| L(s) = 1 | + 2-s + 4-s + 8-s + 16-s + 6·17-s − 8·19-s − 25-s + 32-s + 6·34-s − 8·38-s − 12·41-s + 16·43-s + 2·49-s − 50-s + 64-s − 8·67-s + 6·68-s + 22·73-s − 8·76-s − 12·82-s + 24·83-s + 16·86-s + 6·89-s + 4·97-s + 2·98-s − 100-s − 24·107-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/4·16-s + 1.45·17-s − 1.83·19-s − 1/5·25-s + 0.176·32-s + 1.02·34-s − 1.29·38-s − 1.87·41-s + 2.43·43-s + 2/7·49-s − 0.141·50-s + 1/8·64-s − 0.977·67-s + 0.727·68-s + 2.57·73-s − 0.917·76-s − 1.32·82-s + 2.63·83-s + 1.72·86-s + 0.635·89-s + 0.406·97-s + 0.202·98-s − 0.0999·100-s − 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 839808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 839808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.116533897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.116533897\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.153205859443475808039169365129, −7.79409315011386614496311102735, −7.24345462277249174674167705434, −6.88462097851938764634300699739, −6.18735538198950166736525445007, −6.11563543838000124898248269154, −5.49090131952205180218546644094, −4.97067099118845305707332827681, −4.60568685013129480432704821683, −3.83528584973290190499044501865, −3.69589098584017718744945020548, −2.94903686287780602993956585385, −2.30016225881827778465495235815, −1.78014361920684476225734672011, −0.75418881749976335996165607688,
0.75418881749976335996165607688, 1.78014361920684476225734672011, 2.30016225881827778465495235815, 2.94903686287780602993956585385, 3.69589098584017718744945020548, 3.83528584973290190499044501865, 4.60568685013129480432704821683, 4.97067099118845305707332827681, 5.49090131952205180218546644094, 6.11563543838000124898248269154, 6.18735538198950166736525445007, 6.88462097851938764634300699739, 7.24345462277249174674167705434, 7.79409315011386614496311102735, 8.153205859443475808039169365129
