L(s) = 1 | − 2.05·2-s − 2.81·3-s + 2.23·4-s + 0.650·5-s + 5.80·6-s + 0.324·7-s − 0.481·8-s + 4.95·9-s − 1.33·10-s − 3.65·11-s − 6.29·12-s − 4.76·13-s − 0.667·14-s − 1.83·15-s − 3.47·16-s − 1.11·17-s − 10.1·18-s + 7.62·19-s + 1.45·20-s − 0.914·21-s + 7.51·22-s + 5.96·23-s + 1.35·24-s − 4.57·25-s + 9.80·26-s − 5.50·27-s + 0.724·28-s + ⋯ |
L(s) = 1 | − 1.45·2-s − 1.62·3-s + 1.11·4-s + 0.290·5-s + 2.36·6-s + 0.122·7-s − 0.170·8-s + 1.65·9-s − 0.423·10-s − 1.10·11-s − 1.81·12-s − 1.32·13-s − 0.178·14-s − 0.473·15-s − 0.869·16-s − 0.269·17-s − 2.40·18-s + 1.74·19-s + 0.324·20-s − 0.199·21-s + 1.60·22-s + 1.24·23-s + 0.276·24-s − 0.915·25-s + 1.92·26-s − 1.05·27-s + 0.136·28-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 | \( 1 + 2.05T + 2T^{2} \) |
| 3 | \( 1 + 2.81T + 3T^{2} \) |
| 5 | \( 1 - 0.650T + 5T^{2} \) |
| 7 | \( 1 - 0.324T + 7T^{2} \) |
| 11 | \( 1 + 3.65T + 11T^{2} \) |
| 13 | \( 1 + 4.76T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 - 7.62T + 19T^{2} \) |
| 23 | \( 1 - 5.96T + 23T^{2} \) |
| 29 | \( 1 - 9.04T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 + 3.84T + 37T^{2} \) |
| 41 | \( 1 - 4.18T + 41T^{2} \) |
| 43 | \( 1 - 6.53T + 43T^{2} \) |
| 47 | \( 1 + 1.70T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 5.03T + 61T^{2} \) |
| 67 | \( 1 + 9.87T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 4.36T + 97T^{2} \) |
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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
−9.918838396838255666214176267128, −8.950771470628264846059979069826, −7.67939209306844669573499758927, −7.30532952048052752031056251093, −6.31676641856174550808039122246, −5.20704343852167339938916886152, −4.78976946928943323053628360494, −2.65740311129028304119319437105, −1.18514791276955213778956902876, 0,
1.18514791276955213778956902876, 2.65740311129028304119319437105, 4.78976946928943323053628360494, 5.20704343852167339938916886152, 6.31676641856174550808039122246, 7.30532952048052752031056251093, 7.67939209306844669573499758927, 8.950771470628264846059979069826, 9.918838396838255666214176267128