| L(s) = 1 | − 1.13·2-s − 1.73·3-s − 0.710·4-s − 4.00·5-s + 1.97·6-s + 3.34·7-s + 3.07·8-s + 0.0197·9-s + 4.54·10-s − 4.42·11-s + 1.23·12-s − 5.98·13-s − 3.79·14-s + 6.96·15-s − 2.07·16-s − 5.87·17-s − 0.0223·18-s − 2.64·19-s + 2.84·20-s − 5.81·21-s + 5.02·22-s − 7.22·23-s − 5.34·24-s + 11.0·25-s + 6.79·26-s + 5.17·27-s − 2.37·28-s + ⋯ |
| L(s) = 1 | − 0.802·2-s − 1.00·3-s − 0.355·4-s − 1.79·5-s + 0.805·6-s + 1.26·7-s + 1.08·8-s + 0.00657·9-s + 1.43·10-s − 1.33·11-s + 0.356·12-s − 1.65·13-s − 1.01·14-s + 1.79·15-s − 0.518·16-s − 1.42·17-s − 0.00527·18-s − 0.606·19-s + 0.636·20-s − 1.26·21-s + 1.07·22-s − 1.50·23-s − 1.09·24-s + 2.21·25-s + 1.33·26-s + 0.996·27-s − 0.449·28-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
| \(L(1)\) |
\(\approx\) |
\(0.03949157254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03949157254\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.13T + 2T^{2} \) |
| 3 | \( 1 + 1.73T + 3T^{2} \) |
| 5 | \( 1 + 4.00T + 5T^{2} \) |
| 7 | \( 1 - 3.34T + 7T^{2} \) |
| 11 | \( 1 + 4.42T + 11T^{2} \) |
| 13 | \( 1 + 5.98T + 13T^{2} \) |
| 17 | \( 1 + 5.87T + 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 + 7.22T + 23T^{2} \) |
| 29 | \( 1 + 6.03T + 29T^{2} \) |
| 31 | \( 1 - 0.185T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 - 2.46T + 41T^{2} \) |
| 43 | \( 1 - 6.27T + 43T^{2} \) |
| 47 | \( 1 - 2.90T + 47T^{2} \) |
| 53 | \( 1 - 0.991T + 53T^{2} \) |
| 59 | \( 1 - 8.07T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 8.15T + 67T^{2} \) |
| 71 | \( 1 - 1.43T + 71T^{2} \) |
| 73 | \( 1 + 2.40T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−10.28696065481198250064038053356, −8.935845930632590296226758139660, −8.178948916585263355037268769383, −7.71033978224288019158871719427, −7.04041493187979417724112861742, −5.36175588868401973679518200051, −4.73948817039778854725330220043, −4.13282936974694303708706593135, −2.23360522165523856691283566306, −0.17336759191997231811013801533,
0.17336759191997231811013801533, 2.23360522165523856691283566306, 4.13282936974694303708706593135, 4.73948817039778854725330220043, 5.36175588868401973679518200051, 7.04041493187979417724112861742, 7.71033978224288019158871719427, 8.178948916585263355037268769383, 8.935845930632590296226758139660, 10.28696065481198250064038053356
