| L(s) = 1 | − 4·4-s + 12·16-s − 10·25-s + 14·49-s − 32·64-s − 2·67-s + 10·73-s − 14·79-s
+ 22·97-s + 40·100-s − 26·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s
+ 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s
+ 181-s + 191-s + 193-s + ⋯
|
| L(s) = 1 | − 2·4-s + 3·16-s − 2·25-s + 2·49-s − 4·64-s − 0.244·67-s + 1.17·73-s − 1.57·79-s
+ 2.23·97-s + 4·100-s − 2.49·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s
+ 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s
+ 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 641601 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr
=\mathstrut & \, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 641601 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{3,\;89\}$,
\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;89\}$, then $F_p$ is a polynomial of degree at most 3.
| $p$ | $\Gal(F_p)$ | $F_p$ |
|---|
| bad | 3 | | \( 1 \) |
| 89 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 77 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\[\begin{aligned}
L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Imaginary part of the first few zeros on the critical line
−8.407279713197925052163265155535, −7.974908688946724448869699809349, −7.69045710638473315079962729984, −7.18561155073859624406852722121, −6.45827437551555765686664075317, −5.86917644712850163062864649701, −5.56199420487264253592771873444, −5.15783074656497583469979570467, −4.52330119533499255224262395432, −4.09806919249719650622794122193, −3.79261438973760602015649745895, −3.18616646491018548666489390246, −2.33055703087610667011952854258, −1.41052068632978185547053056838, −0.47170587173855267234238980133,
0.47170587173855267234238980133, 1.41052068632978185547053056838, 2.33055703087610667011952854258, 3.18616646491018548666489390246, 3.79261438973760602015649745895, 4.09806919249719650622794122193, 4.52330119533499255224262395432, 5.15783074656497583469979570467, 5.56199420487264253592771873444, 5.86917644712850163062864649701, 6.45827437551555765686664075317, 7.18561155073859624406852722121, 7.69045710638473315079962729984, 7.974908688946724448869699809349, 8.407279713197925052163265155535
