| L(s) = 1 | − 2·3-s − 4-s + 9-s + 2·12-s + 5·13-s − 3·16-s + 6·17-s − 25-s + 4·27-s + 15·29-s − 36-s − 10·39-s − 2·43-s + 6·48-s − 13·49-s − 12·51-s − 5·52-s + 6·53-s + 19·61-s + 7·64-s − 6·68-s + 2·75-s − 20·79-s − 11·81-s − 30·87-s + 100-s − 6·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 1/3·9-s + 0.577·12-s + 1.38·13-s − 3/4·16-s + 1.45·17-s − 1/5·25-s + 0.769·27-s + 2.78·29-s − 1/6·36-s − 1.60·39-s − 0.304·43-s + 0.866·48-s − 1.85·49-s − 1.68·51-s − 0.693·52-s + 0.824·53-s + 2.43·61-s + 7/8·64-s − 0.727·68-s + 0.230·75-s − 2.25·79-s − 1.22·81-s − 3.21·87-s + 1/10·100-s − 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6751576074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6751576074\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
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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
−11.35602805723046122206829210450, −10.68389180192618923629940403141, −10.15085227464949750149406598643, −9.830828044174449564877761972139, −8.901118960746985838598890682533, −8.417598580717825745872133683050, −8.033008851926041234750244332490, −6.93008699605088141542062183856, −6.52242754969305111991545394487, −5.89619929848292488548421093648, −5.26921996746328915551854401084, −4.67556083454843451240290976455, −3.85603569322153007789294912365, −2.89344923925415568822832195876, −1.09621016892456542075964254605,
1.09621016892456542075964254605, 2.89344923925415568822832195876, 3.85603569322153007789294912365, 4.67556083454843451240290976455, 5.26921996746328915551854401084, 5.89619929848292488548421093648, 6.52242754969305111991545394487, 6.93008699605088141542062183856, 8.033008851926041234750244332490, 8.417598580717825745872133683050, 8.901118960746985838598890682533, 9.830828044174449564877761972139, 10.15085227464949750149406598643, 10.68389180192618923629940403141, 11.35602805723046122206829210450
