| L(s) = 1 | − 2·3-s + 4-s − 4·5-s + 9-s + 4·11-s − 2·12-s + 8·15-s + 16-s − 6·17-s + 8·19-s − 4·20-s − 4·23-s + 3·25-s − 2·27-s + 8·29-s − 8·33-s + 36-s − 4·37-s − 6·43-s + 4·44-s − 4·45-s + 16·47-s − 2·48-s − 13·49-s + 12·51-s + 12·53-s − 16·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s − 1.78·5-s + 1/3·9-s + 1.20·11-s − 0.577·12-s + 2.06·15-s + 1/4·16-s − 1.45·17-s + 1.83·19-s − 0.894·20-s − 0.834·23-s + 3/5·25-s − 0.384·27-s + 1.48·29-s − 1.39·33-s + 1/6·36-s − 0.657·37-s − 0.914·43-s + 0.603·44-s − 0.596·45-s + 2.33·47-s − 0.288·48-s − 1.85·49-s + 1.68·51-s + 1.64·53-s − 2.15·55-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
| \(L(1)\) |
\(\approx\) |
\(0.3201552354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3201552354\) |
| \(L(\frac{3}{2})\) |
|
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_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−19.6933997517, −19.1444693222, −18.3883990965, −17.5187720040, −17.4643953306, −16.3289342707, −16.2105519665, −15.4425168943, −15.2621223361, −14.1081230548, −13.5922863880, −12.1499536241, −12.1081086828, −11.4971267324, −11.2148699926, −10.2802576297, −9.25208611998, −8.28912505160, −7.45985560668, −6.76423132328, −5.88908450916, −4.61153453491, −3.64028761626,
3.64028761626, 4.61153453491, 5.88908450916, 6.76423132328, 7.45985560668, 8.28912505160, 9.25208611998, 10.2802576297, 11.2148699926, 11.4971267324, 12.1081086828, 12.1499536241, 13.5922863880, 14.1081230548, 15.2621223361, 15.4425168943, 16.2105519665, 16.3289342707, 17.4643953306, 17.5187720040, 18.3883990965, 19.1444693222, 19.6933997517
