L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 6·5-s − 4·6-s − 2·7-s − 4·8-s − 3·9-s + 12·10-s + 12·11-s + 6·12-s + 2·13-s + 4·14-s − 12·15-s + 5·16-s − 6·17-s + 6·18-s + 4·19-s − 18·20-s − 4·21-s − 24·22-s − 8·24-s + 17·25-s − 4·26-s − 14·27-s − 6·28-s + 12·29-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 2.68·5-s − 1.63·6-s − 0.755·7-s − 1.41·8-s − 9-s + 3.79·10-s + 3.61·11-s + 1.73·12-s + 0.554·13-s + 1.06·14-s − 3.09·15-s + 5/4·16-s − 1.45·17-s + 1.41·18-s + 0.917·19-s − 4.02·20-s − 0.872·21-s − 5.11·22-s − 1.63·24-s + 17/5·25-s − 0.784·26-s − 2.69·27-s − 1.13·28-s + 2.22·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2658192833\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2658192833\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−19.6933997517, −19.6933997517, −19.1444693222, −19.1444693222, −17.5187720040, −17.5187720040, −16.3289342707, −16.3289342707, −15.2621223361, −15.2621223361, −14.1081230548, −14.1081230548, −12.1081086828, −12.1081086828, −11.2148699926, −11.2148699926, −9.25208611998, −9.25208611998, −8.28912505160, −8.28912505160, −6.76423132328, −6.76423132328, −3.64028761626, −3.64028761626,
3.64028761626, 3.64028761626, 6.76423132328, 6.76423132328, 8.28912505160, 8.28912505160, 9.25208611998, 9.25208611998, 11.2148699926, 11.2148699926, 12.1081086828, 12.1081086828, 14.1081230548, 14.1081230548, 15.2621223361, 15.2621223361, 16.3289342707, 16.3289342707, 17.5187720040, 17.5187720040, 19.1444693222, 19.1444693222, 19.6933997517, 19.6933997517