| L(s) = 1 | − 3·2-s − 3-s + 4·4-s − 4·5-s + 3·6-s − 5·7-s − 3·8-s − 2·9-s + 12·10-s − 2·11-s − 4·12-s − 13-s + 15·14-s + 4·15-s + 3·16-s − 7·17-s + 6·18-s − 16·20-s + 5·21-s + 6·22-s − 6·23-s + 3·24-s + 5·25-s + 3·26-s + 5·27-s − 20·28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 0.577·3-s + 2·4-s − 1.78·5-s + 1.22·6-s − 1.88·7-s − 1.06·8-s − 2/3·9-s + 3.79·10-s − 0.603·11-s − 1.15·12-s − 0.277·13-s + 4.00·14-s + 1.03·15-s + 3/4·16-s − 1.69·17-s + 1.41·18-s − 3.57·20-s + 1.09·21-s + 1.27·22-s − 1.25·23-s + 0.612·24-s + 25-s + 0.588·26-s + 0.962·27-s − 3.77·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + T - 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 43 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 15 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 47 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 152 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 103 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 29 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 139 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 40 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 21 T + 285 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
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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
−16.8527089449, −16.4813354792, −16.0981582693, −15.7493724990, −15.4026936900, −14.8427584626, −13.8686657368, −13.3703590687, −12.6027180033, −12.1501412566, −11.9067517193, −10.9651024459, −10.8466900311, −10.2015961861, −9.56999253132, −9.17611646621, −8.60108098149, −8.05892319427, −7.70035215600, −6.86128105695, −6.46577065804, −5.64411925089, −4.44444096366, −3.60920755098, −2.70035849418, 0, 0,
2.70035849418, 3.60920755098, 4.44444096366, 5.64411925089, 6.46577065804, 6.86128105695, 7.70035215600, 8.05892319427, 8.60108098149, 9.17611646621, 9.56999253132, 10.2015961861, 10.8466900311, 10.9651024459, 11.9067517193, 12.1501412566, 12.6027180033, 13.3703590687, 13.8686657368, 14.8427584626, 15.4026936900, 15.7493724990, 16.0981582693, 16.4813354792, 16.8527089449
