| L(s) = 1 | − 2-s − 2·3-s − 6·5-s + 2·6-s + 8-s − 3·9-s + 6·10-s − 3·11-s − 6·13-s + 12·15-s − 16-s + 3·18-s − 4·19-s + 3·22-s + 6·23-s − 2·24-s + 19·25-s + 6·26-s + 14·27-s − 12·29-s − 12·30-s − 6·31-s + 6·33-s + 8·37-s + 4·38-s + 12·39-s − 6·40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 2.68·5-s + 0.816·6-s + 0.353·8-s − 9-s + 1.89·10-s − 0.904·11-s − 1.66·13-s + 3.09·15-s − 1/4·16-s + 0.707·18-s − 0.917·19-s + 0.639·22-s + 1.25·23-s − 0.408·24-s + 19/5·25-s + 1.17·26-s + 2.69·27-s − 2.22·29-s − 2.19·30-s − 1.07·31-s + 1.04·33-s + 1.31·37-s + 0.648·38-s + 1.92·39-s − 0.948·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 73 | $C_2$ | \( 1 + 17 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 21 T + 206 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−12.59895283481266364816412229728, −12.27719954441586546846105030385, −11.53910483125938074395233905132, −11.32319959735832751266609429960, −10.81946434213386912228406824765, −10.81861148034629771762224560101, −9.732565157905116941574377381543, −9.005539304370804007168266199408, −8.666719081696028592083629942304, −7.86297422010336148723211376042, −7.58553142205687535859490089874, −7.33795555024028488706237695657, −6.38478101174134296391661529267, −5.55593631292815416482399106270, −4.87830021958763730050277928701, −4.50099204841611143485533819483, −3.49549533144765509086144008548, −2.72452869622617230035330060514, 0, 0,
2.72452869622617230035330060514, 3.49549533144765509086144008548, 4.50099204841611143485533819483, 4.87830021958763730050277928701, 5.55593631292815416482399106270, 6.38478101174134296391661529267, 7.33795555024028488706237695657, 7.58553142205687535859490089874, 7.86297422010336148723211376042, 8.666719081696028592083629942304, 9.005539304370804007168266199408, 9.732565157905116941574377381543, 10.81861148034629771762224560101, 10.81946434213386912228406824765, 11.32319959735832751266609429960, 11.53910483125938074395233905132, 12.27719954441586546846105030385, 12.59895283481266364816412229728
