L(s) = 1 | + i·3-s + i·7-s − 9-s + i·13-s + 3·19-s − 21-s − 4i·23-s − i·27-s + 4·29-s − 7·31-s + 6i·37-s − 39-s + 6·41-s + 9i·43-s + 6i·47-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.377i·7-s − 0.333·9-s + 0.277i·13-s + 0.688·19-s − 0.218·21-s − 0.834i·23-s − 0.192i·27-s + 0.742·29-s − 1.25·31-s + 0.986i·37-s − 0.160·39-s + 0.937·41-s + 1.37i·43-s + 0.875i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.511884118\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.511884118\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 9iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 18iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 3iT - 97T^{2} \) |
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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
−8.619447454021176535240103836741, −7.87083354399563097921701828962, −7.07675014168525748225024065445, −6.23717585027297662071734766051, −5.58133535647118716731890918066, −4.76075589202371209914998399297, −4.12669772283713863114032247533, −3.13380876250333608641315971359, −2.42358184763633077055087820008, −1.13227821869206985691070567132,
0.45114119390290028447124511973, 1.52738298164492380055805293306, 2.50951536042092462000759569438, 3.47820096279686066187157957895, 4.19235275388135444557192342561, 5.40479078933068485901212790909, 5.69203359317660362302305299846, 6.83853335680882736108120863477, 7.29141625906304372419516102056, 7.88327540274462329163720754482