| L(s) = 1 | + 2·3-s − 2·4-s + 9-s − 4·12-s + 4·16-s + 4·19-s − 4·27-s − 2·36-s + 20·43-s + 8·48-s + 14·49-s + 8·57-s − 8·64-s − 28·67-s − 4·73-s − 8·76-s − 11·81-s + 20·97-s + 8·108-s + 14·121-s + 127-s + 40·129-s + 131-s + 137-s + 139-s + 4·144-s + 28·147-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s + 1/3·9-s − 1.15·12-s + 16-s + 0.917·19-s − 0.769·27-s − 1/3·36-s + 3.04·43-s + 1.15·48-s + 2·49-s + 1.05·57-s − 64-s − 3.42·67-s − 0.468·73-s − 0.917·76-s − 1.22·81-s + 2.03·97-s + 0.769·108-s + 1.27·121-s + 0.0887·127-s + 3.52·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 2.30·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.072266188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.072266188\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−10.61595755265602291452998266270, −10.38372129917558441687791126916, −9.953455076161035232417059273905, −9.266521253316885301013931231556, −9.129111699901980850780422569453, −8.862850676058498335260499243325, −8.404478465358880357456116114118, −7.65169802598765604280542852865, −7.57678308759711453796633563936, −7.23910810630651099409080134120, −6.16215486645643186765499032966, −5.89936961037098458437174484953, −5.35628999070245721891993393281, −4.75612780775850879758142922509, −4.02659878442431253323237839405, −3.94744695594640439918965092514, −2.97620352145636067292891802234, −2.78019077983702519044124188914, −1.79127616560145264208019356309, −0.796486668414836834978115388766,
0.796486668414836834978115388766, 1.79127616560145264208019356309, 2.78019077983702519044124188914, 2.97620352145636067292891802234, 3.94744695594640439918965092514, 4.02659878442431253323237839405, 4.75612780775850879758142922509, 5.35628999070245721891993393281, 5.89936961037098458437174484953, 6.16215486645643186765499032966, 7.23910810630651099409080134120, 7.57678308759711453796633563936, 7.65169802598765604280542852865, 8.404478465358880357456116114118, 8.862850676058498335260499243325, 9.129111699901980850780422569453, 9.266521253316885301013931231556, 9.953455076161035232417059273905, 10.38372129917558441687791126916, 10.61595755265602291452998266270
