| L(s) = 1 | − 2.79·2-s + 0.695·3-s + 5.80·4-s − 1.43·5-s − 1.94·6-s + 3.16·7-s − 10.6·8-s − 2.51·9-s + 4.00·10-s − 2.99·11-s + 4.03·12-s + 2.89·13-s − 8.84·14-s − 0.996·15-s + 18.0·16-s − 7.99·17-s + 7.03·18-s + 2.23·19-s − 8.32·20-s + 2.19·21-s + 8.37·22-s + 5.98·23-s − 7.38·24-s − 2.94·25-s − 8.08·26-s − 3.83·27-s + 18.3·28-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 0.401·3-s + 2.90·4-s − 0.641·5-s − 0.792·6-s + 1.19·7-s − 3.75·8-s − 0.838·9-s + 1.26·10-s − 0.904·11-s + 1.16·12-s + 0.802·13-s − 2.36·14-s − 0.257·15-s + 4.51·16-s − 1.94·17-s + 1.65·18-s + 0.512·19-s − 1.86·20-s + 0.480·21-s + 1.78·22-s + 1.24·23-s − 1.50·24-s − 0.588·25-s − 1.58·26-s − 0.737·27-s + 3.47·28-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
| \(L(1)\) |
\(\approx\) |
\(0.6296105802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6296105802\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.79T + 2T^{2} \) |
| 3 | \( 1 - 0.695T + 3T^{2} \) |
| 5 | \( 1 + 1.43T + 5T^{2} \) |
| 7 | \( 1 - 3.16T + 7T^{2} \) |
| 11 | \( 1 + 2.99T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 17 | \( 1 + 7.99T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 - 5.98T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 6.69T + 31T^{2} \) |
| 37 | \( 1 - 3.17T + 37T^{2} \) |
| 41 | \( 1 + 2.39T + 41T^{2} \) |
| 43 | \( 1 + 3.70T + 43T^{2} \) |
| 47 | \( 1 - 1.51T + 47T^{2} \) |
| 53 | \( 1 + 2.05T + 53T^{2} \) |
| 59 | \( 1 - 4.95T + 59T^{2} \) |
| 61 | \( 1 + 7.27T + 61T^{2} \) |
| 67 | \( 1 - 5.32T + 67T^{2} \) |
| 71 | \( 1 - 2.28T + 71T^{2} \) |
| 73 | \( 1 + 0.565T + 73T^{2} \) |
| 79 | \( 1 - 3.79T + 79T^{2} \) |
| 83 | \( 1 - 8.06T + 83T^{2} \) |
| 89 | \( 1 + 3.99T + 89T^{2} \) |
| 97 | \( 1 + 3.82T + 97T^{2} \) |
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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
−8.366383423348651456575834748817, −8.142092062052563951529689856410, −7.34900463318872455249761653018, −6.67696949267535567116718376610, −5.75119549934179533886427537810, −4.75960533700159952433078676204, −3.37356852545430103503218406280, −2.56000753386295084713864312271, −1.79029749797141339410180252178, −0.58535445452245110123327122763,
0.58535445452245110123327122763, 1.79029749797141339410180252178, 2.56000753386295084713864312271, 3.37356852545430103503218406280, 4.75960533700159952433078676204, 5.75119549934179533886427537810, 6.67696949267535567116718376610, 7.34900463318872455249761653018, 8.142092062052563951529689856410, 8.366383423348651456575834748817
