| L(s) = 1 | + 2-s + 3-s − 4-s − 2·5-s + 6-s + 7-s − 3·8-s + 9-s − 2·10-s − 12-s + 2·13-s + 14-s − 2·15-s − 16-s + 6·17-s + 18-s − 4·19-s + 2·20-s + 21-s − 3·24-s − 25-s + 2·26-s + 27-s − 28-s + 2·29-s − 2·30-s + 5·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.516·15-s − 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.218·21-s − 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s − 0.365·30-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−19.09596762821552, −18.37611824954840, −17.84486582336623, −16.93940671605389, −16.25565148507381, −15.47199013207948, −14.94760107428179, −14.45416546547210, −13.84332799687643, −13.15796858012131, −12.49597014375472, −11.93911102596347, −11.23727948227840, −10.30302436393845, −9.563733444203339, −8.699165629902587, −8.166185648107227, −7.598605305113226, −6.501047952237606, −5.627673766991474, −4.791352170722338, −3.918052774078995, −3.565996843964124, −2.434703228304793, −0.8605694561736051,
0.8605694561736051, 2.434703228304793, 3.565996843964124, 3.918052774078995, 4.791352170722338, 5.627673766991474, 6.501047952237606, 7.598605305113226, 8.166185648107227, 8.699165629902587, 9.563733444203339, 10.30302436393845, 11.23727948227840, 11.93911102596347, 12.49597014375472, 13.15796858012131, 13.84332799687643, 14.45416546547210, 14.94760107428179, 15.47199013207948, 16.25565148507381, 16.93940671605389, 17.84486582336623, 18.37611824954840, 19.09596762821552
