| L(s) = 1 | + (−1 + 1.41i)3-s + (−1.73 + 1.41i)5-s − 3.46·7-s + (−1.00 − 2.82i)9-s − 4·13-s + (−0.267 − 3.86i)15-s − 6.92·17-s + 6.92·19-s + (3.46 − 4.89i)21-s − 4.89i·23-s + (0.999 − 4.89i)25-s + (5.00 + 1.41i)27-s − 3.46·29-s + 8.48i·31-s + (5.99 − 4.89i)35-s + ⋯ |
| L(s) = 1 | + (−0.577 + 0.816i)3-s + (−0.774 + 0.632i)5-s − 1.30·7-s + (−0.333 − 0.942i)9-s − 1.10·13-s + (−0.0691 − 0.997i)15-s − 1.68·17-s + 1.58·19-s + (0.755 − 1.06i)21-s − 1.02i·23-s + (0.199 − 0.979i)25-s + (0.962 + 0.272i)27-s − 0.643·29-s + 1.52i·31-s + (1.01 − 0.828i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4623888836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4623888836\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 1.41i)T \) |
| 5 | \( 1 + (1.73 - 1.41i)T \) |
| good | 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 + 4.89iT - 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 - 4.89iT - 47T^{2} \) |
| 53 | \( 1 + 2.82iT - 53T^{2} \) |
| 59 | \( 1 + 9.79iT - 59T^{2} \) |
| 61 | \( 1 - 9.79iT - 61T^{2} \) |
| 67 | \( 1 - 8.48iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 8.48iT - 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 11.3iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.434190841566157439519325984883, −8.623246074264046599433362468052, −7.38908626818303302074214273170, −6.79058803672488279626147701446, −6.16103682246704684348476870080, −5.01952055162942933530679123113, −4.29370681968692912922022378637, −3.32916990494010074871021674523, −2.69411563677460750204983311165, −0.32304449282992140779406087289,
0.61250278737152370601670085154, 2.14959076284134416362613747544, 3.25749098564292109044279945196, 4.34748853158236285323119408948, 5.26764827002638977775951727462, 6.02125429198556818268826223766, 7.06861910432741403161128846380, 7.36458979349912863715686652105, 8.261229133847702731498844936745, 9.367003494083508658663698464032
