| L(s) = 1 | − 3-s − 4-s + 2·7-s − 2·8-s − 5·11-s + 12-s − 2·13-s + 16-s + 5·17-s − 2·21-s − 6·23-s + 2·24-s + 2·25-s + 4·27-s − 2·28-s + 6·29-s + 4·31-s + 4·32-s + 5·33-s − 2·37-s + 2·39-s − 4·41-s − 4·43-s + 5·44-s + 6·47-s − 48-s + 2·49-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 0.755·7-s − 0.707·8-s − 1.50·11-s + 0.288·12-s − 0.554·13-s + 1/4·16-s + 1.21·17-s − 0.436·21-s − 1.25·23-s + 0.408·24-s + 2/5·25-s + 0.769·27-s − 0.377·28-s + 1.11·29-s + 0.718·31-s + 0.707·32-s + 0.870·33-s − 0.328·37-s + 0.320·39-s − 0.624·41-s − 0.609·43-s + 0.753·44-s + 0.875·47-s − 0.144·48-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4267240383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4267240383\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 74 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−19.8249414615, −19.0390474825, −18.5182668328, −18.1463541125, −17.6473715784, −17.2059529597, −16.4592368857, −15.8905729982, −15.3033669889, −14.6392536913, −14.0848615523, −13.5268077088, −12.6458342289, −12.0749405057, −11.7974836672, −10.6704551760, −10.2584833158, −9.63358543986, −8.39308893862, −8.17910604684, −7.17880003457, −5.99852528009, −5.30636102241, −4.57838279893, −2.91853435291,
2.91853435291, 4.57838279893, 5.30636102241, 5.99852528009, 7.17880003457, 8.17910604684, 8.39308893862, 9.63358543986, 10.2584833158, 10.6704551760, 11.7974836672, 12.0749405057, 12.6458342289, 13.5268077088, 14.0848615523, 14.6392536913, 15.3033669889, 15.8905729982, 16.4592368857, 17.2059529597, 17.6473715784, 18.1463541125, 18.5182668328, 19.0390474825, 19.8249414615
