| L(s) = 1 | − 2-s + 3-s − 6-s + 8-s + 11-s − 16-s + 2·17-s + 2·19-s − 22-s + 24-s − 25-s − 27-s + 33-s − 2·34-s − 2·38-s − 41-s + 43-s − 48-s + 50-s + 2·51-s + 54-s + 2·57-s − 59-s + 64-s − 66-s + 67-s + 2·73-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 6-s + 8-s + 11-s − 16-s + 2·17-s + 2·19-s − 22-s + 24-s − 25-s − 27-s + 33-s − 2·34-s − 2·38-s − 41-s + 43-s − 48-s + 50-s + 2·51-s + 54-s + 2·57-s − 59-s + 64-s − 66-s + 67-s + 2·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.319634231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.319634231\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_1$ | \( ( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
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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
−9.142792477057806123827918849226, −8.486533330801220667603427185049, −8.114783648399841316673735849006, −7.896955186343717929773410806403, −7.81293167792934109801927586975, −7.26958983436783143438935801586, −6.82424923555793465258863066796, −6.65082042375633154700854803211, −5.74777021942009769818086655632, −5.56120935146871348339114457007, −5.36029001331115272600287279806, −4.72125741956824816919274408551, −4.04260115477821288844358963984, −3.91513948962730953183343170928, −3.26256151468185372240320607177, −3.16202888327544439705383499544, −2.47221110691323698640157847026, −1.78387239858480799404086600067, −1.37474811863420541552851100091, −0.853757497419021379698111168003,
0.853757497419021379698111168003, 1.37474811863420541552851100091, 1.78387239858480799404086600067, 2.47221110691323698640157847026, 3.16202888327544439705383499544, 3.26256151468185372240320607177, 3.91513948962730953183343170928, 4.04260115477821288844358963984, 4.72125741956824816919274408551, 5.36029001331115272600287279806, 5.56120935146871348339114457007, 5.74777021942009769818086655632, 6.65082042375633154700854803211, 6.82424923555793465258863066796, 7.26958983436783143438935801586, 7.81293167792934109801927586975, 7.896955186343717929773410806403, 8.114783648399841316673735849006, 8.486533330801220667603427185049, 9.142792477057806123827918849226
