| L(s) = 1 | + 2·3-s − 5-s − 3·7-s + 9-s + 5·11-s − 4·13-s − 2·15-s − 3·17-s − 19-s − 6·21-s + 8·23-s − 4·25-s − 4·27-s − 2·29-s + 4·31-s + 10·33-s + 3·35-s + 10·37-s − 8·39-s + 10·41-s + 43-s − 45-s − 47-s + 2·49-s − 6·51-s − 4·53-s − 5·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s + 1.50·11-s − 1.10·13-s − 0.516·15-s − 0.727·17-s − 0.229·19-s − 1.30·21-s + 1.66·23-s − 4/5·25-s − 0.769·27-s − 0.371·29-s + 0.718·31-s + 1.74·33-s + 0.507·35-s + 1.64·37-s − 1.28·39-s + 1.56·41-s + 0.152·43-s − 0.149·45-s − 0.145·47-s + 2/7·49-s − 0.840·51-s − 0.549·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−19.52471042637900, −19.13356120455157, −17.40724029957344, −16.52353459774370, −15.24225419288510, −14.63931551152130, −13.52350522314312, −12.52817820768231, −11.31944737144510, −9.553328544165326, −9.084561409834522, −7.624292291523759, −6.443235140922679, −4.155372769559701, −2.825112610707033,
2.825112610707033, 4.155372769559701, 6.443235140922679, 7.624292291523759, 9.084561409834522, 9.553328544165326, 11.31944737144510, 12.52817820768231, 13.52350522314312, 14.63931551152130, 15.24225419288510, 16.52353459774370, 17.40724029957344, 19.13356120455157, 19.52471042637900
