| L(s) = 1 | + 2-s − 3-s − 4-s + 2·5-s − 6-s − 4·7-s − 3·8-s + 9-s + 2·10-s − 11-s + 12-s − 2·13-s − 4·14-s − 2·15-s − 16-s + 2·17-s + 18-s − 2·20-s + 4·21-s − 22-s + 3·24-s − 25-s − 2·26-s − 27-s + 4·28-s − 6·29-s − 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s − 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s − 0.554·13-s − 1.06·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.447·20-s + 0.872·21-s − 0.213·22-s + 0.612·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.755·28-s − 1.11·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17457 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17457 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.67670276289191, −15.41683607326312, −14.53402752733817, −14.16670416481480, −13.41166770660656, −13.17508723344350, −12.58015089406968, −12.30013932460740, −11.57721849670245, −10.68840360243007, −10.18696278925894, −9.582070429337255, −9.334439177640090, −8.684862879703934, −7.587168812136207, −7.060234579246408, −6.279075495554188, −5.786782536432515, −5.468155400475509, −4.753613168561721, −3.858616161173533, −3.359870956490712, −2.603757965429791, −1.652802520486215, −0.2856987873801149,
0.2856987873801149, 1.652802520486215, 2.603757965429791, 3.359870956490712, 3.858616161173533, 4.753613168561721, 5.468155400475509, 5.786782536432515, 6.279075495554188, 7.060234579246408, 7.587168812136207, 8.684862879703934, 9.334439177640090, 9.582070429337255, 10.18696278925894, 10.68840360243007, 11.57721849670245, 12.30013932460740, 12.58015089406968, 13.17508723344350, 13.41166770660656, 14.16670416481480, 14.53402752733817, 15.41683607326312, 15.67670276289191
