| L(s) = 1 | + 2-s + 2·3-s + 2·6-s − 8-s + 3·9-s − 8·13-s − 16-s − 6·17-s + 3·18-s − 2·19-s − 2·24-s + 5·25-s − 8·26-s + 10·27-s − 12·29-s + 4·31-s − 6·34-s − 2·37-s − 2·38-s − 16·39-s + 12·41-s + 16·43-s + 12·47-s − 2·48-s + 5·50-s − 12·51-s − 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 0.816·6-s − 0.353·8-s + 9-s − 2.21·13-s − 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.458·19-s − 0.408·24-s + 25-s − 1.56·26-s + 1.92·27-s − 2.22·29-s + 0.718·31-s − 1.02·34-s − 0.328·37-s − 0.324·38-s − 2.56·39-s + 1.87·41-s + 2.43·43-s + 1.75·47-s − 0.288·48-s + 0.707·50-s − 1.68·51-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.682449691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.682449691\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 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
−14.04145501722198037479383964324, −14.00599460194856240485032040018, −12.89897144252259646211093501170, −12.85516300122271479135410455197, −12.44533047287876342504227065817, −11.74368042716781372037397857686, −10.84462218398902176922880365263, −10.60537085923855477135106016776, −9.630199992252261854634945255997, −9.241789904092050155683227766971, −8.904163825273414140536527508881, −8.083333619699282901638240448876, −7.25391899305612172511439282965, −7.16393847545088947450776580615, −6.12510237641322476348159870666, −5.23488858771214743865463879013, −4.42448063842485448054280162476, −4.12098676425324769033179756617, −2.71904573650174489385286764559, −2.41749920092248519897743465838,
2.41749920092248519897743465838, 2.71904573650174489385286764559, 4.12098676425324769033179756617, 4.42448063842485448054280162476, 5.23488858771214743865463879013, 6.12510237641322476348159870666, 7.16393847545088947450776580615, 7.25391899305612172511439282965, 8.083333619699282901638240448876, 8.904163825273414140536527508881, 9.241789904092050155683227766971, 9.630199992252261854634945255997, 10.60537085923855477135106016776, 10.84462218398902176922880365263, 11.74368042716781372037397857686, 12.44533047287876342504227065817, 12.85516300122271479135410455197, 12.89897144252259646211093501170, 14.00599460194856240485032040018, 14.04145501722198037479383964324
