| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 5-s + 2·6-s − 2·7-s − 2·9-s − 2·10-s
+ 11-s − 2·12-s + 4·13-s + 4·14-s − 15-s − 4·16-s − 2·17-s + 4·18-s
+ 2·20-s + 2·21-s − 2·22-s − 23-s − 4·25-s − 8·26-s + 5·27-s − 4·28-s
+ 2·30-s + 7·31-s + 8·32-s + ⋯
|
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 0.755·7-s − 2/3·9-s − 0.632·10-s
+ 0.301·11-s − 0.577·12-s + 1.10·13-s + 1.06·14-s − 0.258·15-s − 16-s − 0.485·17-s + 0.942·18-s
+ 0.447·20-s + 0.436·21-s − 0.426·22-s − 0.208·23-s − 4/5·25-s − 1.56·26-s + 0.962·27-s − 0.755·28-s
+ 0.365·30-s + 1.25·31-s + 1.41·32-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & \, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \neq 11$,
\[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 11$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 11 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−19.18572497185224, −17.94143357345934, −17.03361032038062, −15.91407260330038, −13.56863905712999, −11.45125861034521, −10.03550909718108, −8.603539619290756, −6.362613894713089,
6.362613894713089, 8.603539619290756, 10.03550909718108, 11.45125861034521, 13.56863905712999, 15.91407260330038, 17.03361032038062, 17.94143357345934, 19.18572497185224
