| L(s) = 1 | − 2-s − 3-s + 6-s + 8-s + 9-s + 11-s + 2·13-s − 16-s + 17-s − 18-s − 22-s − 2·23-s − 24-s − 25-s − 2·26-s − 2·27-s + 2·31-s − 33-s − 34-s − 2·39-s + 2·46-s + 48-s + 50-s − 51-s + 53-s + 2·54-s − 2·62-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 6-s + 8-s + 9-s + 11-s + 2·13-s − 16-s + 17-s − 18-s − 22-s − 2·23-s − 24-s − 25-s − 2·26-s − 2·27-s + 2·31-s − 33-s − 34-s − 2·39-s + 2·46-s + 48-s + 50-s − 51-s + 53-s + 2·54-s − 2·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6443010657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6443010657\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−8.804391706240277141772215986494, −8.717792080139855828873757634081, −8.242594254273728100326010202986, −8.001163121799040453099854773457, −7.44967187896273881270189289222, −7.39442169777394484404913351310, −6.66089263377177453559238509445, −6.30005985605594547415842342790, −6.11161694034564754416256283608, −5.59359611484840553458522136793, −5.55630756354087906755303058675, −4.55256837653007800276850337802, −4.41497327763883080474903611987, −3.99925718411293761089398365115, −3.62515998608347669433834328421, −3.20656767476752308331533792772, −2.09350379397431204661783982583, −1.78776453573208121134216699396, −1.21504330640272431502402553812, −0.72565773226993418656413207620,
0.72565773226993418656413207620, 1.21504330640272431502402553812, 1.78776453573208121134216699396, 2.09350379397431204661783982583, 3.20656767476752308331533792772, 3.62515998608347669433834328421, 3.99925718411293761089398365115, 4.41497327763883080474903611987, 4.55256837653007800276850337802, 5.55630756354087906755303058675, 5.59359611484840553458522136793, 6.11161694034564754416256283608, 6.30005985605594547415842342790, 6.66089263377177453559238509445, 7.39442169777394484404913351310, 7.44967187896273881270189289222, 8.001163121799040453099854773457, 8.242594254273728100326010202986, 8.717792080139855828873757634081, 8.804391706240277141772215986494
