| L(s) = 1 | − 2-s − 3·3-s − 5-s + 3·6-s − 7-s + 8·9-s + 10-s + 11-s − 2·13-s + 14-s + 3·15-s − 17-s − 8·18-s + 8·19-s + 3·21-s − 22-s + 16·23-s + 2·26-s − 20·27-s + 10·29-s − 3·30-s + 2·31-s + 32-s − 3·33-s + 34-s + 35-s + 12·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 0.447·5-s + 1.22·6-s − 0.377·7-s + 8/3·9-s + 0.316·10-s + 0.301·11-s − 0.554·13-s + 0.267·14-s + 0.774·15-s − 0.242·17-s − 1.88·18-s + 1.83·19-s + 0.654·21-s − 0.213·22-s + 3.33·23-s + 0.392·26-s − 3.84·27-s + 1.85·29-s − 0.547·30-s + 0.359·31-s + 0.176·32-s − 0.522·33-s + 0.171·34-s + 0.169·35-s + 1.97·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.128132338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.128132338\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 - T - 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^2:C_4$ | \( 1 + p T + T^{2} - T^{3} + 4 T^{4} - p T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 2 T + 11 T^{2} + 16 T^{3} + 49 T^{4} + 16 p T^{5} + 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + T - T^{2} - 53 T^{3} + 104 T^{4} - 53 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 8 T + 5 T^{2} + 122 T^{3} - 611 T^{4} + 122 p T^{5} + 5 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 10 T + 11 T^{2} + 120 T^{3} - 439 T^{4} + 120 p T^{5} + 11 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 2 T - 7 T^{2} - 154 T^{3} + 1225 T^{4} - 154 p T^{5} - 7 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 12 T + 107 T^{2} - 870 T^{3} + 6661 T^{4} - 870 p T^{5} + 107 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 + 20 T + 199 T^{2} + 1600 T^{3} + 11361 T^{4} + 1600 p T^{5} + 199 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 20 T + 103 T^{2} + 830 T^{3} - 12441 T^{4} + 830 p T^{5} + 103 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 8 T + 11 T^{2} - 404 T^{3} + 5609 T^{4} - 404 p T^{5} + 11 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 2 T + 5 T^{2} - 298 T^{3} + 1169 T^{4} - 298 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 10 T + 99 T^{2} + 1010 T^{3} + 11441 T^{4} + 1010 p T^{5} + 99 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 2 T + 130 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 18 T + 113 T^{2} + 966 T^{3} + 11815 T^{4} + 966 p T^{5} + 113 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 14 T + 123 T^{2} + 1540 T^{3} + 18461 T^{4} + 1540 p T^{5} + 123 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 10 T - 39 T^{2} - 370 T^{3} + 2711 T^{4} - 370 p T^{5} - 39 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 10 T + 77 T^{2} + 250 T^{3} - 2701 T^{4} + 250 p T^{5} + 77 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 14 T + 182 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 7 T - 73 T^{2} + 835 T^{3} + 1816 T^{4} + 835 p T^{5} - 73 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−7.26677921989307027769396333792, −7.24648301636551451409167297408, −7.00294112227483189631005288861, −6.97509028784027638442995139895, −6.36414136328759875192700857919, −6.30422109486960543480442222622, −6.02724495504129424609876595371, −5.93007957669212622717807813368, −5.49164408372385082286634472231, −5.29217576853524540742451375631, −4.92370869163635007077076205438, −4.89170878183644242501974958252, −4.65866979652282064003005619745, −4.40528559429978115272105001685, −4.09895741944768522199052867434, −3.77185289836533125795195071973, −3.47703720975018988266166824690, −2.91837619627508184093047525697, −2.84811801178507111888702903100, −2.79402824520010078336075366722, −1.94618129375991551564381026540, −1.56711545298870206067422054416, −1.12139018908485059017963703793, −0.76671932625391353934574609807, −0.59183579402864273969679052102,
0.59183579402864273969679052102, 0.76671932625391353934574609807, 1.12139018908485059017963703793, 1.56711545298870206067422054416, 1.94618129375991551564381026540, 2.79402824520010078336075366722, 2.84811801178507111888702903100, 2.91837619627508184093047525697, 3.47703720975018988266166824690, 3.77185289836533125795195071973, 4.09895741944768522199052867434, 4.40528559429978115272105001685, 4.65866979652282064003005619745, 4.89170878183644242501974958252, 4.92370869163635007077076205438, 5.29217576853524540742451375631, 5.49164408372385082286634472231, 5.93007957669212622717807813368, 6.02724495504129424609876595371, 6.30422109486960543480442222622, 6.36414136328759875192700857919, 6.97509028784027638442995139895, 7.00294112227483189631005288861, 7.24648301636551451409167297408, 7.26677921989307027769396333792
