| L(s) = 1 | − 2-s − 2·3-s − 4-s − 2·5-s + 2·6-s − 3·7-s + 8-s + 2·10-s − 3·11-s + 2·12-s + 3·14-s + 4·15-s − 16-s − 5·17-s − 3·19-s + 2·20-s + 6·21-s + 3·22-s − 7·23-s − 2·24-s + 2·25-s + 5·27-s + 3·28-s − 3·29-s − 4·30-s − 3·31-s + 5·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.894·5-s + 0.816·6-s − 1.13·7-s + 0.353·8-s + 0.632·10-s − 0.904·11-s + 0.577·12-s + 0.801·14-s + 1.03·15-s − 1/4·16-s − 1.21·17-s − 0.688·19-s + 0.447·20-s + 1.30·21-s + 0.639·22-s − 1.45·23-s − 0.408·24-s + 2/5·25-s + 0.962·27-s + 0.566·28-s − 0.557·29-s − 0.730·30-s − 0.538·31-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36123 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36123 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 12041 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 111 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 77 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T - 65 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 64 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 85 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 133 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 124 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−15.8054561782, −15.3050917045, −14.7930046549, −14.1081535981, −13.5156443632, −13.2144865990, −12.6416186959, −12.2864507856, −11.7135472328, −11.3566509744, −10.7595049459, −10.4161960333, −9.92470551763, −9.21153949275, −8.94428385279, −8.16584366248, −8.00452743785, −7.01815710203, −6.61955297038, −6.04904539127, −5.42813535610, −4.74144416938, −4.10235639375, −3.36226666982, −2.30173033572, 0, 0,
2.30173033572, 3.36226666982, 4.10235639375, 4.74144416938, 5.42813535610, 6.04904539127, 6.61955297038, 7.01815710203, 8.00452743785, 8.16584366248, 8.94428385279, 9.21153949275, 9.92470551763, 10.4161960333, 10.7595049459, 11.3566509744, 11.7135472328, 12.2864507856, 12.6416186959, 13.2144865990, 13.5156443632, 14.1081535981, 14.7930046549, 15.3050917045, 15.8054561782
