| L(s) = 1 | − 1.82·2-s + 2.68·3-s + 1.32·4-s − 4.13·5-s − 4.89·6-s + 5.17·7-s + 1.23·8-s + 4.21·9-s + 7.52·10-s − 3.00·11-s + 3.54·12-s + 0.487·13-s − 9.43·14-s − 11.0·15-s − 4.89·16-s + 1.27·17-s − 7.68·18-s + 5.37·19-s − 5.45·20-s + 13.9·21-s + 5.47·22-s − 0.0884·23-s + 3.32·24-s + 12.0·25-s − 0.887·26-s + 3.26·27-s + 6.83·28-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 1.55·3-s + 0.660·4-s − 1.84·5-s − 1.99·6-s + 1.95·7-s + 0.437·8-s + 1.40·9-s + 2.38·10-s − 0.906·11-s + 1.02·12-s + 0.135·13-s − 2.52·14-s − 2.86·15-s − 1.22·16-s + 0.309·17-s − 1.81·18-s + 1.23·19-s − 1.22·20-s + 3.03·21-s + 1.16·22-s − 0.0184·23-s + 0.678·24-s + 2.41·25-s − 0.174·26-s + 0.628·27-s + 1.29·28-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
| \(L(1)\) |
\(\approx\) |
\(1.496325075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496325075\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.82T + 2T^{2} \) |
| 3 | \( 1 - 2.68T + 3T^{2} \) |
| 5 | \( 1 + 4.13T + 5T^{2} \) |
| 7 | \( 1 - 5.17T + 7T^{2} \) |
| 11 | \( 1 + 3.00T + 11T^{2} \) |
| 13 | \( 1 - 0.487T + 13T^{2} \) |
| 17 | \( 1 - 1.27T + 17T^{2} \) |
| 19 | \( 1 - 5.37T + 19T^{2} \) |
| 23 | \( 1 + 0.0884T + 23T^{2} \) |
| 29 | \( 1 - 2.32T + 29T^{2} \) |
| 31 | \( 1 - 7.57T + 31T^{2} \) |
| 37 | \( 1 + 4.84T + 37T^{2} \) |
| 41 | \( 1 + 1.44T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 1.02T + 47T^{2} \) |
| 53 | \( 1 - 2.55T + 53T^{2} \) |
| 59 | \( 1 + 0.415T + 59T^{2} \) |
| 61 | \( 1 + 4.77T + 61T^{2} \) |
| 67 | \( 1 + 3.55T + 67T^{2} \) |
| 71 | \( 1 + 5.50T + 71T^{2} \) |
| 73 | \( 1 - 4.20T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 7.34T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−8.401280728236478298669228120001, −7.938019106887271706660682463154, −7.60772037153071719913547346466, −7.04876103806521891481222542999, −5.03572154527174686783160112097, −4.61306452712965762009848664808, −3.68450858139528394682990171383, −2.84550774838359319946702894730, −1.77745110315459571451055771230, −0.835525607042836882384023763815,
0.835525607042836882384023763815, 1.77745110315459571451055771230, 2.84550774838359319946702894730, 3.68450858139528394682990171383, 4.61306452712965762009848664808, 5.03572154527174686783160112097, 7.04876103806521891481222542999, 7.60772037153071719913547346466, 7.938019106887271706660682463154, 8.401280728236478298669228120001
