| L(s) = 1 | + 2·3-s − 4·5-s + 3·9-s − 8·15-s + 11·25-s + 4·27-s + 8·31-s − 16·37-s + 20·41-s − 8·43-s − 12·45-s − 2·49-s + 24·53-s + 8·67-s + 22·75-s + 24·79-s + 5·81-s + 8·83-s − 20·89-s + 16·93-s + 24·107-s − 32·111-s + 6·121-s + 40·123-s − 24·125-s + 127-s − 16·129-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 9-s − 2.06·15-s + 11/5·25-s + 0.769·27-s + 1.43·31-s − 2.63·37-s + 3.12·41-s − 1.21·43-s − 1.78·45-s − 2/7·49-s + 3.29·53-s + 0.977·67-s + 2.54·75-s + 2.70·79-s + 5/9·81-s + 0.878·83-s − 2.11·89-s + 1.65·93-s + 2.32·107-s − 3.03·111-s + 6/11·121-s + 3.60·123-s − 2.14·125-s + 0.0887·127-s − 1.40·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.148072913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.148072913\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.438889038947173182413323343330, −8.426220001072337156911335852935, −7.965815070039479399631035195020, −7.68141460742279706607448359768, −7.25741095001970878006871203103, −7.02004650485077925950007305841, −6.67788901736851606326597663012, −6.26345118242283070785353508383, −5.51051806295697786845614915780, −5.30929832302428030216489175673, −4.59340973585300201209166650320, −4.47135175044073417687056955176, −3.94434004355725885132051536689, −3.63925703361201978622807868801, −3.31941608057261962450531382365, −2.83003538722097492246331937631, −2.34943906245832997555044402730, −1.89069860782359118672668043211, −0.955018373414104480512377092868, −0.59737789044940422032016358078,
0.59737789044940422032016358078, 0.955018373414104480512377092868, 1.89069860782359118672668043211, 2.34943906245832997555044402730, 2.83003538722097492246331937631, 3.31941608057261962450531382365, 3.63925703361201978622807868801, 3.94434004355725885132051536689, 4.47135175044073417687056955176, 4.59340973585300201209166650320, 5.30929832302428030216489175673, 5.51051806295697786845614915780, 6.26345118242283070785353508383, 6.67788901736851606326597663012, 7.02004650485077925950007305841, 7.25741095001970878006871203103, 7.68141460742279706607448359768, 7.965815070039479399631035195020, 8.426220001072337156911335852935, 8.438889038947173182413323343330
