| L(s) = 1 | + 1.85·3-s + 1.13·5-s + 7-s + 0.457·9-s − 11-s − 13-s + 2.11·15-s − 2.43·17-s
− 4.36·19-s + 1.85·21-s + 4.06·23-s − 3.70·25-s − 4.72·27-s − 0.412·29-s − 8.12·31-s − 1.85·33-s
+ 1.13·35-s − 7.51·37-s − 1.85·39-s − 0.210·41-s + 8.57·43-s + 0.520·45-s − 8.98·47-s + 49-s
− 4.53·51-s + 8.77·53-s − 1.13·55-s + ⋯
|
| L(s) = 1 | + 1.07·3-s + 0.509·5-s + 0.377·7-s + 0.152·9-s − 0.301·11-s − 0.277·13-s + 0.546·15-s − 0.591·17-s
− 1.00·19-s + 0.405·21-s + 0.846·23-s − 0.740·25-s − 0.909·27-s − 0.0765·29-s − 1.45·31-s − 0.323·33-s
+ 0.192·35-s − 1.23·37-s − 0.297·39-s − 0.0328·41-s + 1.30·43-s + 0.0776·45-s − 1.31·47-s + 0.142·49-s
− 0.635·51-s + 1.20·53-s − 0.153·55-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & -\, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & -\, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{2,\;7,\;11,\;13\}$,
\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.85T + 3T^{2} \) |
| 5 | \( 1 - 1.13T + 5T^{2} \) |
| 17 | \( 1 + 2.43T + 17T^{2} \) |
| 19 | \( 1 + 4.36T + 19T^{2} \) |
| 23 | \( 1 - 4.06T + 23T^{2} \) |
| 29 | \( 1 + 0.412T + 29T^{2} \) |
| 31 | \( 1 + 8.12T + 31T^{2} \) |
| 37 | \( 1 + 7.51T + 37T^{2} \) |
| 41 | \( 1 + 0.210T + 41T^{2} \) |
| 43 | \( 1 - 8.57T + 43T^{2} \) |
| 47 | \( 1 + 8.98T + 47T^{2} \) |
| 53 | \( 1 - 8.77T + 53T^{2} \) |
| 59 | \( 1 + 2.23T + 59T^{2} \) |
| 61 | \( 1 + 6.23T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 7.84T + 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 79 | \( 1 + 5.38T + 79T^{2} \) |
| 83 | \( 1 + 5.74T + 83T^{2} \) |
| 89 | \( 1 - 0.577T + 89T^{2} \) |
| 97 | \( 1 + 2.41T + 97T^{2} \) |
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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
−7.47865980501557681686401065008, −7.04742792987735290478100362361, −6.02463174906376930170778242389, −5.44382882765204344314957451287, −4.57490110289721165905106171533, −3.81710460902389320505682722854, −2.98114516768278791631868626841, −2.21932419101583945130557265545, −1.66942785009857548197048305449, 0,
1.66942785009857548197048305449, 2.21932419101583945130557265545, 2.98114516768278791631868626841, 3.81710460902389320505682722854, 4.57490110289721165905106171533, 5.44382882765204344314957451287, 6.02463174906376930170778242389, 7.04742792987735290478100362361, 7.47865980501557681686401065008
