L(s) = 1 | + (0.707 + 0.707i)2-s + (0.267 − 0.267i)3-s + 1.00i·4-s + 0.378·6-s + (0.709 − 0.709i)7-s + (−0.707 + 0.707i)8-s + 2.85i·9-s − 5.59·11-s + (0.267 + 0.267i)12-s + (−2.99 + 2.99i)13-s + 1.00·14-s − 1.00·16-s + (−2.20 + 2.20i)17-s + (−2.02 + 2.02i)18-s + (−3.84 + 2.05i)19-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.154 − 0.154i)3-s + 0.500i·4-s + 0.154·6-s + (0.268 − 0.268i)7-s + (−0.250 + 0.250i)8-s + 0.952i·9-s − 1.68·11-s + (0.0772 + 0.0772i)12-s + (−0.830 + 0.830i)13-s + 0.268·14-s − 0.250·16-s + (−0.534 + 0.534i)17-s + (−0.476 + 0.476i)18-s + (−0.881 + 0.472i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.528827 + 1.35076i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.528827 + 1.35076i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (3.84 - 2.05i)T \) |
good | 3 | \( 1 + (-0.267 + 0.267i)T - 3iT^{2} \) |
| 7 | \( 1 + (-0.709 + 0.709i)T - 7iT^{2} \) |
| 11 | \( 1 + 5.59T + 11T^{2} \) |
| 13 | \( 1 + (2.99 - 2.99i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.20 - 2.20i)T - 17iT^{2} \) |
| 23 | \( 1 + (-5.89 - 5.89i)T + 23iT^{2} \) |
| 29 | \( 1 - 9.34T + 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + (-6.61 - 6.61i)T + 37iT^{2} \) |
| 41 | \( 1 + 9.14iT - 41T^{2} \) |
| 43 | \( 1 + (0.927 + 0.927i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.20 - 4.20i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.44 + 2.44i)T - 53iT^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + (5.72 + 5.72i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.9iT - 71T^{2} \) |
| 73 | \( 1 + (-4.33 - 4.33i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.47T + 79T^{2} \) |
| 83 | \( 1 + (-5.53 - 5.53i)T + 83iT^{2} \) |
| 89 | \( 1 + 8.99T + 89T^{2} \) |
| 97 | \( 1 + (-9.16 - 9.16i)T + 97iT^{2} \) |
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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
−10.56181609858061836236193946274, −9.466884676248519085995163463361, −8.318612273351737829376060546130, −7.81217566841626540614998370776, −7.07800020248918786795255352269, −6.03517548620426252684821704793, −4.92015012104630578739644450996, −4.54273505248011488930198503719, −2.97344579429785937412110166245, −2.02820704147790226496791272705,
0.52657832821610413113101222214, 2.58467724890846670974494930534, 2.93560321761433570583598016382, 4.58880747120507637416157162208, 4.99730728293386924106927902592, 6.17038183435000233725105701150, 7.07873076407270816277187435598, 8.207388756192276786584606156339, 8.921245601121044257713704052652, 9.965856537234457434090426656535