| L(s) = 1 | − 11-s − 4·16-s − 11·31-s + 9·41-s − 61-s + 19·71-s − 9·81-s + 101-s
+ 103-s + 107-s + 109-s + 113-s + 10·121-s + 127-s + 131-s + 137-s
+ 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 4·176-s
+ 179-s + 181-s + 191-s + ⋯
|
| L(s) = 1 | − 0.301·11-s − 16-s − 1.97·31-s + 1.40·41-s − 0.128·61-s + 2.25·71-s − 81-s + 0.0995·101-s
+ 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.909·121-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 + 0.0760·173-s + 0.301·176-s
+ 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 3125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr
=\mathstrut & \, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 3125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \neq 5$,
\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 5$, then $F_p$ is a polynomial of degree at most 3.
| $p$ | $\Gal(F_p)$ | $F_p$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + T - 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_4$ | \( 1 + 11 T + 61 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_4$ | \( 1 + T + 91 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 19 T + 201 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| show more | | |
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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
−18.2513478698, −17.794030318, −17.0572865553, −16.6516421521, −15.9944202971, −15.6665478707, −15.0412242767, −14.3272056793, −14.0663480476, −13.2082005437, −12.8575532333, −12.2756492446, −11.4301570976, −11.07580254, −10.4664572499, −9.62470469199, −9.14526688174, −8.48931188846, −7.62285385181, −7.1021858939, −6.22989238233, −5.43292897511, −4.57292147662, −3.58578684104, −2.26074119601,
2.26074119601, 3.58578684104, 4.57292147662, 5.43292897511, 6.22989238233, 7.1021858939, 7.62285385181, 8.48931188846, 9.14526688174, 9.62470469199, 10.4664572499, 11.07580254, 11.4301570976, 12.2756492446, 12.8575532333, 13.2082005437, 14.0663480476, 14.3272056793, 15.0412242767, 15.6665478707, 15.9944202971, 16.6516421521, 17.0572865553, 17.794030318, 18.2513478698
