| L(s) = 1 | + 3-s + 4-s + 9-s + 12-s − 4·13-s + 16-s − 3·19-s − 6·25-s
+ 27-s − 8·31-s + 36-s + 4·37-s − 4·39-s − 8·43-s + 48-s − 14·49-s
− 4·52-s − 3·57-s + 12·61-s + 64-s − 8·67-s + 20·73-s − 6·75-s − 3·76-s
− 16·79-s + 81-s − 8·93-s + ⋯
|
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 1/3·9-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.688·19-s − 6/5·25-s
+ 0.192·27-s − 1.43·31-s + 1/6·36-s + 0.657·37-s − 0.640·39-s − 1.21·43-s + 0.144·48-s − 2·49-s
− 0.554·52-s − 0.397·57-s + 1.53·61-s + 1/8·64-s − 0.977·67-s + 2.34·73-s − 0.692·75-s − 0.344·76-s
− 1.80·79-s + 1/9·81-s − 0.829·93-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 494532 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr
=\mathstrut & -\, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 494532 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr
=\mathstrut & -\, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{2,\;3,\;19,\;241\}$,
\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;19,\;241\}$, then $F_p$ is a polynomial of degree at most 3.
| $p$ | $\Gal(F_p)$ | $F_p$ |
|---|
| bad | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 241 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 10 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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\[\begin{aligned}
L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Imaginary part of the first few zeros on the critical line
−8.338164554822696432012230609046, −7.81889340785255776097334613180, −7.38879730017847074689252726547, −7.06988173517309928211546772496, −6.47580018567473015770058088105, −6.11018254741667048905502674181, −5.45056991976219356650618740437, −4.97131879112196049533659345780, −4.47183151255355436922809419089, −3.73595363601605577624472726058, −3.42097706965918733694572888967, −2.57113327388630744520336208429, −2.15377838381868246197286512820, −1.51401256103575461547756344719, 0,
1.51401256103575461547756344719, 2.15377838381868246197286512820, 2.57113327388630744520336208429, 3.42097706965918733694572888967, 3.73595363601605577624472726058, 4.47183151255355436922809419089, 4.97131879112196049533659345780, 5.45056991976219356650618740437, 6.11018254741667048905502674181, 6.47580018567473015770058088105, 7.06988173517309928211546772496, 7.38879730017847074689252726547, 7.81889340785255776097334613180, 8.338164554822696432012230609046
