| L(s) = 1 | − 2.40·2-s − 1.91·3-s + 3.76·4-s − 2.41·5-s + 4.60·6-s − 1.95·7-s − 4.24·8-s + 0.684·9-s + 5.80·10-s − 0.661·11-s − 7.23·12-s + 3.28·13-s + 4.69·14-s + 4.63·15-s + 2.66·16-s − 6.82·17-s − 1.64·18-s + 3.26·19-s − 9.10·20-s + 3.74·21-s + 1.58·22-s − 6.36·23-s + 8.14·24-s + 0.836·25-s − 7.87·26-s + 4.44·27-s − 7.35·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s − 1.10·3-s + 1.88·4-s − 1.08·5-s + 1.88·6-s − 0.738·7-s − 1.50·8-s + 0.228·9-s + 1.83·10-s − 0.199·11-s − 2.08·12-s + 0.909·13-s + 1.25·14-s + 1.19·15-s + 0.665·16-s − 1.65·17-s − 0.387·18-s + 0.749·19-s − 2.03·20-s + 0.818·21-s + 0.338·22-s − 1.32·23-s + 1.66·24-s + 0.167·25-s − 1.54·26-s + 0.855·27-s − 1.39·28-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
| \(L(1)\) |
\(\approx\) |
\(0.02167762121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02167762121\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.40T + 2T^{2} \) |
| 3 | \( 1 + 1.91T + 3T^{2} \) |
| 5 | \( 1 + 2.41T + 5T^{2} \) |
| 7 | \( 1 + 1.95T + 7T^{2} \) |
| 11 | \( 1 + 0.661T + 11T^{2} \) |
| 13 | \( 1 - 3.28T + 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 19 | \( 1 - 3.26T + 19T^{2} \) |
| 23 | \( 1 + 6.36T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 + 3.18T + 31T^{2} \) |
| 37 | \( 1 - 5.91T + 37T^{2} \) |
| 41 | \( 1 - 0.877T + 41T^{2} \) |
| 43 | \( 1 - 4.37T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 6.93T + 53T^{2} \) |
| 59 | \( 1 + 7.14T + 59T^{2} \) |
| 61 | \( 1 - 2.14T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 7.19T + 71T^{2} \) |
| 73 | \( 1 + 9.65T + 73T^{2} \) |
| 79 | \( 1 + 0.353T + 79T^{2} \) |
| 83 | \( 1 - 0.410T + 83T^{2} \) |
| 89 | \( 1 - 1.94T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−8.378919540857101763320559956577, −7.88470502825955148664783176750, −7.11091019379693684521963196383, −6.37409685775846312107290616587, −5.96927901935104876037316008635, −4.69321758160890798030035128613, −3.79275514318568368338771059967, −2.70401507248788386120253968697, −1.42338319046344788177454802161, −0.11966607665734543396567789393,
0.11966607665734543396567789393, 1.42338319046344788177454802161, 2.70401507248788386120253968697, 3.79275514318568368338771059967, 4.69321758160890798030035128613, 5.96927901935104876037316008635, 6.37409685775846312107290616587, 7.11091019379693684521963196383, 7.88470502825955148664783176750, 8.378919540857101763320559956577
