| L(s) = 1 | − 4-s + 4·11-s + 16-s + 5·19-s + 5·29-s − 31-s − 41-s − 4·44-s
− 10·49-s − 15·59-s − 6·61-s − 64-s − 11·71-s − 5·76-s + 10·79-s − 9·81-s
+ 20·89-s + 101-s + 103-s + 107-s + 109-s + 113-s − 5·116-s − 10·121-s
+ 124-s + 127-s + 131-s + ⋯
|
| L(s) = 1 | − 1/2·4-s + 1.20·11-s + 1/4·16-s + 1.14·19-s + 0.928·29-s − 0.179·31-s − 0.156·41-s − 0.603·44-s
− 1.42·49-s − 1.95·59-s − 0.768·61-s − 1/8·64-s − 1.30·71-s − 0.573·76-s + 1.12·79-s − 81-s
+ 2.11·89-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.464·116-s − 0.909·121-s
+ 0.0898·124-s + 0.0887·127-s + 0.0873·131-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 12500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr
=\mathstrut & \, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 12500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{2,\;5\}$,
\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
| $p$ | $\Gal(F_p)$ | $F_p$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 13 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 15 T + 143 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 6 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
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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
−16.3486478295, −15.9168177788, −15.2498750496, −14.8387821617, −14.2448627481, −13.9071977484, −13.5499526755, −12.7792910762, −12.3928787507, −11.7248272699, −11.5232216023, −10.7002898501, −10.160141098, −9.5492928378, −9.1231458947, −8.6602377903, −7.8358810525, −7.41712954412, −6.49364126284, −6.13939202948, −5.15084205654, −4.60495208272, −3.74290124925, −2.99781820026, −1.42931298829,
1.42931298829, 2.99781820026, 3.74290124925, 4.60495208272, 5.15084205654, 6.13939202948, 6.49364126284, 7.41712954412, 7.8358810525, 8.6602377903, 9.1231458947, 9.5492928378, 10.160141098, 10.7002898501, 11.5232216023, 11.7248272699, 12.3928787507, 12.7792910762, 13.5499526755, 13.9071977484, 14.2448627481, 14.8387821617, 15.2498750496, 15.9168177788, 16.3486478295
