| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s + 3·11-s + 2·13-s − 2·14-s + 16-s + 3·17-s − 19-s − 3·22-s + 6·23-s − 5·25-s − 2·26-s + 2·28-s − 6·29-s − 4·31-s − 32-s − 3·34-s − 4·37-s + 38-s − 9·41-s − 43-s + 3·44-s − 6·46-s + 6·47-s − 3·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 0.904·11-s + 0.554·13-s − 0.534·14-s + 1/4·16-s + 0.727·17-s − 0.229·19-s − 0.639·22-s + 1.25·23-s − 25-s − 0.392·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.514·34-s − 0.657·37-s + 0.162·38-s − 1.40·41-s − 0.152·43-s + 0.452·44-s − 0.884·46-s + 0.875·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−19.00688277120738, −18.42746631330654, −17.26128934152204, −16.93118765065978, −15.70555831584594, −14.82782448622343, −13.98195129207708, −12.70434390882728, −11.57619177441088, −10.96157598355786, −9.711604924732819, −8.789487803962223, −7.806708544758987, −6.686197134032186, −5.343839769637045, −3.650760604853421, −1.606932511063427,
1.606932511063427, 3.650760604853421, 5.343839769637045, 6.686197134032186, 7.806708544758987, 8.789487803962223, 9.711604924732819, 10.96157598355786, 11.57619177441088, 12.70434390882728, 13.98195129207708, 14.82782448622343, 15.70555831584594, 16.93118765065978, 17.26128934152204, 18.42746631330654, 19.00688277120738
