| L(s) = 1 | − 2-s + 3-s − 4-s + 2·5-s − 6-s + 7-s + 3·8-s + 9-s − 2·10-s − 4·11-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 + 4·22-s + 3·24-s − 25-s + 2·26-s + 27-s − 28-s − 2·29-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 − 1.20·11-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 + 0.852·22-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 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
−16.54889524378139, −16.12721077917573, −15.25366721666811, −14.66925374684455, −14.20722956314196, −13.66376007614563, −13.15015001604370, −12.63613257600360, −11.99688166384301, −10.88631993171876, −10.48082909631966, −9.911067176852529, −9.565274802747360, −8.825779665838237, −8.258459326199391, −7.700706042617390, −7.294053011772538, −6.210073091367256, −5.376923206001885, −5.026507186152663, −4.125245764550922, −3.262243590819377, −2.278617602622956, −1.761238006117855, −0.6355519755353798,
0.6355519755353798, 1.761238006117855, 2.278617602622956, 3.262243590819377, 4.125245764550922, 5.026507186152663, 5.376923206001885, 6.210073091367256, 7.294053011772538, 7.700706042617390, 8.258459326199391, 8.825779665838237, 9.565274802747360, 9.911067176852529, 10.48082909631966, 10.88631993171876, 11.99688166384301, 12.63613257600360, 13.15015001604370, 13.66376007614563, 14.20722956314196, 14.66925374684455, 15.25366721666811, 16.12721077917573, 16.54889524378139
