| L(s) = 1 | − 2·2-s − 3-s + 4-s − 2·5-s + 2·6-s + 2·8-s + 4·10-s + 4·11-s − 12-s − 6·13-s + 2·15-s − 3·16-s + 6·19-s − 2·20-s − 8·22-s − 2·24-s − 6·25-s + 12·26-s + 4·27-s − 8·29-s − 4·30-s − 4·31-s − 2·32-s − 4·33-s + 8·37-s − 12·38-s + 6·39-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s + 0.707·8-s + 1.26·10-s + 1.20·11-s − 0.288·12-s − 1.66·13-s + 0.516·15-s − 3/4·16-s + 1.37·19-s − 0.447·20-s − 1.70·22-s − 0.408·24-s − 6/5·25-s + 2.35·26-s + 0.769·27-s − 1.48·29-s − 0.730·30-s − 0.718·31-s − 0.353·32-s − 0.696·33-s + 1.31·37-s − 1.94·38-s + 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1489689799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1489689799\) |
| \(L(\frac{3}{2})\) |
|
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_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 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$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−19.8970146357, −19.6678488176, −19.5800270198, −18.4192282680, −18.2120541317, −17.2128532162, −17.1598241820, −16.3370810014, −15.8734598001, −14.6077730657, −14.5135323404, −13.1937925053, −12.3052609901, −11.5559508175, −11.2313614144, −9.76554711946, −9.72466121073, −8.67236519668, −7.57571100089, −7.25047783803, −5.57928681743, −4.13559084051,
4.13559084051, 5.57928681743, 7.25047783803, 7.57571100089, 8.67236519668, 9.72466121073, 9.76554711946, 11.2313614144, 11.5559508175, 12.3052609901, 13.1937925053, 14.5135323404, 14.6077730657, 15.8734598001, 16.3370810014, 17.1598241820, 17.2128532162, 18.2120541317, 18.4192282680, 19.5800270198, 19.6678488176, 19.8970146357
