L(s) = 1 | + i·3-s + i·7-s − 9-s − 11-s − i·13-s − i·17-s − 21-s − i·23-s − i·27-s + 29-s + 31-s − i·33-s + i·37-s + 39-s − 41-s + ⋯ |
L(s) = 1 | + i·3-s + i·7-s − 9-s − 11-s − i·13-s − i·17-s − 21-s − i·23-s − i·27-s + 29-s + 31-s − i·33-s + i·37-s + 39-s − 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.076746606 - 0.3058811465i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.076746606 - 0.3058811465i\) |
\(L(1)\) |
\(\approx\) |
\(0.8872725076 + 0.2094566263i\) |
\(L(1)\) |
\(\approx\) |
\(0.8872725076 + 0.2094566263i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 - iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 \) |
| 73 | \( 1 \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.21968702520285144942333376564, −23.44709429752069951647165780840, −23.2044702759179545045264876435, −21.714154010136134419374241137587, −20.88437654138870778751128090573, −19.7805504620641128595481171565, −19.25602876097835217788755393193, −18.24807271130062615206343280722, −17.41145798182367960950997378147, −16.69021328664430217859330246211, −15.55657888041836122892223377679, −14.302741733374395646467385780095, −13.61510873257668584605715418645, −12.90196065515974056743782956978, −11.86204737566106742688883983955, −10.92135182918894488977690084412, −9.97062951619419209722344805171, −8.59271611841181568235891590522, −7.74303367325734781462442089019, −6.92474696228286967913612152019, −5.994318778429037793335631643629, −4.70435748750243661352031037816, −3.43490222525270259821992509657, −2.12738247460264372583628007959, −1.03403719670332481660453019984,
0.35166839167439624994601327648, 2.53285296626793703180326361915, 3.13454769864309824474584108248, 4.756566635747933732155791353904, 5.29552324582213053609393702746, 6.402938466778331048840646255223, 8.02774222645886969018697409910, 8.67245382594252661365655406390, 9.83536792009014357753668110662, 10.47714402155996228231880545157, 11.570823747884138182475543713154, 12.45141610398539712745484804013, 13.615849761025835443544812881281, 14.683927967484963425192247855010, 15.62730782377802465031168511064, 15.8880578531410780927445147250, 17.16169983411012461202488474772, 18.10364311499436029008372343090, 18.90178388674697182804330827415, 20.20670916148701101144377857153, 20.78541096188507422055211926428, 21.62927781380062761239658485867, 22.47390616470362002979317647077, 23.08538148998178102316777965940, 24.37240463656695504097299819068