| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s + 8-s + 9-s − 2·10-s
− 4·11-s + 12-s − 2·13-s − 2·15-s + 16-s + 17-s + 18-s + 4·19-s
− 2·20-s − 4·22-s + 24-s − 25-s − 2·26-s + 27-s − 10·29-s − 2·30-s
+ 8·31-s + 32-s − 4·33-s + ⋯
|
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s
− 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s
− 0.447·20-s − 0.852·22-s + 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 1.85·29-s − 0.365·30-s
+ 1.43·31-s + 0.176·32-s − 0.696·33-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & \, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{2,\;3,\;17\}$,
\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\[\begin{aligned}
L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Imaginary part of the first few zeros on the critical line
−19.60787398221141, −18.97481261132692, −17.79286854934361, −16.33123300834676, −15.62682950030683, −14.86810353896462, −13.82453782976521, −12.87130622556280, −11.93324918288480, −10.84353554600873, −9.570394810938146, −7.958882066985846, −7.364407200731999, −5.514578829260743, −4.138750900059067, −2.763098134093487,
2.763098134093487, 4.138750900059067, 5.514578829260743, 7.364407200731999, 7.958882066985846, 9.570394810938146, 10.84353554600873, 11.93324918288480, 12.87130622556280, 13.82453782976521, 14.86810353896462, 15.62682950030683, 16.33123300834676, 17.79286854934361, 18.97481261132692, 19.60787398221141
