| L(s) = 1 | − 2.69·2-s + 5.26·4-s + 3.35·5-s − 7-s − 8.81·8-s − 9.05·10-s − 2.89·11-s − 2.10·13-s
+ 2.69·14-s + 13.2·16-s + 6.29·17-s − 0.167·19-s + 17.6·20-s + 7.80·22-s + 4.03·23-s + 6.28·25-s
+ 5.68·26-s − 5.26·28-s − 0.392·29-s − 7.54·31-s − 18.0·32-s − 16.9·34-s − 3.35·35-s − 2.42·37-s
+ 0.450·38-s − 29.6·40-s + 0.160·41-s + ⋯
|
| L(s) = 1 | − 1.90·2-s + 2.63·4-s + 1.50·5-s − 0.377·7-s − 3.11·8-s − 2.86·10-s − 0.872·11-s − 0.584·13-s
+ 0.720·14-s + 3.30·16-s + 1.52·17-s − 0.0383·19-s + 3.95·20-s + 1.66·22-s + 0.841·23-s + 1.25·25-s
+ 1.11·26-s − 0.995·28-s − 0.0727·29-s − 1.35·31-s − 3.18·32-s − 2.90·34-s − 0.567·35-s − 0.399·37-s
+ 0.0731·38-s − 4.68·40-s + 0.0251·41-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 | \( 1 + 2.69T + 2T^{2} \) |
| 5 | \( 1 - 3.35T + 5T^{2} \) |
| 11 | \( 1 + 2.89T + 11T^{2} \) |
| 13 | \( 1 + 2.10T + 13T^{2} \) |
| 17 | \( 1 - 6.29T + 17T^{2} \) |
| 19 | \( 1 + 0.167T + 19T^{2} \) |
| 23 | \( 1 - 4.03T + 23T^{2} \) |
| 29 | \( 1 + 0.392T + 29T^{2} \) |
| 31 | \( 1 + 7.54T + 31T^{2} \) |
| 37 | \( 1 + 2.42T + 37T^{2} \) |
| 41 | \( 1 - 0.160T + 41T^{2} \) |
| 43 | \( 1 + 6.15T + 43T^{2} \) |
| 47 | \( 1 + 8.81T + 47T^{2} \) |
| 53 | \( 1 - 2.87T + 53T^{2} \) |
| 59 | \( 1 - 3.45T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 9.36T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 3.36T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 1.23T + 97T^{2} \) |
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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
−7.56734841155529496080696267410, −7.03395660043013078314971111989, −6.34450089249562463294815539038, −5.58321091302913426031762943851, −5.17254428232854490536115960669, −3.32955328759689101793908242379, −2.71646524715798009941569587156, −1.92498085767698358500431788361, −1.23109092902642923581321582119, 0,
1.23109092902642923581321582119, 1.92498085767698358500431788361, 2.71646524715798009941569587156, 3.32955328759689101793908242379, 5.17254428232854490536115960669, 5.58321091302913426031762943851, 6.34450089249562463294815539038, 7.03395660043013078314971111989, 7.56734841155529496080696267410
