L(s) = 1 | + i·7-s − 4.24·11-s + 2·13-s − 2.82i·17-s + 4i·19-s − 1.41·23-s + 5·25-s − 7.07i·29-s − 8i·31-s − 8·37-s + 5.65i·41-s + 4i·43-s + 2.82·47-s − 49-s − 7.07i·53-s + ⋯ |
L(s) = 1 | + 0.377i·7-s − 1.27·11-s + 0.554·13-s − 0.685i·17-s + 0.917i·19-s − 0.294·23-s + 25-s − 1.31i·29-s − 1.43i·31-s − 1.31·37-s + 0.883i·41-s + 0.609i·43-s + 0.412·47-s − 0.142·49-s − 0.971i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.418415064\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.418415064\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 7.07iT - 29T^{2} \) |
| 31 | \( 1 + 8iT - 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 7.07iT - 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + 11.3iT - 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−8.135409885858276448790410107266, −7.85225452523527299161281492288, −6.84494350768207595297882132723, −6.03159415370751041319645720500, −5.39946766422453506327710266592, −4.63989637901656283314016243023, −3.65365226636829543038216837041, −2.75650701594980597046193598101, −1.94542266879150701101708026921, −0.48114363465582557680818607695,
0.956525192328075346381689201982, 2.15795225530334895685131655374, 3.14456048093976067517817961885, 3.87116340803381978732974183137, 5.05722061423159098304405762604, 5.30844715111272461731754665859, 6.53250937703357502980314475950, 7.02924363304841303261418784468, 7.83139131680750401553014403654, 8.648971805428006050617047375060