| L(s) = 1 | − 2-s + 3-s − 6-s − 7-s + 8-s + 9-s − 11-s − 13-s
+ 14-s − 16-s − 17-s − 18-s − 21-s + 22-s + 24-s + 25-s
+ 26-s + 27-s + 29-s − 33-s + 34-s − 39-s + 2·41-s + 42-s
− 47-s − 48-s − 50-s + ⋯
|
| L(s) = 1 | − 2-s + 3-s − 6-s − 7-s + 8-s + 9-s − 11-s − 13-s
+ 14-s − 16-s − 17-s − 18-s − 21-s + 22-s + 24-s + 25-s
+ 26-s + 27-s + 29-s − 33-s + 34-s − 39-s + 2·41-s + 42-s
− 47-s − 48-s − 50-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{3,\;29\}$,
\(F_p\) is a polynomial of degree 2. If $p \in \{3,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 3 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−14.45102409066779657045769585783, −13.32162862510749184354432270374, −12.66997768785203913828312408994, −10.65716487846662969380774941347, −9.805231985523799033951222248495, −9.004858706215995835885407289726, −7.939979829085170543073834513148, −6.89861709118401082750268414601, −4.60444267458741594170483479006, −2.68368590772453422884570720998,
2.68368590772453422884570720998, 4.60444267458741594170483479006, 6.89861709118401082750268414601, 7.939979829085170543073834513148, 9.004858706215995835885407289726, 9.805231985523799033951222248495, 10.65716487846662969380774941347, 12.66997768785203913828312408994, 13.32162862510749184354432270374, 14.45102409066779657045769585783
