L(s) = 1 | + 1.41i·2-s + (−2.79 + 1.07i)3-s − 2.00·4-s + (−1.52 − 3.95i)6-s + 2.64·7-s − 2.82i·8-s + (6.67 − 6.03i)9-s + 5.98i·11-s + (5.59 − 2.15i)12-s − 19.0·13-s + 3.74i·14-s + 4.00·16-s − 2.02i·17-s + (8.53 + 9.44i)18-s + 21.1·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.933 + 0.359i)3-s − 0.500·4-s + (−0.253 − 0.659i)6-s + 0.377·7-s − 0.353i·8-s + (0.742 − 0.670i)9-s + 0.544i·11-s + (0.466 − 0.179i)12-s − 1.46·13-s + 0.267i·14-s + 0.250·16-s − 0.119i·17-s + (0.473 + 0.524i)18-s + 1.11·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.359 - 0.933i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.100492812\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.100492812\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (2.79 - 1.07i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
good | 11 | \( 1 - 5.98iT - 121T^{2} \) |
| 13 | \( 1 + 19.0T + 169T^{2} \) |
| 17 | \( 1 + 2.02iT - 289T^{2} \) |
| 19 | \( 1 - 21.1T + 361T^{2} \) |
| 23 | \( 1 + 32.2iT - 529T^{2} \) |
| 29 | \( 1 + 12.2iT - 841T^{2} \) |
| 31 | \( 1 - 39.3T + 961T^{2} \) |
| 37 | \( 1 - 20.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 34.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 72.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 57.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 71.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 81.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 81.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 7.70iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 12.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 131.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 115. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 8.19T + 9.40e3T^{2} \) |
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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
−9.814623597984799415922475361275, −9.403280092602595549441463803630, −8.090020932677390153112969046887, −7.35482548690947913345007382319, −6.58862654711400253208643951330, −5.68516592161616641615482837611, −4.74898003787715907094382213816, −4.38073864611477236582790952912, −2.68580064503752648988923567429, −0.888669198092309305712521829059,
0.53839281238522377461888785298, 1.71086758381668163282378999879, 2.92699108442357774403848825804, 4.24338782718221550825459139199, 5.19783633952035402839463067167, 5.74038763917713257510205142953, 7.09930147958858495789804914376, 7.63925707913396915781677413316, 8.735805696715042758418440642775, 9.841357668860059827094196175382