| L(s) = 1 | + 0.959·2-s − 1.07·4-s + 2.40·5-s − 7-s − 2.95·8-s + 2.30·10-s + 4.25·11-s + 3.54·13-s
− 0.959·14-s − 0.679·16-s + 0.302·17-s − 6.75·19-s − 2.59·20-s + 4.08·22-s − 6.51·23-s + 0.782·25-s
+ 3.40·26-s + 1.07·28-s − 3.90·29-s − 5.48·31-s + 5.25·32-s + 0.290·34-s − 2.40·35-s − 6.55·37-s
− 6.48·38-s − 7.10·40-s − 2.17·41-s + ⋯
|
| L(s) = 1 | + 0.678·2-s − 0.539·4-s + 1.07·5-s − 0.377·7-s − 1.04·8-s + 0.729·10-s + 1.28·11-s + 0.982·13-s
− 0.256·14-s − 0.169·16-s + 0.0733·17-s − 1.55·19-s − 0.579·20-s + 0.871·22-s − 1.35·23-s + 0.156·25-s
+ 0.667·26-s + 0.203·28-s − 0.725·29-s − 0.985·31-s + 0.929·32-s + 0.0497·34-s − 0.406·35-s − 1.07·37-s
− 1.05·38-s − 1.12·40-s − 0.339·41-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 | \( 1 - 0.959T + 2T^{2} \) |
| 5 | \( 1 - 2.40T + 5T^{2} \) |
| 11 | \( 1 - 4.25T + 11T^{2} \) |
| 13 | \( 1 - 3.54T + 13T^{2} \) |
| 17 | \( 1 - 0.302T + 17T^{2} \) |
| 19 | \( 1 + 6.75T + 19T^{2} \) |
| 23 | \( 1 + 6.51T + 23T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 + 5.48T + 31T^{2} \) |
| 37 | \( 1 + 6.55T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 - 9.47T + 43T^{2} \) |
| 47 | \( 1 + 1.43T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 7.97T + 59T^{2} \) |
| 61 | \( 1 - 1.05T + 61T^{2} \) |
| 67 | \( 1 + 4.72T + 67T^{2} \) |
| 71 | \( 1 + 6.25T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 4.55T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 8.24T + 89T^{2} \) |
| 97 | \( 1 - 1.74T + 97T^{2} \) |
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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
−7.30825551135353269068417059002, −6.37274282934683146819670465513, −6.04542123953429389365568850070, −5.61750444289831304683868519013, −4.52490600343899763293110784493, −3.92572718379283316569497910306, −3.41114466482564541916485872879, −2.19709594708874365379859769453, −1.47420693669790440193520748477, 0,
1.47420693669790440193520748477, 2.19709594708874365379859769453, 3.41114466482564541916485872879, 3.92572718379283316569497910306, 4.52490600343899763293110784493, 5.61750444289831304683868519013, 6.04542123953429389365568850070, 6.37274282934683146819670465513, 7.30825551135353269068417059002
