| L(s) = 1 | − 3-s + 4-s − 4·7-s + 9-s − 12-s − 12·13-s + 16-s + 8·19-s + 4·21-s + 6·25-s − 27-s − 4·28-s − 12·31-s + 36-s − 8·37-s + 12·39-s − 8·43-s − 48-s − 2·49-s − 12·52-s − 8·57-s − 8·61-s − 4·63-s + 64-s − 24·67-s + 4·73-s − 6·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 1.51·7-s + 1/3·9-s − 0.288·12-s − 3.32·13-s + 1/4·16-s + 1.83·19-s + 0.872·21-s + 6/5·25-s − 0.192·27-s − 0.755·28-s − 2.15·31-s + 1/6·36-s − 1.31·37-s + 1.92·39-s − 1.21·43-s − 0.144·48-s − 2/7·49-s − 1.66·52-s − 1.05·57-s − 1.02·61-s − 0.503·63-s + 1/8·64-s − 2.93·67-s + 0.468·73-s − 0.692·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31212 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31212 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 + T \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−10.13636675520507553497406453887, −9.809680223299837765262582433989, −9.330186221363971309576111465163, −8.951168622566808298087877447389, −7.58460004774890445683046626123, −7.50224155158573118874050462938, −6.99640780173569532110131873676, −6.52498883063792512928187502617, −5.71929507199710618712172159386, −5.00716852164944696644803170219, −4.84625070630626219217033988858, −3.34589616522309311397390467846, −3.08824258991321588799800323044, −2.01050304782327789083256558357, 0,
2.01050304782327789083256558357, 3.08824258991321588799800323044, 3.34589616522309311397390467846, 4.84625070630626219217033988858, 5.00716852164944696644803170219, 5.71929507199710618712172159386, 6.52498883063792512928187502617, 6.99640780173569532110131873676, 7.50224155158573118874050462938, 7.58460004774890445683046626123, 8.951168622566808298087877447389, 9.330186221363971309576111465163, 9.809680223299837765262582433989, 10.13636675520507553497406453887
