| L(s) = 1 | + 2.08·2-s + 1.62·3-s + 2.35·4-s − 2.03·5-s + 3.39·6-s + 1.75·7-s + 0.741·8-s − 0.361·9-s − 4.25·10-s + 3.26·11-s + 3.82·12-s + 6.15·13-s + 3.66·14-s − 3.30·15-s − 3.16·16-s + 3.48·17-s − 0.753·18-s + 6.73·19-s − 4.79·20-s + 2.85·21-s + 6.82·22-s − 8.95·23-s + 1.20·24-s − 0.850·25-s + 12.8·26-s − 5.46·27-s + 4.13·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 0.937·3-s + 1.17·4-s − 0.911·5-s + 1.38·6-s + 0.664·7-s + 0.262·8-s − 0.120·9-s − 1.34·10-s + 0.985·11-s + 1.10·12-s + 1.70·13-s + 0.980·14-s − 0.854·15-s − 0.790·16-s + 0.846·17-s − 0.177·18-s + 1.54·19-s − 1.07·20-s + 0.622·21-s + 1.45·22-s − 1.86·23-s + 0.245·24-s − 0.170·25-s + 2.51·26-s − 1.05·27-s + 0.782·28-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
| \(L(1)\) |
\(\approx\) |
\(4.443952474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.443952474\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.08T + 2T^{2} \) |
| 3 | \( 1 - 1.62T + 3T^{2} \) |
| 5 | \( 1 + 2.03T + 5T^{2} \) |
| 7 | \( 1 - 1.75T + 7T^{2} \) |
| 11 | \( 1 - 3.26T + 11T^{2} \) |
| 13 | \( 1 - 6.15T + 13T^{2} \) |
| 17 | \( 1 - 3.48T + 17T^{2} \) |
| 19 | \( 1 - 6.73T + 19T^{2} \) |
| 23 | \( 1 + 8.95T + 23T^{2} \) |
| 29 | \( 1 - 2.74T + 29T^{2} \) |
| 31 | \( 1 - 3.44T + 31T^{2} \) |
| 37 | \( 1 + 5.67T + 37T^{2} \) |
| 41 | \( 1 + 7.10T + 41T^{2} \) |
| 43 | \( 1 - 3.43T + 43T^{2} \) |
| 47 | \( 1 + 3.01T + 47T^{2} \) |
| 53 | \( 1 + 2.00T + 53T^{2} \) |
| 59 | \( 1 + 9.39T + 59T^{2} \) |
| 61 | \( 1 - 7.17T + 61T^{2} \) |
| 67 | \( 1 - 0.0881T + 67T^{2} \) |
| 71 | \( 1 + 4.83T + 71T^{2} \) |
| 73 | \( 1 + 8.89T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 4.17T + 83T^{2} \) |
| 89 | \( 1 - 0.927T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−10.02392174238775317848395434637, −8.897330443611336320425906308938, −8.243924581853612477297640469699, −7.55210932245212273333024336297, −6.32181828114026925995499812017, −5.58817077341443356430359750147, −4.40117190846940068814380893080, −3.60787415741773737543720915832, −3.24491540147114122951420054988, −1.62330405833784839552077295051,
1.62330405833784839552077295051, 3.24491540147114122951420054988, 3.60787415741773737543720915832, 4.40117190846940068814380893080, 5.58817077341443356430359750147, 6.32181828114026925995499812017, 7.55210932245212273333024336297, 8.243924581853612477297640469699, 8.897330443611336320425906308938, 10.02392174238775317848395434637
