| L(s) = 1 | − 2.77·2-s − 0.970·3-s + 5.72·4-s + 0.973·5-s + 2.69·6-s + 0.392·7-s − 10.3·8-s − 2.05·9-s − 2.70·10-s − 4.34·11-s − 5.55·12-s + 2.35·13-s − 1.09·14-s − 0.944·15-s + 17.3·16-s + 6.89·17-s + 5.72·18-s + 6.69·19-s + 5.57·20-s − 0.380·21-s + 12.0·22-s − 2.69·23-s + 10.0·24-s − 4.05·25-s − 6.54·26-s + 4.90·27-s + 2.24·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s − 0.560·3-s + 2.86·4-s + 0.435·5-s + 1.10·6-s + 0.148·7-s − 3.66·8-s − 0.686·9-s − 0.855·10-s − 1.31·11-s − 1.60·12-s + 0.653·13-s − 0.291·14-s − 0.243·15-s + 4.33·16-s + 1.67·17-s + 1.34·18-s + 1.53·19-s + 1.24·20-s − 0.0831·21-s + 2.57·22-s − 0.562·23-s + 2.05·24-s − 0.810·25-s − 1.28·26-s + 0.944·27-s + 0.424·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5864513727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5864513727\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4001 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 3 | \( 1 + 0.970T + 3T^{2} \) |
| 5 | \( 1 - 0.973T + 5T^{2} \) |
| 7 | \( 1 - 0.392T + 7T^{2} \) |
| 11 | \( 1 + 4.34T + 11T^{2} \) |
| 13 | \( 1 - 2.35T + 13T^{2} \) |
| 17 | \( 1 - 6.89T + 17T^{2} \) |
| 19 | \( 1 - 6.69T + 19T^{2} \) |
| 23 | \( 1 + 2.69T + 23T^{2} \) |
| 29 | \( 1 - 0.233T + 29T^{2} \) |
| 31 | \( 1 - 9.20T + 31T^{2} \) |
| 37 | \( 1 + 6.23T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 - 5.31T + 43T^{2} \) |
| 47 | \( 1 + 5.36T + 47T^{2} \) |
| 53 | \( 1 + 5.62T + 53T^{2} \) |
| 59 | \( 1 + 7.14T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 + 2.47T + 67T^{2} \) |
| 71 | \( 1 + 7.84T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 9.57T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 9.52T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.378830790007563084331380466124, −7.79729317935365697892393236305, −7.45542656715980417766322511427, −6.17304086164720656106587303627, −5.89701997861666301690386343726, −5.16478186231926115966986720912, −3.24770961926772566677677465416, −2.70104075061646875415535455990, −1.51751194863065072550666476652, −0.63145340172050522994436707051,
0.63145340172050522994436707051, 1.51751194863065072550666476652, 2.70104075061646875415535455990, 3.24770961926772566677677465416, 5.16478186231926115966986720912, 5.89701997861666301690386343726, 6.17304086164720656106587303627, 7.45542656715980417766322511427, 7.79729317935365697892393236305, 8.378830790007563084331380466124
