| L(s) = 1 | − 2-s + 3-s − 4-s + 5-s − 6-s − 5·7-s + 8-s − 2·9-s − 10-s + 2·11-s − 12-s + 2·13-s + 5·14-s + 15-s + 3·16-s + 3·17-s + 2·18-s − 19-s − 20-s − 5·21-s − 2·22-s − 6·23-s + 24-s + 3·25-s − 2·26-s − 2·27-s + 5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.88·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.603·11-s − 0.288·12-s + 0.554·13-s + 1.33·14-s + 0.258·15-s + 3/4·16-s + 0.727·17-s + 0.471·18-s − 0.229·19-s − 0.223·20-s − 1.09·21-s − 0.426·22-s − 1.25·23-s + 0.204·24-s + 3/5·25-s − 0.392·26-s − 0.384·27-s + 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1042 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1042 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3755928000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3755928000\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 521 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 27 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - 2 T - 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - T - 29 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 86 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T - 84 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−19.6781314958, −19.3929182151, −18.8984548881, −18.2858579855, −17.8864635819, −16.9890230889, −16.7202226407, −16.1888901358, −15.4696001249, −14.5492069125, −14.2791613599, −13.5365955240, −12.9348196603, −12.5008281054, −11.6144360394, −10.6204892492, −9.82811939675, −9.52630879597, −8.93538205655, −8.31673286485, −7.36129064644, −6.16244578557, −5.82812580818, −3.94953303477, −3.01952760481,
3.01952760481, 3.94953303477, 5.82812580818, 6.16244578557, 7.36129064644, 8.31673286485, 8.93538205655, 9.52630879597, 9.82811939675, 10.6204892492, 11.6144360394, 12.5008281054, 12.9348196603, 13.5365955240, 14.2791613599, 14.5492069125, 15.4696001249, 16.1888901358, 16.7202226407, 16.9890230889, 17.8864635819, 18.2858579855, 18.8984548881, 19.3929182151, 19.6781314958
