| 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
−7.88327540274462329163720754482, −7.29141625906304372419516102056, −6.83853335680882736108120863477, −5.69203359317660362302305299846, −5.40479078933068485901212790909, −4.19235275388135444557192342561, −3.47820096279686066187157957895, −2.50951536042092462000759569438, −1.52738298164492380055805293306, −0.45114119390290028447124511973,
1.13227821869206985691070567132, 2.42358184763633077055087820008, 3.13380876250333608641315971359, 4.12669772283713863114032247533, 4.76075589202371209914998399297, 5.58133535647118716731890918066, 6.23717585027297662071734766051, 7.07675014168525748225024065445, 7.87083354399563097921701828962, 8.619447454021176535240103836741
