L(s) = 1 | + (−0.173 + 0.984i)2-s + (−0.639 − 0.536i)3-s + (−0.939 − 0.342i)4-s + (0.639 − 0.536i)6-s + (2.09 + 3.62i)7-s + (0.5 − 0.866i)8-s + (−0.399 − 2.26i)9-s + (−2.69 + 4.66i)11-s + (0.417 + 0.723i)12-s + (3.01 − 2.52i)13-s + (−3.93 + 1.43i)14-s + (0.766 + 0.642i)16-s + (0.371 − 2.10i)17-s + 2.30·18-s + (−2.92 + 3.22i)19-s + ⋯ |
L(s) = 1 | + (−0.122 + 0.696i)2-s + (−0.369 − 0.309i)3-s + (−0.469 − 0.171i)4-s + (0.261 − 0.219i)6-s + (0.790 + 1.36i)7-s + (0.176 − 0.306i)8-s + (−0.133 − 0.755i)9-s + (−0.812 + 1.40i)11-s + (0.120 + 0.208i)12-s + (0.835 − 0.700i)13-s + (−1.05 + 0.382i)14-s + (0.191 + 0.160i)16-s + (0.0901 − 0.511i)17-s + 0.542·18-s + (−0.671 + 0.740i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.466069 + 0.932931i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.466069 + 0.932931i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (2.92 - 3.22i)T \) |
good | 3 | \( 1 + (0.639 + 0.536i)T + (0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (-2.09 - 3.62i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.69 - 4.66i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.01 + 2.52i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.371 + 2.10i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.68 - 0.976i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.34 - 7.65i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.86 + 4.96i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.84T + 37T^{2} \) |
| 41 | \( 1 + (-4.67 - 3.92i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (11.8 - 4.31i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.760 - 4.31i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-12.2 - 4.44i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (1.36 - 7.75i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (10.3 + 3.74i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.82 - 10.3i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (12.6 - 4.61i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-2.58 - 2.16i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (3.96 + 3.32i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-4.36 - 7.56i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.52 - 2.12i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.150 + 0.851i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.22866663828520709318622453117, −9.278285987725931499551845161704, −8.589946037986013113614994717274, −7.79765708150402808380134562414, −6.96390900498201539347363421769, −5.89772815692284235797628310648, −5.42259201668802981027839411499, −4.44996650575398522266934206099, −2.89792511129944636775804318195, −1.51012635886404558961198129801,
0.55895082021510627576996112749, 1.98586369695712219445047399313, 3.42880869742335138219725758917, 4.34292809026233750700374698151, 5.10431925654265962508265981432, 6.19295616608496446490688138214, 7.40156322515710429291427075737, 8.287031092359184841953212180002, 8.784302572033193760289475478001, 10.23290524957699147668999561875