| L(s) = 1 | − 0.875·2-s + 0.413·3-s − 1.23·4-s + 0.141·5-s − 0.361·6-s − 0.646·7-s + 2.83·8-s − 2.82·9-s − 0.123·10-s − 0.501·11-s − 0.509·12-s + 1.72·13-s + 0.566·14-s + 0.0582·15-s − 0.0105·16-s + 4.80·17-s + 2.47·18-s + 3.13·19-s − 0.174·20-s − 0.267·21-s + 0.438·22-s − 4.27·23-s + 1.16·24-s − 4.98·25-s − 1.50·26-s − 2.40·27-s + 0.797·28-s + ⋯ |
| L(s) = 1 | − 0.618·2-s + 0.238·3-s − 0.616·4-s + 0.0630·5-s − 0.147·6-s − 0.244·7-s + 1.00·8-s − 0.943·9-s − 0.0390·10-s − 0.151·11-s − 0.147·12-s + 0.477·13-s + 0.151·14-s + 0.0150·15-s − 0.00263·16-s + 1.16·17-s + 0.583·18-s + 0.718·19-s − 0.0389·20-s − 0.0583·21-s + 0.0935·22-s − 0.891·23-s + 0.238·24-s − 0.996·25-s − 0.295·26-s − 0.463·27-s + 0.150·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 983 | \( 1 + T \) |
| good | 2 | \( 1 + 0.875T + 2T^{2} \) |
| 3 | \( 1 - 0.413T + 3T^{2} \) |
| 5 | \( 1 - 0.141T + 5T^{2} \) |
| 7 | \( 1 + 0.646T + 7T^{2} \) |
| 11 | \( 1 + 0.501T + 11T^{2} \) |
| 13 | \( 1 - 1.72T + 13T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 - 3.13T + 19T^{2} \) |
| 23 | \( 1 + 4.27T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 4.85T + 31T^{2} \) |
| 37 | \( 1 - 1.57T + 37T^{2} \) |
| 41 | \( 1 - 2.34T + 41T^{2} \) |
| 43 | \( 1 + 3.02T + 43T^{2} \) |
| 47 | \( 1 - 9.52T + 47T^{2} \) |
| 53 | \( 1 + 1.35T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 3.19T + 61T^{2} \) |
| 67 | \( 1 + 3.20T + 67T^{2} \) |
| 71 | \( 1 + 8.67T + 71T^{2} \) |
| 73 | \( 1 + 9.90T + 73T^{2} \) |
| 79 | \( 1 - 3.51T + 79T^{2} \) |
| 83 | \( 1 + 0.509T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 5.28T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.430496688305555763055677498550, −8.924463510990756726005034164868, −7.85228126142684144472542870737, −7.57293664274347001908861198381, −5.92666278237194890974458213616, −5.44224650013328968106128387442, −4.04775187976541194393560627894, −3.20890734454257972765970307856, −1.65480008594678269563427715231, 0,
1.65480008594678269563427715231, 3.20890734454257972765970307856, 4.04775187976541194393560627894, 5.44224650013328968106128387442, 5.92666278237194890974458213616, 7.57293664274347001908861198381, 7.85228126142684144472542870737, 8.924463510990756726005034164868, 9.430496688305555763055677498550
