L(s) = 1 | + i·3-s + 2·4-s + i·7-s − 9-s − 11-s + 2i·12-s − i·13-s + 4·16-s + 6i·17-s + 7·19-s − 21-s + 6i·23-s − i·27-s + 2i·28-s + 6·29-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 4-s + 0.377i·7-s − 0.333·9-s − 0.301·11-s + 0.577i·12-s − 0.277i·13-s + 16-s + 1.45i·17-s + 1.60·19-s − 0.218·21-s + 1.25i·23-s − 0.192i·27-s + 0.377i·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.64717 + 1.01800i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.64717 + 1.01800i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 2T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 5iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 17iT - 97T^{2} \) |
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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
−10.36467142458194665773856575474, −9.743266103787576039722236131918, −8.654585951585341645052169791964, −7.80395394648162199615220509504, −6.96603780975736194533674849360, −5.80294756829669589198140087384, −5.32250589294213183809342614364, −3.77362824829262836002606185941, −2.95609141318275865831659086845, −1.63213325808392530808587876579,
1.01773788594283783118718283700, 2.42296671286617807117088775567, 3.27189933961853533642033831821, 4.83828686299959817477009400813, 5.80813584391321442846332812547, 6.90391979850415290068963605508, 7.26126166237811198057229744141, 8.151687095490208766420391886401, 9.256996825265998262809995670799, 10.21764228163477797669494668790