| L(s) = 1 | + 3-s + 9-s + 12·13-s − 8·19-s + 25-s + 27-s − 16·31-s − 4·37-s + 12·39-s + 24·43-s − 14·49-s − 8·57-s + 28·61-s + 8·67-s − 12·73-s + 75-s − 16·79-s + 81-s − 16·93-s + 4·97-s − 36·109-s − 4·111-s + 12·117-s − 6·121-s + 127-s + 24·129-s + 131-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 3.32·13-s − 1.83·19-s + 1/5·25-s + 0.192·27-s − 2.87·31-s − 0.657·37-s + 1.92·39-s + 3.65·43-s − 2·49-s − 1.05·57-s + 3.58·61-s + 0.977·67-s − 1.40·73-s + 0.115·75-s − 1.80·79-s + 1/9·81-s − 1.65·93-s + 0.406·97-s − 3.44·109-s − 0.379·111-s + 1.10·117-s − 0.545·121-s + 0.0887·127-s + 2.11·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.768510500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.768510500\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.45550448318589263597286780431, −9.487642964123562756489741043280, −9.042960162718976648409639393560, −8.663099716937980035576099714984, −8.323057177226078640880314002796, −7.76560916456884806442341251241, −6.89270810769313104844169224865, −6.58285993178476791834968599347, −5.74168392413099570963868387782, −5.59585158632682227968586342616, −4.19803603229033482796144552370, −3.95303834850677407956821293662, −3.38984665882479408513314051846, −2.25002432004047770108316173263, −1.36054514292875137606341288502,
1.36054514292875137606341288502, 2.25002432004047770108316173263, 3.38984665882479408513314051846, 3.95303834850677407956821293662, 4.19803603229033482796144552370, 5.59585158632682227968586342616, 5.74168392413099570963868387782, 6.58285993178476791834968599347, 6.89270810769313104844169224865, 7.76560916456884806442341251241, 8.323057177226078640880314002796, 8.663099716937980035576099714984, 9.042960162718976648409639393560, 9.487642964123562756489741043280, 10.45550448318589263597286780431
