| L(s) = 1 | + 0.801·2-s + 1.69·3-s − 1.35·4-s − 5-s + 1.35·6-s − 7-s − 2.69·8-s − 0.137·9-s − 0.801·10-s − 3.80·11-s − 2.29·12-s − 4.54·13-s − 0.801·14-s − 1.69·15-s + 0.554·16-s + 2.91·17-s − 0.109·18-s − 3.35·19-s + 1.35·20-s − 1.69·21-s − 3.04·22-s − 23-s − 4.55·24-s + 25-s − 3.64·26-s − 5.30·27-s + 1.35·28-s + ⋯ |
| L(s) = 1 | + 0.567·2-s + 0.976·3-s − 0.678·4-s − 0.447·5-s + 0.553·6-s − 0.377·7-s − 0.951·8-s − 0.0456·9-s − 0.253·10-s − 1.14·11-s − 0.662·12-s − 1.25·13-s − 0.214·14-s − 0.436·15-s + 0.138·16-s + 0.706·17-s − 0.0259·18-s − 0.770·19-s + 0.303·20-s − 0.369·21-s − 0.650·22-s − 0.208·23-s − 0.929·24-s + 0.200·25-s − 0.714·26-s − 1.02·27-s + 0.256·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 0.801T + 2T^{2} \) |
| 3 | \( 1 - 1.69T + 3T^{2} \) |
| 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 + 4.54T + 13T^{2} \) |
| 17 | \( 1 - 2.91T + 17T^{2} \) |
| 19 | \( 1 + 3.35T + 19T^{2} \) |
| 29 | \( 1 - 2.09T + 29T^{2} \) |
| 31 | \( 1 - 0.246T + 31T^{2} \) |
| 37 | \( 1 - 5.40T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 + 0.801T + 43T^{2} \) |
| 47 | \( 1 - 1.97T + 47T^{2} \) |
| 53 | \( 1 + 9.78T + 53T^{2} \) |
| 59 | \( 1 + 6.66T + 59T^{2} \) |
| 61 | \( 1 + 5.02T + 61T^{2} \) |
| 67 | \( 1 + 5.03T + 67T^{2} \) |
| 71 | \( 1 - 5.05T + 71T^{2} \) |
| 73 | \( 1 + 0.972T + 73T^{2} \) |
| 79 | \( 1 - 3.14T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 7.18T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.692231273254223181823404317905, −8.984847345582920926673361043674, −8.048140721938519622070096694679, −7.59960726974466287279060004096, −6.16210499881621762735684485022, −5.15398351663416817176671487953, −4.30294272305617922110948871446, −3.20990569499169791234434488287, −2.54665906620521375314917062626, 0,
2.54665906620521375314917062626, 3.20990569499169791234434488287, 4.30294272305617922110948871446, 5.15398351663416817176671487953, 6.16210499881621762735684485022, 7.59960726974466287279060004096, 8.048140721938519622070096694679, 8.984847345582920926673361043674, 9.692231273254223181823404317905
