| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s + 2·13-s
+ 14-s + 16-s − 2·17-s + 4·19-s + 20-s − 22-s + 4·23-s + 25-s
− 2·26-s − 28-s + 6·29-s − 32-s + 2·34-s − 35-s + 2·37-s − 4·38-s
− 40-s − 6·41-s + 12·43-s + ⋯
|
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.554·13-s
+ 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s
− 0.392·26-s − 0.188·28-s + 1.11·29-s − 0.176·32-s + 0.342·34-s − 0.169·35-s + 0.328·37-s − 0.648·38-s
− 0.158·40-s − 0.937·41-s + 1.82·43-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & \, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$,
\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−17.10736794933669, −16.81609009727152, −16.03811678727238, −15.59375454244738, −15.04389787266324, −14.03975328305960, −13.85818149152602, −12.92773482217876, −12.48412617025114, −11.60936232388057, −11.17142559960118, −10.41660357199356, −9.908261552962809, −9.155778960343793, −8.843879932863189, −7.999269703901244, −7.270064039136368, −6.586646178706958, −6.077154111900921, −5.239904256823969, −4.376488489100932, −3.328324091579576, −2.699492473662413, −1.619000253471359, −0.7612365854525905,
0.7612365854525905, 1.619000253471359, 2.699492473662413, 3.328324091579576, 4.376488489100932, 5.239904256823969, 6.077154111900921, 6.586646178706958, 7.270064039136368, 7.999269703901244, 8.843879932863189, 9.155778960343793, 9.908261552962809, 10.41660357199356, 11.17142559960118, 11.60936232388057, 12.48412617025114, 12.92773482217876, 13.85818149152602, 14.03975328305960, 15.04389787266324, 15.59375454244738, 16.03811678727238, 16.81609009727152, 17.10736794933669
