| L(s) = 1 | − 2-s − 4-s + 2·5-s + 7-s + 3·8-s − 2·10-s + 2·13-s − 14-s − 16-s − 6·17-s − 4·19-s − 2·20-s − 25-s − 2·26-s − 28-s − 2·29-s − 5·32-s + 6·34-s + 2·35-s + 6·37-s + 4·38-s + 6·40-s + 2·41-s + 4·43-s + 49-s + 50-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s + 1.06·8-s − 0.632·10-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.447·20-s − 1/5·25-s − 0.392·26-s − 0.188·28-s − 0.371·29-s − 0.883·32-s + 1.02·34-s + 0.338·35-s + 0.986·37-s + 0.648·38-s + 0.948·40-s + 0.312·41-s + 0.609·43-s + 1/7·49-s + 0.141·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]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 | \( 1 \) |
| 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 | \( 1 - 18 T + p T^{2} \) |
| show more | |
| show less | |
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]
Imaginary part of the first few zeros on the critical line
−17.14055573366832, −16.83575571401204, −15.93027713845859, −15.40965513850226, −14.61498799125813, −14.03507831529025, −13.53481514762512, −13.02391570779124, −12.53968112568611, −11.34220857640261, −11.01049711886279, −10.36427958838071, −9.730563286935959, −9.056096586309823, −8.783200717465471, −7.976525779912434, −7.393988504131659, −6.345180936532752, −6.015810616969177, −4.924035533657760, −4.467926406572259, −3.620887797848250, −2.280591623076023, −1.776112969349438, −0.6333466336966226,
0.6333466336966226, 1.776112969349438, 2.280591623076023, 3.620887797848250, 4.467926406572259, 4.924035533657760, 6.015810616969177, 6.345180936532752, 7.393988504131659, 7.976525779912434, 8.783200717465471, 9.056096586309823, 9.730563286935959, 10.36427958838071, 11.01049711886279, 11.34220857640261, 12.53968112568611, 13.02391570779124, 13.53481514762512, 14.03507831529025, 14.61498799125813, 15.40965513850226, 15.93027713845859, 16.83575571401204, 17.14055573366832
