| L(s) = 1 | − 2-s + 2·3-s + 5-s − 2·6-s − 7-s + 8-s + 3·9-s − 10-s − 2·13-s + 14-s + 2·15-s − 16-s − 3·18-s − 2·19-s − 2·21-s + 23-s + 2·24-s + 25-s + 2·26-s + 4·27-s − 2·30-s − 35-s + 2·38-s − 4·39-s + 40-s + 2·42-s + 3·45-s + ⋯ |
| L(s) = 1 | − 2-s + 2·3-s + 5-s − 2·6-s − 7-s + 8-s + 3·9-s − 10-s − 2·13-s + 14-s + 2·15-s − 16-s − 3·18-s − 2·19-s − 2·21-s + 23-s + 2·24-s + 25-s + 2·26-s + 4·27-s − 2·30-s − 35-s + 2·38-s − 4·39-s + 40-s + 2·42-s + 3·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8428147775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8428147775\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−10.85615308179037527096033707483, −10.63730981310339958248220375650, −10.05336237641633738941287371096, −9.913468260948056977983502797938, −9.548843887652243704977109060494, −9.135801004569564371320108454935, −8.734186844481811898744008869376, −8.569730251692255414767893088893, −7.78741213173594247147192461213, −7.50429823732128498679165418175, −6.90648135148348130200714294975, −6.73256908674013255759844335816, −6.00478283645380134632712850492, −4.97231901674417063433174229573, −4.64030483812028349803504844570, −4.12076537595443995934849092890, −3.24584509188937312741968202860, −2.67046259978795969709661450435, −2.24204347859469371287362129720, −1.55295682418007375271480264696,
1.55295682418007375271480264696, 2.24204347859469371287362129720, 2.67046259978795969709661450435, 3.24584509188937312741968202860, 4.12076537595443995934849092890, 4.64030483812028349803504844570, 4.97231901674417063433174229573, 6.00478283645380134632712850492, 6.73256908674013255759844335816, 6.90648135148348130200714294975, 7.50429823732128498679165418175, 7.78741213173594247147192461213, 8.569730251692255414767893088893, 8.734186844481811898744008869376, 9.135801004569564371320108454935, 9.548843887652243704977109060494, 9.913468260948056977983502797938, 10.05336237641633738941287371096, 10.63730981310339958248220375650, 10.85615308179037527096033707483
