| L(s) = 1 | − 2-s − 2·3-s − 4-s − 3·5-s + 2·6-s − 7-s + 8-s + 2·9-s + 3·10-s + 3·11-s + 2·12-s − 2·13-s + 14-s + 6·15-s − 16-s − 2·17-s − 2·18-s + 5·19-s + 3·20-s + 2·21-s − 3·22-s + 4·23-s − 2·24-s + 8·25-s + 2·26-s − 6·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s − 1.34·5-s + 0.816·6-s − 0.377·7-s + 0.353·8-s + 2/3·9-s + 0.948·10-s + 0.904·11-s + 0.577·12-s − 0.554·13-s + 0.267·14-s + 1.54·15-s − 1/4·16-s − 0.485·17-s − 0.471·18-s + 1.14·19-s + 0.670·20-s + 0.436·21-s − 0.639·22-s + 0.834·23-s − 0.408·24-s + 8/5·25-s + 0.392·26-s − 1.15·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10115 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10115 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2597890920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2597890920\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 58 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 80 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 120 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T - 72 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−16.6184544633, −16.2742023730, −15.6532697359, −15.3550308361, −14.7716107836, −14.1047592170, −13.5032384646, −13.0129633516, −12.2258690788, −11.8377523088, −11.6759954344, −11.0751601396, −10.3085617918, −9.92504657781, −9.16796356861, −8.78782268281, −8.14779691586, −7.39776653989, −6.83370247039, −6.35935325285, −5.27690616464, −4.70911048028, −4.04432489402, −3.05421372790, −0.750661867126,
0.750661867126, 3.05421372790, 4.04432489402, 4.70911048028, 5.27690616464, 6.35935325285, 6.83370247039, 7.39776653989, 8.14779691586, 8.78782268281, 9.16796356861, 9.92504657781, 10.3085617918, 11.0751601396, 11.6759954344, 11.8377523088, 12.2258690788, 13.0129633516, 13.5032384646, 14.1047592170, 14.7716107836, 15.3550308361, 15.6532697359, 16.2742023730, 16.6184544633
