| L(s) = 1 | + 3-s − 2·4-s + 6·5-s + 9-s − 2·12-s + 6·15-s + 4·16-s − 2·19-s − 12·20-s + 18·23-s + 17·25-s + 27-s + 12·29-s − 2·36-s − 14·43-s + 6·45-s − 12·47-s + 4·48-s + 2·49-s − 12·53-s − 2·57-s − 12·60-s − 8·64-s − 8·67-s + 18·69-s + 24·71-s + 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 2.68·5-s + 1/3·9-s − 0.577·12-s + 1.54·15-s + 16-s − 0.458·19-s − 2.68·20-s + 3.75·23-s + 17/5·25-s + 0.192·27-s + 2.22·29-s − 1/3·36-s − 2.13·43-s + 0.894·45-s − 1.75·47-s + 0.577·48-s + 2/7·49-s − 1.64·53-s − 0.264·57-s − 1.54·60-s − 64-s − 0.977·67-s + 2.16·69-s + 2.84·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 499392 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 499392 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.623782528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.623782528\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−8.618966621189813626434174913816, −8.314506166371985811982082864189, −7.79303952698470481568778832395, −6.76945764827107790687114381372, −6.66463575826783700250676263362, −6.40959827015436577970274639195, −5.52928744974309714451608173806, −5.14306131647545403565477997005, −4.96593914963688452623551493731, −4.43400572211820912324912288374, −3.33065163930804441307691290385, −3.00632648437536992753290178800, −2.44673583930559801289396227405, −1.54383370192840503594716272369, −1.15713044816513337869513126279,
1.15713044816513337869513126279, 1.54383370192840503594716272369, 2.44673583930559801289396227405, 3.00632648437536992753290178800, 3.33065163930804441307691290385, 4.43400572211820912324912288374, 4.96593914963688452623551493731, 5.14306131647545403565477997005, 5.52928744974309714451608173806, 6.40959827015436577970274639195, 6.66463575826783700250676263362, 6.76945764827107790687114381372, 7.79303952698470481568778832395, 8.314506166371985811982082864189, 8.618966621189813626434174913816
