| L(s) = 1 | − 2-s + 2·3-s + 4-s − 5-s − 2·6-s − 7-s − 3·8-s + 2·9-s + 10-s − 4·11-s + 2·12-s − 2·13-s + 14-s − 2·15-s + 16-s + 3·17-s − 2·18-s + 4·19-s − 20-s − 2·21-s + 4·22-s − 4·23-s − 6·24-s + 25-s + 2·26-s + 6·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s − 0.377·7-s − 1.06·8-s + 2/3·9-s + 0.316·10-s − 1.20·11-s + 0.577·12-s − 0.554·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.727·17-s − 0.471·18-s + 0.917·19-s − 0.223·20-s − 0.436·21-s + 0.852·22-s − 0.834·23-s − 1.22·24-s + 1/5·25-s + 0.392·26-s + 1.15·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2371 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2371 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5999372526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5999372526\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2371 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 68 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 52 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 94 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 6 T - 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 7 T + 72 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−18.6222506980, −18.2102730790, −17.7945163442, −16.9955053976, −16.2757241837, −15.8768728889, −15.4862758142, −14.9066032847, −14.3450789248, −13.7504628039, −13.1878519301, −12.2123822295, −12.1819006952, −11.2233693778, −10.3838846045, −9.88196142198, −9.32189704112, −8.67314065284, −7.88861466382, −7.70345107475, −6.71892606316, −5.78326498426, −4.62853316642, −3.21047103397, −2.65352551709,
2.65352551709, 3.21047103397, 4.62853316642, 5.78326498426, 6.71892606316, 7.70345107475, 7.88861466382, 8.67314065284, 9.32189704112, 9.88196142198, 10.3838846045, 11.2233693778, 12.1819006952, 12.2123822295, 13.1878519301, 13.7504628039, 14.3450789248, 14.9066032847, 15.4862758142, 15.8768728889, 16.2757241837, 16.9955053976, 17.7945163442, 18.2102730790, 18.6222506980
