| L(s) = 1 | − 0.870·2-s − 2.53·3-s − 1.24·4-s + 1.03·5-s + 2.20·6-s + 5.14·7-s + 2.82·8-s + 3.43·9-s − 0.900·10-s + 4.42·11-s + 3.15·12-s − 6.28·13-s − 4.48·14-s − 2.62·15-s + 0.0296·16-s + 1.27·17-s − 2.98·18-s − 1.62·19-s − 1.28·20-s − 13.0·21-s − 3.84·22-s + 6.30·23-s − 7.15·24-s − 3.92·25-s + 5.46·26-s − 1.09·27-s − 6.39·28-s + ⋯ |
| L(s) = 1 | − 0.615·2-s − 1.46·3-s − 0.621·4-s + 0.462·5-s + 0.901·6-s + 1.94·7-s + 0.997·8-s + 1.14·9-s − 0.284·10-s + 1.33·11-s + 0.909·12-s − 1.74·13-s − 1.19·14-s − 0.677·15-s + 0.00741·16-s + 0.309·17-s − 0.703·18-s − 0.373·19-s − 0.287·20-s − 2.84·21-s − 0.820·22-s + 1.31·23-s − 1.46·24-s − 0.785·25-s + 1.07·26-s − 0.210·27-s − 1.20·28-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
| \(L(1)\) |
\(\approx\) |
\(0.8006849911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8006849911\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.870T + 2T^{2} \) |
| 3 | \( 1 + 2.53T + 3T^{2} \) |
| 5 | \( 1 - 1.03T + 5T^{2} \) |
| 7 | \( 1 - 5.14T + 7T^{2} \) |
| 11 | \( 1 - 4.42T + 11T^{2} \) |
| 13 | \( 1 + 6.28T + 13T^{2} \) |
| 17 | \( 1 - 1.27T + 17T^{2} \) |
| 19 | \( 1 + 1.62T + 19T^{2} \) |
| 23 | \( 1 - 6.30T + 23T^{2} \) |
| 29 | \( 1 + 0.236T + 29T^{2} \) |
| 31 | \( 1 - 0.563T + 31T^{2} \) |
| 37 | \( 1 - 7.81T + 37T^{2} \) |
| 41 | \( 1 + 2.30T + 41T^{2} \) |
| 43 | \( 1 + 9.06T + 43T^{2} \) |
| 47 | \( 1 + 1.64T + 47T^{2} \) |
| 53 | \( 1 - 4.58T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 1.88T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 - 5.62T + 73T^{2} \) |
| 79 | \( 1 - 3.10T + 79T^{2} \) |
| 83 | \( 1 - 2.05T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−9.994058861785954114424736388364, −9.332427986175623256556359521918, −8.346670326390617025632767616795, −7.51532968918279574901036555252, −6.65173306454741360725912770177, −5.38096678347089700944644908995, −4.94747144774762142718624090643, −4.25198295570693165155377698307, −1.89177414741896318930004447820, −0.890038412149039410318342168695,
0.890038412149039410318342168695, 1.89177414741896318930004447820, 4.25198295570693165155377698307, 4.94747144774762142718624090643, 5.38096678347089700944644908995, 6.65173306454741360725912770177, 7.51532968918279574901036555252, 8.346670326390617025632767616795, 9.332427986175623256556359521918, 9.994058861785954114424736388364
