| L(s) = 1 | − 2·2-s + 4-s − 2·7-s + 2·8-s + 8·13-s + 4·14-s − 4·16-s − 4·19-s − 6·23-s + 10·25-s − 16·26-s − 2·28-s − 10·31-s + 2·32-s + 8·37-s + 8·38-s + 12·41-s + 8·43-s + 12·46-s − 18·47-s + 7·49-s − 20·50-s + 8·52-s − 4·56-s − 4·61-s + 20·62-s + 3·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 0.755·7-s + 0.707·8-s + 2.21·13-s + 1.06·14-s − 16-s − 0.917·19-s − 1.25·23-s + 2·25-s − 3.13·26-s − 0.377·28-s − 1.79·31-s + 0.353·32-s + 1.31·37-s + 1.29·38-s + 1.87·41-s + 1.21·43-s + 1.76·46-s − 2.62·47-s + 49-s − 2.82·50-s + 1.10·52-s − 0.534·56-s − 0.512·61-s + 2.54·62-s + 3/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5514759225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5514759225\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 4 T^{2} - 105 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} - 56 T^{3} - 89 T^{4} - 56 p T^{5} - 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - T^{2} - 54 T^{3} + 12 T^{4} - 54 p T^{5} - p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 37 | $C_2^2$ | \( ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 18 T + 167 T^{2} + 1134 T^{3} + 7212 T^{4} + 1134 p T^{5} + 167 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 88 T^{2} + 4935 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T - 92 T^{2} - 56 T^{3} + 6967 T^{4} - 56 p T^{5} - 92 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T - 68 T^{2} + 16 T^{3} + 8647 T^{4} + 16 p T^{5} - 68 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 6 T + 133 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 10 T - 65 T^{2} + 70 T^{3} + 13084 T^{4} + 70 p T^{5} - 65 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 24 T + 292 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T - 79 T^{2} - 378 T^{3} + 2100 T^{4} - 378 p T^{5} - 79 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−6.82914186007369605637861282646, −6.80718423710511291996804024902, −6.75133963205791592699768043907, −6.35809594633612898727779916516, −6.35778261817163863332274959546, −5.90498297820400869521403364448, −5.68084113679543115289865774415, −5.59843955587482128614009909544, −5.43667127218926352701723971465, −4.92544106224331987487947099075, −4.66758419912387318865709790653, −4.47202764934999347029831912716, −4.05935626035062936441210310642, −4.01216089199578239137811993596, −3.89175939020385398928351116114, −3.30777001380764061484798867366, −3.24303472018210787681044175020, −3.01401528721515711702608633035, −2.35591193833139393105188660904, −2.34149059121705305705294097767, −1.94504933193175269348730085057, −1.46691246463502151763816491415, −1.04421651691205194529517752704, −0.982168178589997154039320241182, −0.25554154876626425051116562597,
0.25554154876626425051116562597, 0.982168178589997154039320241182, 1.04421651691205194529517752704, 1.46691246463502151763816491415, 1.94504933193175269348730085057, 2.34149059121705305705294097767, 2.35591193833139393105188660904, 3.01401528721515711702608633035, 3.24303472018210787681044175020, 3.30777001380764061484798867366, 3.89175939020385398928351116114, 4.01216089199578239137811993596, 4.05935626035062936441210310642, 4.47202764934999347029831912716, 4.66758419912387318865709790653, 4.92544106224331987487947099075, 5.43667127218926352701723971465, 5.59843955587482128614009909544, 5.68084113679543115289865774415, 5.90498297820400869521403364448, 6.35778261817163863332274959546, 6.35809594633612898727779916516, 6.75133963205791592699768043907, 6.80718423710511291996804024902, 6.82914186007369605637861282646
