L(s) = 1 | + (0.580 − 0.814i)2-s + (−0.327 − 0.945i)4-s + (−0.235 + 0.971i)5-s + (−0.928 + 0.371i)7-s + (−0.959 − 0.281i)8-s + (0.654 + 0.755i)10-s + (0.327 + 0.945i)13-s + (−0.235 + 0.971i)14-s + (−0.786 + 0.618i)16-s + (0.841 − 0.540i)17-s + (−0.841 − 0.540i)19-s + (0.995 − 0.0950i)20-s + (−0.928 − 0.371i)23-s + (−0.888 − 0.458i)25-s + (0.959 + 0.281i)26-s + ⋯ |
L(s) = 1 | + (0.580 − 0.814i)2-s + (−0.327 − 0.945i)4-s + (−0.235 + 0.971i)5-s + (−0.928 + 0.371i)7-s + (−0.959 − 0.281i)8-s + (0.654 + 0.755i)10-s + (0.327 + 0.945i)13-s + (−0.235 + 0.971i)14-s + (−0.786 + 0.618i)16-s + (0.841 − 0.540i)17-s + (−0.841 − 0.540i)19-s + (0.995 − 0.0950i)20-s + (−0.928 − 0.371i)23-s + (−0.888 − 0.458i)25-s + (0.959 + 0.281i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1263934543 - 0.7549630120i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1263934543 - 0.7549630120i\) |
\(L(1)\) |
\(\approx\) |
\(0.8818000482 - 0.4062870054i\) |
\(L(1)\) |
\(\approx\) |
\(0.8818000482 - 0.4062870054i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.580 - 0.814i)T \) |
| 5 | \( 1 + (-0.235 + 0.971i)T \) |
| 7 | \( 1 + (-0.928 + 0.371i)T \) |
| 13 | \( 1 + (0.327 + 0.945i)T \) |
| 17 | \( 1 + (0.841 - 0.540i)T \) |
| 19 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (-0.928 - 0.371i)T \) |
| 29 | \( 1 + (0.0475 - 0.998i)T \) |
| 31 | \( 1 + (0.981 - 0.189i)T \) |
| 37 | \( 1 + (-0.654 - 0.755i)T \) |
| 41 | \( 1 + (-0.995 - 0.0950i)T \) |
| 43 | \( 1 + (-0.235 - 0.971i)T \) |
| 47 | \( 1 + (0.995 - 0.0950i)T \) |
| 53 | \( 1 + (0.142 - 0.989i)T \) |
| 59 | \( 1 + (-0.580 - 0.814i)T \) |
| 61 | \( 1 + (0.995 - 0.0950i)T \) |
| 67 | \( 1 + (-0.995 - 0.0950i)T \) |
| 71 | \( 1 + (-0.841 - 0.540i)T \) |
| 73 | \( 1 + (0.142 + 0.989i)T \) |
| 79 | \( 1 + (-0.723 - 0.690i)T \) |
| 83 | \( 1 + (0.928 - 0.371i)T \) |
| 89 | \( 1 + (-0.841 + 0.540i)T \) |
| 97 | \( 1 + (0.235 + 0.971i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.91916878200519441061298402473, −21.055724000197690606703870583266, −20.36329084578884483979321886392, −19.58556080396701193163444826481, −18.62948322865824279008471474766, −17.54650087454883507126117287254, −16.89406916320466441018460853689, −16.293127260474110635268871438654, −15.62799385624993163524913640360, −14.89821207764900102593942063894, −13.78741947572983147294602404453, −13.21787038396830123750273434781, −12.41898795043890727004559602951, −12.04669728804018593204794871056, −10.52689412345386568572898718740, −9.73047409414829787799697308084, −8.60680118721104248592619313427, −8.12553981791460048777311108375, −7.20825118791579855917291917393, −6.13038082992694790663167133000, −5.60992252790320122101371798100, −4.53382225351119358485089197377, −3.74333841544591459719109359320, −2.99413868394954257269703884855, −1.27267127801966313851158181894,
0.26503094263149747124417194648, 1.99753434910539153076552976194, 2.68456027920041254868624308176, 3.60477347030498151205735476701, 4.26844369205900594345111643046, 5.56054566045959780572906145403, 6.389594818607498903988078849404, 6.95217138682703889429150839038, 8.36978160134766191388337814690, 9.38616181197663472250625934702, 10.08950470995319592825567573165, 10.759521892354974483390502825461, 11.869466968957772611784676818797, 12.059682912217050750648087313236, 13.3273916668959693183765776011, 13.877935529203795530307983143754, 14.68130311460632627459325790636, 15.49782084384699908130136001740, 16.1240189631344952955353402963, 17.369531580797363434578466847188, 18.42681944308455246336255350607, 19.03123952412485433597155615526, 19.29055748402684707726385172593, 20.37585356138762161869595956499, 21.22442042830008816172108447118