| L(s) = 1 | + 3-s + 4-s + 3·5-s − 2·9-s − 6·11-s + 12-s + 3·15-s + 16-s + 3·20-s + 3·23-s − 25-s − 5·27-s + 31-s − 6·33-s − 2·36-s + 13·37-s − 6·44-s − 6·45-s − 12·47-s + 48-s − 4·49-s − 9·53-s − 18·55-s − 6·59-s + 3·60-s + 64-s + 67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 1.34·5-s − 2/3·9-s − 1.80·11-s + 0.288·12-s + 0.774·15-s + 1/4·16-s + 0.670·20-s + 0.625·23-s − 1/5·25-s − 0.962·27-s + 0.179·31-s − 1.04·33-s − 1/3·36-s + 2.13·37-s − 0.904·44-s − 0.894·45-s − 1.75·47-s + 0.144·48-s − 4/7·49-s − 1.23·53-s − 2.42·55-s − 0.781·59-s + 0.387·60-s + 1/8·64-s + 0.122·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13068 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13068 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.479285710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.479285710\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−11.18067797813160868987581050763, −10.75172005791015318217940061044, −9.992641543815054519789795250540, −9.693722725011220009511555723373, −9.212631067126484442668935106900, −8.289489758689356273997742747512, −7.970101528782990270282392395746, −7.43708855270899865292282347095, −6.37023347721565950641306665109, −6.05124154415542722856367510876, −5.33394036579808536136977182112, −4.76620532469427021089205770108, −3.33607365157126774404099005106, −2.65375288180146450346973583884, −2.01187532805195793674257978390,
2.01187532805195793674257978390, 2.65375288180146450346973583884, 3.33607365157126774404099005106, 4.76620532469427021089205770108, 5.33394036579808536136977182112, 6.05124154415542722856367510876, 6.37023347721565950641306665109, 7.43708855270899865292282347095, 7.970101528782990270282392395746, 8.289489758689356273997742747512, 9.212631067126484442668935106900, 9.693722725011220009511555723373, 9.992641543815054519789795250540, 10.75172005791015318217940061044, 11.18067797813160868987581050763
