| L(s) = 1 | − 3-s − 4-s + 5-s − 2·7-s − 2·11-s + 12-s − 2·13-s − 15-s − 3·16-s + 6·17-s + 19-s − 20-s + 2·21-s − 23-s − 3·25-s + 4·27-s + 2·28-s + 6·29-s + 2·33-s − 2·35-s − 37-s + 2·39-s + 41-s + 6·43-s + 2·44-s + 3·48-s − 2·49-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.755·7-s − 0.603·11-s + 0.288·12-s − 0.554·13-s − 0.258·15-s − 3/4·16-s + 1.45·17-s + 0.229·19-s − 0.223·20-s + 0.436·21-s − 0.208·23-s − 3/5·25-s + 0.769·27-s + 0.377·28-s + 1.11·29-s + 0.348·33-s − 0.338·35-s − 0.164·37-s + 0.320·39-s + 0.156·41-s + 0.914·43-s + 0.301·44-s + 0.433·48-s − 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1137 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1137 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4311764803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4311764803\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 379 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 20 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + T + 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - T - 32 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 17 T + 178 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−19.6913841141, −19.2102154628, −18.5259503192, −18.1141356753, −17.6334822270, −17.0014939014, −16.4921780809, −15.9234669386, −15.4477908146, −14.4514652967, −14.0326794237, −13.4388243459, −12.7647778685, −12.1982214071, −11.6720927939, −10.6345473836, −10.1191881801, −9.58829886790, −8.82281367960, −7.87091721878, −7.04333554952, −6.07659273981, −5.41456897800, −4.44071838240, −2.93168366950,
2.93168366950, 4.44071838240, 5.41456897800, 6.07659273981, 7.04333554952, 7.87091721878, 8.82281367960, 9.58829886790, 10.1191881801, 10.6345473836, 11.6720927939, 12.1982214071, 12.7647778685, 13.4388243459, 14.0326794237, 14.4514652967, 15.4477908146, 15.9234669386, 16.4921780809, 17.0014939014, 17.6334822270, 18.1141356753, 18.5259503192, 19.2102154628, 19.6913841141
