866912698569379408310025305, 1.20480822413186362905384904556, 1.65682659682459983161283038721, 2.30999976352384243564119616520, 2.90428787350610435929061426855, 3.08653713917297007128156976107, 3.21324240661074982387176291190, 4.01124921930819702221244830145, 4.05923055466294356652616400216, 4.19524433718368433358398962132, 4.21947527178244387581857858706, 4.67891716524379161904037165412, 4.93081218936936467509974121311, 4.96656706775210278815858309841, 5.08176404446748032357770544102, 5.45576548110866131389305971815, 5.85056887388041110113384980556, 6.04400112278807799385136038317, 6.43155063641086419350441862929, 6.49098677832349941816677725236, 6.79299653239344968462066310351, 6.91436885347375842537284156254, 7.04238113089590116829006909756, 7.32515830043046227134114113916
