| L(s) = 1 | − 3·3-s + 4-s − 2·5-s + 4·9-s − 4·11-s − 3·12-s + 2·13-s + 6·15-s + 16-s + 8·17-s − 2·19-s − 2·20-s + 8·23-s − 6·25-s − 8·29-s − 4·31-s + 12·33-s + 4·36-s − 8·37-s − 6·39-s + 4·43-s − 4·44-s − 8·45-s − 12·47-s − 3·48-s + 49-s − 24·51-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1/2·4-s − 0.894·5-s + 4/3·9-s − 1.20·11-s − 0.866·12-s + 0.554·13-s + 1.54·15-s + 1/4·16-s + 1.94·17-s − 0.458·19-s − 0.447·20-s + 1.66·23-s − 6/5·25-s − 1.48·29-s − 0.718·31-s + 2.08·33-s + 2/3·36-s − 1.31·37-s − 0.960·39-s + 0.609·43-s − 0.603·44-s − 1.19·45-s − 1.75·47-s − 0.433·48-s + 1/7·49-s − 3.36·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2869187443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2869187443\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−19.5800270198, −19.2406986299, −18.4485210975, −18.2120541317, −17.2128532162, −16.8741179353, −16.3370810014, −15.8196150869, −15.2472562838, −14.6077730657, −13.4821859369, −12.6521323881, −12.3052609901, −11.3731892768, −11.2313614144, −10.5195197796, −9.76554711946, −8.35691896663, −7.57571100089, −6.84552480603, −5.57928681743, −5.33985014788, −3.62482887082,
3.62482887082, 5.33985014788, 5.57928681743, 6.84552480603, 7.57571100089, 8.35691896663, 9.76554711946, 10.5195197796, 11.2313614144, 11.3731892768, 12.3052609901, 12.6521323881, 13.4821859369, 14.6077730657, 15.2472562838, 15.8196150869, 16.3370810014, 16.8741179353, 17.2128532162, 18.2120541317, 18.4485210975, 19.2406986299, 19.5800270198
