| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s + 2·7-s − 8-s + 3·9-s + 3·11-s + 12-s + 2·13-s + 2·14-s + 3·16-s + 3·18-s + 2·19-s − 2·21-s + 3·22-s + 5·23-s + 24-s + 4·25-s + 2·26-s − 8·27-s − 2·28-s − 6·29-s − 2·31-s + 3·32-s − 3·33-s − 3·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 9-s + 0.904·11-s + 0.288·12-s + 0.554·13-s + 0.534·14-s + 3/4·16-s + 0.707·18-s + 0.458·19-s − 0.436·21-s + 0.639·22-s + 1.04·23-s + 0.204·24-s + 4/5·25-s + 0.392·26-s − 1.53·27-s − 0.377·28-s − 1.11·29-s − 0.359·31-s + 0.530·32-s − 0.522·33-s − 1/2·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100154 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100154 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.005802168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.005802168\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 50077 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 236 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 26 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 90 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 54 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 44 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 13 T + 112 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 2 T - 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 21 T + 268 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
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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
−13.9008749922, −13.5804953487, −13.2037311156, −12.8705166238, −12.3258393978, −11.9270287067, −11.4332526887, −11.1220560026, −10.5507096599, −10.1588591846, −9.51202846187, −9.09537364252, −8.71621644871, −7.96576518927, −7.58379655288, −6.82490764039, −6.67233344866, −5.59858568380, −5.47490932504, −4.85701446572, −4.27230489305, −3.80882402041, −3.23901093676, −1.83531329234, −1.12223110384,
1.12223110384, 1.83531329234, 3.23901093676, 3.80882402041, 4.27230489305, 4.85701446572, 5.47490932504, 5.59858568380, 6.67233344866, 6.82490764039, 7.58379655288, 7.96576518927, 8.71621644871, 9.09537364252, 9.51202846187, 10.1588591846, 10.5507096599, 11.1220560026, 11.4332526887, 11.9270287067, 12.3258393978, 12.8705166238, 13.2037311156, 13.5804953487, 13.9008749922
