| L(s) = 1 | + 2·3-s − 2·5-s + 3·7-s + 9-s + 5·11-s + 13-s − 4·15-s + 6·21-s + 5·23-s − 2·25-s − 4·27-s + 29-s + 3·31-s + 10·33-s − 6·35-s + 4·37-s + 2·39-s − 3·41-s − 9·43-s − 2·45-s − 47-s + 5·49-s + 4·53-s − 10·55-s + 19·59-s − 11·61-s + 3·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1.13·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s − 1.03·15-s + 1.30·21-s + 1.04·23-s − 2/5·25-s − 0.769·27-s + 0.185·29-s + 0.538·31-s + 1.74·33-s − 1.01·35-s + 0.657·37-s + 0.320·39-s − 0.468·41-s − 1.37·43-s − 0.298·45-s − 0.145·47-s + 5/7·49-s + 0.549·53-s − 1.34·55-s + 2.47·59-s − 1.40·61-s + 0.377·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103680 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103680 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.400543890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.400543890\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 5 T + 20 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 9 T + 96 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 19 T + 190 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 128 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 120 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T - 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3 T + 48 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−14.1492903264, −13.6528734811, −13.3225653842, −12.8239713177, −12.0133090824, −11.8111833404, −11.4946984846, −11.1226883929, −10.4620741799, −9.92978737824, −9.31512218984, −8.95441301931, −8.49937929669, −8.16670325835, −7.76118513312, −7.13556067943, −6.75749715231, −6.01350946986, −5.32278340218, −4.55887366567, −4.16063944879, −3.56511831113, −3.00716125667, −2.04032819968, −1.22953033266,
1.22953033266, 2.04032819968, 3.00716125667, 3.56511831113, 4.16063944879, 4.55887366567, 5.32278340218, 6.01350946986, 6.75749715231, 7.13556067943, 7.76118513312, 8.16670325835, 8.49937929669, 8.95441301931, 9.31512218984, 9.92978737824, 10.4620741799, 11.1226883929, 11.4946984846, 11.8111833404, 12.0133090824, 12.8239713177, 13.3225653842, 13.6528734811, 14.1492903264
