| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 2·9-s − 4·11-s − 12-s + 2·13-s + 16-s + 17-s + 2·18-s − 3·19-s − 4·22-s + 3·23-s − 24-s − 25-s + 2·26-s − 6·27-s − 11·29-s − 3·31-s + 32-s + 4·33-s + 34-s + 2·36-s − 37-s − 3·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 2/3·9-s − 1.20·11-s − 0.288·12-s + 0.554·13-s + 1/4·16-s + 0.242·17-s + 0.471·18-s − 0.688·19-s − 0.852·22-s + 0.625·23-s − 0.204·24-s − 1/5·25-s + 0.392·26-s − 1.15·27-s − 2.04·29-s − 0.538·31-s + 0.176·32-s + 0.696·33-s + 0.171·34-s + 1/3·36-s − 0.164·37-s − 0.486·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.037416584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.037416584\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 27 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 11 T + 63 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 21 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 218 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−17.2668998568, −16.9282138924, −16.3866106561, −15.7855805820, −15.5211177685, −14.8897067168, −14.4878665375, −13.6170375153, −13.1594209316, −12.8761614887, −12.3782746515, −11.5640977760, −11.0022065418, −10.8005429325, −9.99846306305, −9.39540622258, −8.55496538283, −7.67039011182, −7.32746655252, −6.42968339517, −5.66320558443, −5.25953523916, −4.26905822276, −3.47603887909, −2.09763675662,
2.09763675662, 3.47603887909, 4.26905822276, 5.25953523916, 5.66320558443, 6.42968339517, 7.32746655252, 7.67039011182, 8.55496538283, 9.39540622258, 9.99846306305, 10.8005429325, 11.0022065418, 11.5640977760, 12.3782746515, 12.8761614887, 13.1594209316, 13.6170375153, 14.4878665375, 14.8897067168, 15.5211177685, 15.7855805820, 16.3866106561, 16.9282138924, 17.2668998568
