| L(s) = 1 | + 3-s − 4-s − 3·5-s − 2·7-s + 9-s − 12-s + 13-s − 3·15-s − 3·16-s + 5·17-s + 19-s + 3·20-s − 2·21-s − 3·23-s + 5·25-s + 27-s + 2·28-s + 3·29-s + 4·31-s + 6·35-s − 36-s − 8·37-s + 39-s + 3·41-s + 43-s − 3·45-s − 6·47-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s − 1.34·5-s − 0.755·7-s + 1/3·9-s − 0.288·12-s + 0.277·13-s − 0.774·15-s − 3/4·16-s + 1.21·17-s + 0.229·19-s + 0.670·20-s − 0.436·21-s − 0.625·23-s + 25-s + 0.192·27-s + 0.377·28-s + 0.557·29-s + 0.718·31-s + 1.01·35-s − 1/6·36-s − 1.31·37-s + 0.160·39-s + 0.468·41-s + 0.152·43-s − 0.447·45-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1377 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1377 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5167999514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5167999514\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 66 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T - 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−19.5367522933, −19.0306859253, −18.4229221956, −18.1239681892, −17.2233214186, −16.4906919493, −16.0660389417, −15.5670704383, −15.1407017115, −14.2634140717, −13.8801434825, −13.2547937413, −12.4368992852, −12.0935413783, −11.4038825068, −10.4911639385, −9.87809728127, −9.12028158071, −8.46244447903, −7.82828798573, −7.13674898544, −6.19695490302, −4.88551320983, −3.94175293048, −3.14748754461,
3.14748754461, 3.94175293048, 4.88551320983, 6.19695490302, 7.13674898544, 7.82828798573, 8.46244447903, 9.12028158071, 9.87809728127, 10.4911639385, 11.4038825068, 12.0935413783, 12.4368992852, 13.2547937413, 13.8801434825, 14.2634140717, 15.1407017115, 15.5670704383, 16.0660389417, 16.4906919493, 17.2233214186, 18.1239681892, 18.4229221956, 19.0306859253, 19.5367522933
