| L(s) = 1 | − 2.73·2-s − 2.15·3-s + 5.46·4-s − 2.36·5-s + 5.88·6-s − 3.03·7-s − 9.45·8-s + 1.64·9-s
+ 6.46·10-s − 1.23·11-s − 11.7·12-s + 0.935·13-s + 8.30·14-s + 5.09·15-s + 14.8·16-s + 3.71·17-s
− 4.48·18-s + 6.12·19-s − 12.9·20-s + 6.54·21-s + 3.37·22-s − 3.46·23-s + 20.3·24-s + 0.603·25-s
− 2.55·26-s + 2.92·27-s − 16.5·28-s + ⋯
|
| L(s) = 1 | − 1.93·2-s − 1.24·3-s + 2.73·4-s − 1.05·5-s + 2.40·6-s − 1.14·7-s − 3.34·8-s + 0.547·9-s
+ 2.04·10-s − 0.372·11-s − 3.39·12-s + 0.259·13-s + 2.21·14-s + 1.31·15-s + 3.72·16-s + 0.901·17-s
− 1.05·18-s + 1.40·19-s − 2.89·20-s + 1.42·21-s + 0.720·22-s − 0.722·23-s + 4.15·24-s + 0.120·25-s
− 0.501·26-s + 0.563·27-s − 3.13·28-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & -\, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & -\, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{71,\;113\}$,
\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{71,\;113\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 71 | \( 1 + T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 3 | \( 1 + 2.15T + 3T^{2} \) |
| 5 | \( 1 + 2.36T + 5T^{2} \) |
| 7 | \( 1 + 3.03T + 7T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 - 0.935T + 13T^{2} \) |
| 17 | \( 1 - 3.71T + 17T^{2} \) |
| 19 | \( 1 - 6.12T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 5.46T + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 + 1.25T + 37T^{2} \) |
| 41 | \( 1 + 5.00T + 41T^{2} \) |
| 43 | \( 1 + 8.04T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 1.52T + 53T^{2} \) |
| 59 | \( 1 + 2.78T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 5.50T + 67T^{2} \) |
| 73 | \( 1 + 7.75T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 1.58T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 8.08T + 97T^{2} \) |
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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
−7.64368174547166190429119211164, −6.98141420483696511734256135745, −6.28574982030495648566035656705, −5.84320514265553038980625732207, −4.92606663524495476537400991104, −3.38524302277908480650659990412, −3.18046167313570893469716680450, −1.69898743349362802874496522289, −0.65483075070968887689993378536, 0,
0.65483075070968887689993378536, 1.69898743349362802874496522289, 3.18046167313570893469716680450, 3.38524302277908480650659990412, 4.92606663524495476537400991104, 5.84320514265553038980625732207, 6.28574982030495648566035656705, 6.98141420483696511734256135745, 7.64368174547166190429119211164
