| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s + 3·9-s − 5·11-s − 12-s − 3·13-s + 2·14-s + 16-s − 2·17-s − 3·18-s + 3·19-s + 2·21-s + 5·22-s + 7·23-s + 24-s + 2·25-s + 3·26-s − 8·27-s − 2·28-s − 4·29-s + 10·31-s − 32-s + 5·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 9-s − 1.50·11-s − 0.288·12-s − 0.832·13-s + 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 0.688·19-s + 0.436·21-s + 1.06·22-s + 1.45·23-s + 0.204·24-s + 2/5·25-s + 0.588·26-s − 1.53·27-s − 0.377·28-s − 0.742·29-s + 1.79·31-s − 0.176·32-s + 0.870·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3146367453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3146367453\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 107 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 22 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 44 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 122 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.4401044307, −19.0367325770, −18.6843684576, −17.9642187988, −17.5483821484, −16.8503152364, −16.3711991211, −15.6822658836, −15.4275930479, −14.7155408454, −13.5450861890, −12.9660110983, −12.6917671223, −11.6624098100, −11.1336118914, −10.2750759455, −9.87168924838, −9.22793360657, −8.08542896447, −7.35201006126, −6.73805954848, −5.63002623595, −4.67944316802, −2.85514339534,
2.85514339534, 4.67944316802, 5.63002623595, 6.73805954848, 7.35201006126, 8.08542896447, 9.22793360657, 9.87168924838, 10.2750759455, 11.1336118914, 11.6624098100, 12.6917671223, 12.9660110983, 13.5450861890, 14.7155408454, 15.4275930479, 15.6822658836, 16.3711991211, 16.8503152364, 17.5483821484, 17.9642187988, 18.6843684576, 19.0367325770, 19.4401044307
