L(s) = 1 | − 2-s + 2.74·3-s + 4-s + 3.70·5-s − 2.74·6-s − 4.88·7-s − 8-s + 4.54·9-s − 3.70·10-s + 3.59·11-s + 2.74·12-s − 2.37·13-s + 4.88·14-s + 10.1·15-s + 16-s + 2.63·17-s − 4.54·18-s − 3.85·19-s + 3.70·20-s − 13.4·21-s − 3.59·22-s + 4.08·23-s − 2.74·24-s + 8.71·25-s + 2.37·26-s + 4.25·27-s − 4.88·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.58·3-s + 0.5·4-s + 1.65·5-s − 1.12·6-s − 1.84·7-s − 0.353·8-s + 1.51·9-s − 1.17·10-s + 1.08·11-s + 0.793·12-s − 0.659·13-s + 1.30·14-s + 2.62·15-s + 0.250·16-s + 0.639·17-s − 1.07·18-s − 0.884·19-s + 0.828·20-s − 2.93·21-s − 0.765·22-s + 0.852·23-s − 0.560·24-s + 1.74·25-s + 0.466·26-s + 0.819·27-s − 0.923·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.050123353\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.050123353\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
good | 3 | \( 1 - 2.74T + 3T^{2} \) |
| 5 | \( 1 - 3.70T + 5T^{2} \) |
| 7 | \( 1 + 4.88T + 7T^{2} \) |
| 11 | \( 1 - 3.59T + 11T^{2} \) |
| 13 | \( 1 + 2.37T + 13T^{2} \) |
| 17 | \( 1 - 2.63T + 17T^{2} \) |
| 19 | \( 1 + 3.85T + 19T^{2} \) |
| 23 | \( 1 - 4.08T + 23T^{2} \) |
| 29 | \( 1 - 5.73T + 29T^{2} \) |
| 31 | \( 1 - 6.73T + 31T^{2} \) |
| 37 | \( 1 - 7.28T + 37T^{2} \) |
| 41 | \( 1 - 1.25T + 41T^{2} \) |
| 43 | \( 1 + 5.50T + 43T^{2} \) |
| 47 | \( 1 + 0.769T + 47T^{2} \) |
| 53 | \( 1 + 5.32T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 6.47T + 61T^{2} \) |
| 67 | \( 1 + 5.54T + 67T^{2} \) |
| 71 | \( 1 - 2.23T + 71T^{2} \) |
| 73 | \( 1 - 16.8T + 73T^{2} \) |
| 79 | \( 1 - 1.69T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 1.40T + 97T^{2} \) |
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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
−8.744409966192089775536840828240, −7.930985571054059842079231988347, −6.79125603568764663254324331814, −6.59155086843993766925309415730, −5.85117130440113018258984101220, −4.50195740573731333537601115165, −3.31548995602629627060612802002, −2.82560576968726288152823965649, −2.15156052172808205605687937948, −1.07535522464031925438302655332,
1.07535522464031925438302655332, 2.15156052172808205605687937948, 2.82560576968726288152823965649, 3.31548995602629627060612802002, 4.50195740573731333537601115165, 5.85117130440113018258984101220, 6.59155086843993766925309415730, 6.79125603568764663254324331814, 7.930985571054059842079231988347, 8.744409966192089775536840828240