| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 9-s − 12·13-s + 5·16-s − 17-s − 2·18-s + 8·19-s + 6·25-s + 24·26-s − 6·32-s + 2·34-s + 3·36-s − 16·38-s − 8·43-s + 8·47-s − 10·49-s − 12·50-s − 36·52-s − 4·53-s + 24·59-s + 7·64-s − 24·67-s − 3·68-s − 4·72-s + 24·76-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 1/3·9-s − 3.32·13-s + 5/4·16-s − 0.242·17-s − 0.471·18-s + 1.83·19-s + 6/5·25-s + 4.70·26-s − 1.06·32-s + 0.342·34-s + 1/2·36-s − 2.59·38-s − 1.21·43-s + 1.16·47-s − 1.42·49-s − 1.69·50-s − 4.99·52-s − 0.549·53-s + 3.12·59-s + 7/8·64-s − 2.93·67-s − 0.363·68-s − 0.471·72-s + 2.75·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176868 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176868 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.951168622566808298087877447389, −8.623210362724267034247608731207, −7.84253913019476823710858358599, −7.50224155158573118874050462938, −7.10043852506497274232713842564, −6.99640780173569532110131873676, −6.15744346307609821396804060542, −5.34777681077972405320527300661, −5.00716852164944696644803170219, −4.49371969105134698855653078527, −3.34589616522309311397390467846, −2.75000607216763176095328952178, −2.25996527018388001879662209200, −1.24659789595230381621816494252, 0,
1.24659789595230381621816494252, 2.25996527018388001879662209200, 2.75000607216763176095328952178, 3.34589616522309311397390467846, 4.49371969105134698855653078527, 5.00716852164944696644803170219, 5.34777681077972405320527300661, 6.15744346307609821396804060542, 6.99640780173569532110131873676, 7.10043852506497274232713842564, 7.50224155158573118874050462938, 7.84253913019476823710858358599, 8.623210362724267034247608731207, 8.951168622566808298087877447389
