| L(s) = 1 | + 3-s + 3·11-s − 4·17-s + 12·19-s − 8·25-s − 4·27-s + 3·33-s − 2·43-s + 2·49-s − 4·51-s + 12·57-s − 8·59-s − 4·67-s + 12·73-s − 8·75-s − 7·81-s + 8·83-s − 8·89-s − 20·97-s − 14·107-s − 20·113-s − 4·121-s + 127-s − 2·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.904·11-s − 0.970·17-s + 2.75·19-s − 8/5·25-s − 0.769·27-s + 0.522·33-s − 0.304·43-s + 2/7·49-s − 0.560·51-s + 1.58·57-s − 1.04·59-s − 0.488·67-s + 1.40·73-s − 0.923·75-s − 7/9·81-s + 0.878·83-s − 0.847·89-s − 2.03·97-s − 1.35·107-s − 1.88·113-s − 0.363·121-s + 0.0887·127-s − 0.176·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.363237899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.363237899\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−11.18812270659567359755308919701, −10.43231428638302670158827106281, −9.596536434466715130054090425699, −9.488701847378449676854826599003, −9.033981437251853927033736588062, −8.139886427150089678972980095711, −7.78282963012088444603309411023, −7.14269184950230406807663506994, −6.54405344339948600611364323223, −5.73189326791197338159543524755, −5.20012652223148230211464038162, −4.18429820413759524649736672972, −3.60735988899299721259761764631, −2.76397338394847805714609402284, −1.57660829497846344239914860314,
1.57660829497846344239914860314, 2.76397338394847805714609402284, 3.60735988899299721259761764631, 4.18429820413759524649736672972, 5.20012652223148230211464038162, 5.73189326791197338159543524755, 6.54405344339948600611364323223, 7.14269184950230406807663506994, 7.78282963012088444603309411023, 8.139886427150089678972980095711, 9.033981437251853927033736588062, 9.488701847378449676854826599003, 9.596536434466715130054090425699, 10.43231428638302670158827106281, 11.18812270659567359755308919701
