| L(s) = 1 | + 3-s − 2·4-s − 2·9-s − 2·12-s + 4·16-s − 14·19-s + 25-s − 5·27-s + 4·36-s − 20·43-s + 4·48-s + 5·49-s − 14·57-s − 8·64-s − 8·67-s + 4·73-s + 75-s + 28·76-s + 81-s − 20·97-s − 2·100-s + 10·108-s + 121-s + 127-s − 20·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 2/3·9-s − 0.577·12-s + 16-s − 3.21·19-s + 1/5·25-s − 0.962·27-s + 2/3·36-s − 3.04·43-s + 0.577·48-s + 5/7·49-s − 1.85·57-s − 64-s − 0.977·67-s + 0.468·73-s + 0.115·75-s + 3.21·76-s + 1/9·81-s − 2.03·97-s − 1/5·100-s + 0.962·108-s + 1/11·121-s + 0.0887·127-s − 1.76·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5372535941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5372535941\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.040460516550001471734519470308, −7.60236473112536637929154511987, −6.84816340029153899150109659868, −6.56032601094134442591874935435, −6.16406908249237227197626963659, −5.59543933610979103036970417314, −5.20456374238785512192638384626, −4.62110378161723352051570314008, −4.26606672523529396586203843914, −3.82773842606571010136543224538, −3.30313980739090856719676600044, −2.74608694741698517148860572607, −2.11157504068281887405456332947, −1.57196597364400704736238581312, −0.28799972540674795587320194896,
0.28799972540674795587320194896, 1.57196597364400704736238581312, 2.11157504068281887405456332947, 2.74608694741698517148860572607, 3.30313980739090856719676600044, 3.82773842606571010136543224538, 4.26606672523529396586203843914, 4.62110378161723352051570314008, 5.20456374238785512192638384626, 5.59543933610979103036970417314, 6.16406908249237227197626963659, 6.56032601094134442591874935435, 6.84816340029153899150109659868, 7.60236473112536637929154511987, 8.040460516550001471734519470308
