| L(s) = 1 | − 3-s − 4-s + 5-s + 2·7-s + 11-s + 12-s − 3·13-s − 15-s − 3·16-s + 2·17-s − 6·19-s − 20-s − 2·21-s − 6·23-s − 4·25-s + 4·27-s − 2·28-s + 4·29-s − 2·31-s − 33-s + 2·35-s + 4·37-s + 3·39-s + 10·41-s + 4·43-s − 44-s − 2·47-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.755·7-s + 0.301·11-s + 0.288·12-s − 0.832·13-s − 0.258·15-s − 3/4·16-s + 0.485·17-s − 1.37·19-s − 0.223·20-s − 0.436·21-s − 1.25·23-s − 4/5·25-s + 0.769·27-s − 0.377·28-s + 0.742·29-s − 0.359·31-s − 0.174·33-s + 0.338·35-s + 0.657·37-s + 0.480·39-s + 1.56·41-s + 0.609·43-s − 0.150·44-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2145 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2145 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5617574965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5617574965\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 2 T - 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 4 T - 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−18.5075956674, −18.0167097669, −17.8068348035, −17.1430521064, −16.8763241303, −16.1133045678, −15.6003083508, −14.7171367896, −14.3850321251, −13.9396737942, −13.2075929759, −12.5717910570, −11.9910186998, −11.4921325199, −10.7104149355, −10.2096468399, −9.47002573931, −8.83943604641, −8.07780655870, −7.36614050114, −6.32436144697, −5.77267355219, −4.73341965688, −4.19458292121, −2.28073156075,
2.28073156075, 4.19458292121, 4.73341965688, 5.77267355219, 6.32436144697, 7.36614050114, 8.07780655870, 8.83943604641, 9.47002573931, 10.2096468399, 10.7104149355, 11.4921325199, 11.9910186998, 12.5717910570, 13.2075929759, 13.9396737942, 14.3850321251, 14.7171367896, 15.6003083508, 16.1133045678, 16.8763241303, 17.1430521064, 17.8068348035, 18.0167097669, 18.5075956674
