L(s) = 1 | + (1.23 + 1.23i)3-s + (−0.571 + 0.571i)5-s − i·7-s + 0.0594i·9-s + (−4.55 + 4.55i)11-s + (−3.65 − 3.65i)13-s − 1.41·15-s + 2.47·17-s + (−5.54 − 5.54i)19-s + (1.23 − 1.23i)21-s − 4.08i·23-s + 4.34i·25-s + (3.63 − 3.63i)27-s + (−4.30 − 4.30i)29-s − 1.02·31-s + ⋯ |
L(s) = 1 | + (0.714 + 0.714i)3-s + (−0.255 + 0.255i)5-s − 0.377i·7-s + 0.0198i·9-s + (−1.37 + 1.37i)11-s + (−1.01 − 1.01i)13-s − 0.365·15-s + 0.599·17-s + (−1.27 − 1.27i)19-s + (0.269 − 0.269i)21-s − 0.851i·23-s + 0.869i·25-s + (0.699 − 0.699i)27-s + (−0.798 − 0.798i)29-s − 0.184·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5691757503\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5691757503\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (-1.23 - 1.23i)T + 3iT^{2} \) |
| 5 | \( 1 + (0.571 - 0.571i)T - 5iT^{2} \) |
| 11 | \( 1 + (4.55 - 4.55i)T - 11iT^{2} \) |
| 13 | \( 1 + (3.65 + 3.65i)T + 13iT^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 + (5.54 + 5.54i)T + 19iT^{2} \) |
| 23 | \( 1 + 4.08iT - 23T^{2} \) |
| 29 | \( 1 + (4.30 + 4.30i)T + 29iT^{2} \) |
| 31 | \( 1 + 1.02T + 31T^{2} \) |
| 37 | \( 1 + (-5.91 + 5.91i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.47iT - 41T^{2} \) |
| 43 | \( 1 + (4.35 - 4.35i)T - 43iT^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 + (1.94 - 1.94i)T - 53iT^{2} \) |
| 59 | \( 1 + (6.90 - 6.90i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.84 + 7.84i)T + 61iT^{2} \) |
| 67 | \( 1 + (-3.88 - 3.88i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.44iT - 71T^{2} \) |
| 73 | \( 1 + 1.06iT - 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + (-5.68 - 5.68i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.81iT - 89T^{2} \) |
| 97 | \( 1 + 1.74T + 97T^{2} \) |
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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
−9.232249710952348984505929049651, −8.141989657338919742212405653392, −7.56556078707759016404169067413, −6.93183753638504044637491161279, −5.60937085211253993312239154843, −4.70124485106611759415609498288, −4.11187230494352599696268447910, −2.89057978884609739255938481689, −2.36786759138558752387130254869, −0.17535958429806206975419124703,
1.64086060012169212690409968926, 2.55186737171242122452418280282, 3.40380017246812332173869936671, 4.64256219533322383989569081135, 5.53377434223212171457559543008, 6.35808094583347576373855360131, 7.46191452425557494218732594291, 8.021890957215383936857492052231, 8.469695215820944214304775151047, 9.331120683731496914215625357225