| L(s) = 1 | + 1.66·2-s + 0.783·4-s + 3.76·5-s − 7-s − 2.02·8-s + 6.28·10-s − 3.16·11-s − 3.02·13-s
− 1.66·14-s − 4.95·16-s + 1.05·17-s + 3.14·19-s + 2.95·20-s − 5.27·22-s − 7.12·23-s + 9.17·25-s
− 5.04·26-s − 0.783·28-s + 2.67·29-s − 8.88·31-s − 4.20·32-s + 1.76·34-s − 3.76·35-s + 0.943·37-s
+ 5.24·38-s − 7.64·40-s + 9.90·41-s + ⋯
|
| L(s) = 1 | + 1.17·2-s + 0.391·4-s + 1.68·5-s − 0.377·7-s − 0.717·8-s + 1.98·10-s − 0.953·11-s − 0.838·13-s
− 0.445·14-s − 1.23·16-s + 0.256·17-s + 0.721·19-s + 0.659·20-s − 1.12·22-s − 1.48·23-s + 1.83·25-s
− 0.988·26-s − 0.148·28-s + 0.496·29-s − 1.59·31-s − 0.743·32-s + 0.302·34-s − 0.636·35-s + 0.155·37-s
+ 0.851·38-s − 1.20·40-s + 1.54·41-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & -\, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & -\, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{3,\;7,\;127\}$,
\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;127\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 1.66T + 2T^{2} \) |
| 5 | \( 1 - 3.76T + 5T^{2} \) |
| 11 | \( 1 + 3.16T + 11T^{2} \) |
| 13 | \( 1 + 3.02T + 13T^{2} \) |
| 17 | \( 1 - 1.05T + 17T^{2} \) |
| 19 | \( 1 - 3.14T + 19T^{2} \) |
| 23 | \( 1 + 7.12T + 23T^{2} \) |
| 29 | \( 1 - 2.67T + 29T^{2} \) |
| 31 | \( 1 + 8.88T + 31T^{2} \) |
| 37 | \( 1 - 0.943T + 37T^{2} \) |
| 41 | \( 1 - 9.90T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 4.25T + 47T^{2} \) |
| 53 | \( 1 + 5.14T + 53T^{2} \) |
| 59 | \( 1 + 4.15T + 59T^{2} \) |
| 61 | \( 1 - 1.23T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 7.29T + 73T^{2} \) |
| 79 | \( 1 - 5.09T + 79T^{2} \) |
| 83 | \( 1 + 8.93T + 83T^{2} \) |
| 89 | \( 1 - 5.97T + 89T^{2} \) |
| 97 | \( 1 + 9.75T + 97T^{2} \) |
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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
−7.28821749229893206104276375544, −6.44684884288908041031852658295, −5.84601630466999523181439960556, −5.43310496175598323108190940318, −4.88796705639650252968083440612, −3.99417749180424320252982865655, −2.98392948112995424146751911316, −2.52720431443314530036587494409, −1.65778769295668435153481966279, 0,
1.65778769295668435153481966279, 2.52720431443314530036587494409, 2.98392948112995424146751911316, 3.99417749180424320252982865655, 4.88796705639650252968083440612, 5.43310496175598323108190940318, 5.84601630466999523181439960556, 6.44684884288908041031852658295, 7.28821749229893206104276375544
