| L(s) = 1 | − 4·2-s + 2·3-s + 10·4-s − 3·5-s − 8·6-s − 20·8-s − 3·9-s + 12·10-s − 3·11-s + 20·12-s − 4·13-s − 6·15-s + 35·16-s + 3·17-s + 12·18-s − 10·19-s − 30·20-s + 12·22-s + 9·23-s − 40·24-s + 4·25-s + 16·26-s − 14·27-s + 6·29-s + 24·30-s + 8·31-s − 56·32-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 1.15·3-s + 5·4-s − 1.34·5-s − 3.26·6-s − 7.07·8-s − 9-s + 3.79·10-s − 0.904·11-s + 5.77·12-s − 1.10·13-s − 1.54·15-s + 35/4·16-s + 0.727·17-s + 2.82·18-s − 2.29·19-s − 6.70·20-s + 2.55·22-s + 1.87·23-s − 8.16·24-s + 4/5·25-s + 3.13·26-s − 2.69·27-s + 1.11·29-s + 4.38·30-s + 1.43·31-s − 9.89·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2011787736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2011787736\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + 3 T + p T^{2} )^{2}( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 18 p T^{5} - 7 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 3 T - 19 T^{2} + 18 T^{3} + 342 T^{4} + 18 p T^{5} - 19 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 - 9 T + p T^{2} )^{2}( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $D_4\times C_2$ | \( 1 - 6 T + 2 T^{2} + 144 T^{3} - 729 T^{4} + 144 p T^{5} + 2 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 15 T + 95 T^{2} + 720 T^{3} + 5994 T^{4} + 720 p T^{5} + 95 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + T - 11 T^{2} - 74 T^{3} - 1748 T^{4} - 74 p T^{5} - 11 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T - 46 T^{2} + 144 T^{3} + 2007 T^{4} + 144 p T^{5} - 46 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 7 T - 35 T^{2} - 434 T^{3} - 1850 T^{4} - 434 p T^{5} - 35 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 7 T + 96 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 12 T + 74 T^{2} - 1152 T^{3} - 13941 T^{4} - 1152 p T^{5} + 74 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 18 T + 98 T^{2} + 864 T^{3} + 14319 T^{4} + 864 p T^{5} + 98 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + T - 119 T^{2} - 74 T^{3} + 4894 T^{4} - 74 p T^{5} - 119 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−7.45212130261850003651368919876, −7.20522880343029496510321895500, −6.94113952076478391115287803848, −6.85177736274537856809789061654, −6.60574952439824204582245878961, −6.34965658173622514145795234122, −5.99163455981608958712422138955, −5.97922389491566618933190716438, −5.37362729261151268202039248911, −5.23492794995606086474644387144, −4.88867815302078554668618865899, −4.88131699890072881416083461128, −4.26668683659216218235463115051, −4.11883595147716406648718460210, −3.58716830092482537799373396611, −3.32663057106721408303186125694, −3.17659171318759441331865818555, −2.93794216901806269454716678468, −2.53131245205573829950108739792, −2.37287115006211072697219471491, −2.28577084803254638252556945450, −1.57554304970466362172548748094, −1.38059435506006490010756870256, −0.54591743175343111291134740029, −0.28275972629578904859996732317,
0.28275972629578904859996732317, 0.54591743175343111291134740029, 1.38059435506006490010756870256, 1.57554304970466362172548748094, 2.28577084803254638252556945450, 2.37287115006211072697219471491, 2.53131245205573829950108739792, 2.93794216901806269454716678468, 3.17659171318759441331865818555, 3.32663057106721408303186125694, 3.58716830092482537799373396611, 4.11883595147716406648718460210, 4.26668683659216218235463115051, 4.88131699890072881416083461128, 4.88867815302078554668618865899, 5.23492794995606086474644387144, 5.37362729261151268202039248911, 5.97922389491566618933190716438, 5.99163455981608958712422138955, 6.34965658173622514145795234122, 6.60574952439824204582245878961, 6.85177736274537856809789061654, 6.94113952076478391115287803848, 7.20522880343029496510321895500, 7.45212130261850003651368919876
