| L(s) = 1 | − 2·3-s + 3·9-s + 4·11-s + 8·19-s + 6·25-s − 4·27-s − 8·33-s − 8·41-s + 16·43-s + 49-s − 16·57-s − 16·59-s − 8·67-s + 12·73-s − 12·75-s + 5·81-s + 24·83-s + 8·89-s − 4·97-s + 12·99-s + 20·107-s + 4·113-s − 10·121-s + 16·123-s + 127-s − 32·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 1.20·11-s + 1.83·19-s + 6/5·25-s − 0.769·27-s − 1.39·33-s − 1.24·41-s + 2.43·43-s + 1/7·49-s − 2.11·57-s − 2.08·59-s − 0.977·67-s + 1.40·73-s − 1.38·75-s + 5/9·81-s + 2.63·83-s + 0.847·89-s − 0.406·97-s + 1.20·99-s + 1.93·107-s + 0.376·113-s − 0.909·121-s + 1.44·123-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.683749695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683749695\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.052896223091606570950311686774, −7.53935193216942884829954158220, −7.27987334109443228148867323301, −6.82996158982273698048374301846, −6.24984441387476424451277097244, −6.07877945965578593249857952220, −5.51173614948836766111548042093, −4.83790832179161095688226596322, −4.82087120360185900706726704905, −4.03196371863461298187155314836, −3.48798700664607672936542140881, −3.02829428846360456459960342661, −2.08673280409477301775619832385, −1.25803993641506171158754813946, −0.77310464999562111477653259452,
0.77310464999562111477653259452, 1.25803993641506171158754813946, 2.08673280409477301775619832385, 3.02829428846360456459960342661, 3.48798700664607672936542140881, 4.03196371863461298187155314836, 4.82087120360185900706726704905, 4.83790832179161095688226596322, 5.51173614948836766111548042093, 6.07877945965578593249857952220, 6.24984441387476424451277097244, 6.82996158982273698048374301846, 7.27987334109443228148867323301, 7.53935193216942884829954158220, 8.052896223091606570950311686774
