| L(s) = 1 | + 2·3-s − 4-s − 3·9-s − 2·12-s − 4·13-s + 16-s − 6·17-s − 12·23-s + 25-s − 14·27-s + 3·36-s − 8·39-s − 2·43-s + 2·48-s − 12·51-s + 4·52-s − 12·53-s + 16·61-s − 64-s + 6·68-s − 24·69-s + 2·75-s + 20·79-s − 4·81-s + 12·92-s − 100-s − 24·101-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s − 9-s − 0.577·12-s − 1.10·13-s + 1/4·16-s − 1.45·17-s − 2.50·23-s + 1/5·25-s − 2.69·27-s + 1/2·36-s − 1.28·39-s − 0.304·43-s + 0.288·48-s − 1.68·51-s + 0.554·52-s − 1.64·53-s + 2.04·61-s − 1/8·64-s + 0.727·68-s − 2.88·69-s + 0.230·75-s + 2.25·79-s − 4/9·81-s + 1.25·92-s − 0.0999·100-s − 2.38·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1623076 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1623076 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5407956638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5407956638\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−10.10258568979940088599983144930, −9.303174547707715162015696818145, −9.247505587401910499150306161007, −8.596184817503442915513823967274, −8.146136132144137611464316936936, −7.965012769356168601218124801149, −7.903981653463609263376344440170, −6.86749199959062239285900991994, −6.75484292360192518643024258134, −6.14828936113467308883189587520, −5.54200561231411218410755730686, −5.36118951509767446478862873306, −4.69998363903456048616341944583, −4.01188494039394235887480642625, −3.97375178412471925947689655489, −3.15240974857690971116427223197, −2.69900066446135815829022500786, −2.21328283889656046556872531294, −1.82401273910339278189209075082, −0.26443151766956092446487921182,
0.26443151766956092446487921182, 1.82401273910339278189209075082, 2.21328283889656046556872531294, 2.69900066446135815829022500786, 3.15240974857690971116427223197, 3.97375178412471925947689655489, 4.01188494039394235887480642625, 4.69998363903456048616341944583, 5.36118951509767446478862873306, 5.54200561231411218410755730686, 6.14828936113467308883189587520, 6.75484292360192518643024258134, 6.86749199959062239285900991994, 7.903981653463609263376344440170, 7.965012769356168601218124801149, 8.146136132144137611464316936936, 8.596184817503442915513823967274, 9.247505587401910499150306161007, 9.303174547707715162015696818145, 10.10258568979940088599983144930
