L(s) = 1 | − 2.80·2-s − 3.24·3-s + 5.84·4-s − 0.362·5-s + 9.08·6-s + 0.683·7-s − 10.7·8-s + 7.51·9-s + 1.01·10-s − 2.13·11-s − 18.9·12-s − 1.68·13-s − 1.91·14-s + 1.17·15-s + 18.4·16-s − 5.09·17-s − 21.0·18-s − 0.801·19-s − 2.11·20-s − 2.21·21-s + 5.97·22-s − 8.57·23-s + 34.9·24-s − 4.86·25-s + 4.72·26-s − 14.6·27-s + 3.99·28-s + ⋯ |
L(s) = 1 | − 1.98·2-s − 1.87·3-s + 2.92·4-s − 0.161·5-s + 3.70·6-s + 0.258·7-s − 3.80·8-s + 2.50·9-s + 0.320·10-s − 0.642·11-s − 5.47·12-s − 0.467·13-s − 0.511·14-s + 0.303·15-s + 4.62·16-s − 1.23·17-s − 4.96·18-s − 0.183·19-s − 0.473·20-s − 0.483·21-s + 1.27·22-s − 1.78·23-s + 7.13·24-s − 0.973·25-s + 0.925·26-s − 2.81·27-s + 0.754·28-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
\(L(1)\) |
\(\approx\) |
\(0.0003369582061\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0003369582061\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.80T + 2T^{2} \) |
| 3 | \( 1 + 3.24T + 3T^{2} \) |
| 5 | \( 1 + 0.362T + 5T^{2} \) |
| 7 | \( 1 - 0.683T + 7T^{2} \) |
| 11 | \( 1 + 2.13T + 11T^{2} \) |
| 13 | \( 1 + 1.68T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 + 0.801T + 19T^{2} \) |
| 23 | \( 1 + 8.57T + 23T^{2} \) |
| 29 | \( 1 + 3.50T + 29T^{2} \) |
| 31 | \( 1 - 1.80T + 31T^{2} \) |
| 37 | \( 1 + 5.00T + 37T^{2} \) |
| 41 | \( 1 + 9.31T + 41T^{2} \) |
| 43 | \( 1 + 7.60T + 43T^{2} \) |
| 47 | \( 1 - 1.24T + 47T^{2} \) |
| 53 | \( 1 - 7.78T + 53T^{2} \) |
| 59 | \( 1 + 1.98T + 59T^{2} \) |
| 61 | \( 1 + 3.99T + 61T^{2} \) |
| 67 | \( 1 - 5.71T + 67T^{2} \) |
| 71 | \( 1 - 8.29T + 71T^{2} \) |
| 73 | \( 1 + 1.66T + 73T^{2} \) |
| 79 | \( 1 - 1.42T + 79T^{2} \) |
| 83 | \( 1 + 1.86T + 83T^{2} \) |
| 89 | \( 1 + 6.35T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.304098637508850692431864306131, −7.81450108577376570839460464391, −6.95715352519918415668640991397, −6.54022205900463081228475607119, −5.78703455372106833400207771551, −5.07091438320978455950713194720, −3.84975293036613543694670327293, −2.22962248604289780971814110162, −1.56846978684756033099330802537, −0.01540561967072693204999701714,
0.01540561967072693204999701714, 1.56846978684756033099330802537, 2.22962248604289780971814110162, 3.84975293036613543694670327293, 5.07091438320978455950713194720, 5.78703455372106833400207771551, 6.54022205900463081228475607119, 6.95715352519918415668640991397, 7.81450108577376570839460464391, 8.304098637508850692431864306131