L(s) = 1 | − 2-s + 1.75·3-s + 4-s − 1.84·5-s − 1.75·6-s − 3.49·7-s − 8-s + 0.0945·9-s + 1.84·10-s − 3.47·11-s + 1.75·12-s − 5.30·13-s + 3.49·14-s − 3.25·15-s + 16-s − 4.38·17-s − 0.0945·18-s − 1.62·19-s − 1.84·20-s − 6.14·21-s + 3.47·22-s + 7.38·23-s − 1.75·24-s − 1.58·25-s + 5.30·26-s − 5.11·27-s − 3.49·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.01·3-s + 0.5·4-s − 0.826·5-s − 0.718·6-s − 1.31·7-s − 0.353·8-s + 0.0315·9-s + 0.584·10-s − 1.04·11-s + 0.507·12-s − 1.47·13-s + 0.933·14-s − 0.839·15-s + 0.250·16-s − 1.06·17-s − 0.0222·18-s − 0.373·19-s − 0.413·20-s − 1.34·21-s + 0.740·22-s + 1.54·23-s − 0.359·24-s − 0.316·25-s + 1.04·26-s − 0.983·27-s − 0.659·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5435498254\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5435498254\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
good | 3 | \( 1 - 1.75T + 3T^{2} \) |
| 5 | \( 1 + 1.84T + 5T^{2} \) |
| 7 | \( 1 + 3.49T + 7T^{2} \) |
| 11 | \( 1 + 3.47T + 11T^{2} \) |
| 13 | \( 1 + 5.30T + 13T^{2} \) |
| 17 | \( 1 + 4.38T + 17T^{2} \) |
| 19 | \( 1 + 1.62T + 19T^{2} \) |
| 23 | \( 1 - 7.38T + 23T^{2} \) |
| 29 | \( 1 - 4.23T + 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 + 4.37T + 37T^{2} \) |
| 41 | \( 1 - 9.78T + 41T^{2} \) |
| 43 | \( 1 + 3.77T + 43T^{2} \) |
| 47 | \( 1 - 5.05T + 47T^{2} \) |
| 53 | \( 1 + 3.83T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 0.262T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 6.06T + 71T^{2} \) |
| 73 | \( 1 - 9.88T + 73T^{2} \) |
| 79 | \( 1 + 8.31T + 79T^{2} \) |
| 83 | \( 1 - 1.64T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 3.67T + 97T^{2} \) |
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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
−8.443267387207279273193920789872, −7.85388088878850565749604094626, −7.13560954563933923237046925814, −6.67113808157999714275407113142, −5.50638913528817037338681635657, −4.52722361980263015107776456020, −3.52794382804938542024858375841, −2.70675982263190892626641513354, −2.39689657945001954551947471539, −0.40676484911100966985049140259,
0.40676484911100966985049140259, 2.39689657945001954551947471539, 2.70675982263190892626641513354, 3.52794382804938542024858375841, 4.52722361980263015107776456020, 5.50638913528817037338681635657, 6.67113808157999714275407113142, 7.13560954563933923237046925814, 7.85388088878850565749604094626, 8.443267387207279273193920789872