| L(s) = 1 | − 2·2-s − 3-s + 2·6-s − 3·7-s + 4·8-s − 2·9-s − 2·11-s − 2·13-s + 6·14-s − 4·16-s − 8·17-s + 4·18-s + 3·19-s + 3·21-s + 4·22-s + 23-s − 4·24-s − 5·25-s + 4·26-s + 2·27-s + 10·29-s − 31-s + 2·33-s + 16·34-s − 3·37-s − 6·38-s + 2·39-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 0.816·6-s − 1.13·7-s + 1.41·8-s − 2/3·9-s − 0.603·11-s − 0.554·13-s + 1.60·14-s − 16-s − 1.94·17-s + 0.942·18-s + 0.688·19-s + 0.654·21-s + 0.852·22-s + 0.208·23-s − 0.816·24-s − 25-s + 0.784·26-s + 0.384·27-s + 1.85·29-s − 0.179·31-s + 0.348·33-s + 2.74·34-s − 0.493·37-s − 0.973·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2725 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2725 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + p T^{2} \) |
| 109 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 16 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - T - 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 64 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 9 T + 76 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 67 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 44 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 13 T + 135 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5 T + 94 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 23 T + 292 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.4885434876, −17.9522497223, −17.6640069878, −17.3202276772, −16.7647874561, −16.1892660007, −15.6533593488, −15.3024570715, −14.1189431047, −13.8032135525, −13.1893204241, −12.6509443538, −11.8897318060, −11.2643160742, −10.5929616240, −9.98935839268, −9.60565775532, −8.85340351774, −8.52744224766, −7.66073957965, −6.86947665809, −6.12064664496, −5.17386453005, −4.28755467732, −2.76793982171, 0,
2.76793982171, 4.28755467732, 5.17386453005, 6.12064664496, 6.86947665809, 7.66073957965, 8.52744224766, 8.85340351774, 9.60565775532, 9.98935839268, 10.5929616240, 11.2643160742, 11.8897318060, 12.6509443538, 13.1893204241, 13.8032135525, 14.1189431047, 15.3024570715, 15.6533593488, 16.1892660007, 16.7647874561, 17.3202276772, 17.6640069878, 17.9522497223, 18.4885434876
