| L(s) = 1 | + 3-s + 2·5-s + 9-s + 2·15-s − 8·19-s + 3·25-s + 27-s − 4·29-s + 24·43-s + 2·45-s + 16·47-s − 14·49-s + 12·53-s − 8·57-s + 8·67-s + 16·71-s − 12·73-s + 3·75-s + 81-s − 4·87-s − 16·95-s + 4·97-s + 12·101-s − 6·121-s + 4·125-s + 127-s + 24·129-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.516·15-s − 1.83·19-s + 3/5·25-s + 0.192·27-s − 0.742·29-s + 3.65·43-s + 0.298·45-s + 2.33·47-s − 2·49-s + 1.64·53-s − 1.05·57-s + 0.977·67-s + 1.89·71-s − 1.40·73-s + 0.346·75-s + 1/9·81-s − 0.428·87-s − 1.64·95-s + 0.406·97-s + 1.19·101-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 2.11·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.631757453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.631757453\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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
−8.979673560126252996904408641448, −8.323057177226078640880314002796, −7.85971332456036455091052470017, −7.36860587905514204016946354425, −6.89270810769313104844169224865, −6.38799072784883331024834438481, −5.83633635666208002119272737589, −5.59585158632682227968586342616, −4.81681980421746127667781868965, −4.14224381134792067626093257234, −3.95303834850677407956821293662, −3.00246618464052305255387261031, −2.25002432004047770108316173263, −2.12011231568422406855465196308, −0.910745729015585711280898694007,
0.910745729015585711280898694007, 2.12011231568422406855465196308, 2.25002432004047770108316173263, 3.00246618464052305255387261031, 3.95303834850677407956821293662, 4.14224381134792067626093257234, 4.81681980421746127667781868965, 5.59585158632682227968586342616, 5.83633635666208002119272737589, 6.38799072784883331024834438481, 6.89270810769313104844169224865, 7.36860587905514204016946354425, 7.85971332456036455091052470017, 8.323057177226078640880314002796, 8.979673560126252996904408641448
