| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 2·11-s + 12-s + 16-s − 6·17-s − 2·19-s + 2·22-s + 24-s − 2·25-s − 4·27-s + 32-s + 2·33-s − 6·34-s − 2·38-s + 6·41-s − 3·43-s + 2·44-s + 48-s − 6·49-s − 2·50-s − 6·51-s − 4·54-s − 2·57-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 0.603·11-s + 0.288·12-s + 1/4·16-s − 1.45·17-s − 0.458·19-s + 0.426·22-s + 0.204·24-s − 2/5·25-s − 0.769·27-s + 0.176·32-s + 0.348·33-s − 1.02·34-s − 0.324·38-s + 0.937·41-s − 0.457·43-s + 0.301·44-s + 0.144·48-s − 6/7·49-s − 0.282·50-s − 0.840·51-s − 0.544·54-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16512 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16512 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.913557752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.913557752\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−11.15764697423832092629656698211, −10.67136403399062790356512055328, −9.909076421806283658556403704994, −9.368934585027175237456558882449, −8.878487181898600773253591021709, −8.239180776558953806117975468422, −7.73699388756759732856986788580, −6.86982087565189684898918237954, −6.55344107624164518577758001146, −5.81181489196694236310664252549, −5.03782858744093749503641094641, −4.21663308740010569812633206873, −3.77864551379947278145381265608, −2.73395216637921125377630311503, −1.91456809004942474737018798589,
1.91456809004942474737018798589, 2.73395216637921125377630311503, 3.77864551379947278145381265608, 4.21663308740010569812633206873, 5.03782858744093749503641094641, 5.81181489196694236310664252549, 6.55344107624164518577758001146, 6.86982087565189684898918237954, 7.73699388756759732856986788580, 8.239180776558953806117975468422, 8.878487181898600773253591021709, 9.368934585027175237456558882449, 9.909076421806283658556403704994, 10.67136403399062790356512055328, 11.15764697423832092629656698211
