| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 5·7-s − 8-s − 2·9-s
+ 10-s + 11-s + 12-s + 2·13-s − 5·14-s − 15-s + 16-s + 3·17-s
+ 2·18-s − 7·19-s − 20-s + 5·21-s − 22-s − 6·23-s − 24-s + 25-s
− 2·26-s − 5·27-s + 5·28-s + ⋯
|
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.88·7-s − 0.353·8-s − 2/3·9-s
+ 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.554·13-s − 1.33·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s
+ 0.471·18-s − 1.60·19-s − 0.223·20-s + 1.09·21-s − 0.213·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s
− 0.392·26-s − 0.962·27-s + 0.944·28-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & \, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{2,\;5,\;11\}$,
\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−19.32704181196344, −18.46000402182116, −17.52251662599698, −16.84579337197409, −15.52740236396597, −14.57255006605518, −14.13397147037926, −12.37150177127700, −11.32146429554651, −10.70422776242305, −8.996361573480265, −8.285662434899333, −7.569811964690275, −5.738764390858290, −4.011731357475171, −1.971708425879564,
1.971708425879564, 4.011731357475171, 5.738764390858290, 7.569811964690275, 8.285662434899333, 8.996361573480265, 10.70422776242305, 11.32146429554651, 12.37150177127700, 14.13397147037926, 14.57255006605518, 15.52740236396597, 16.84579337197409, 17.52251662599698, 18.46000402182116, 19.32704181196344
