L(s) = 1 | + i·3-s + i·7-s − 9-s + 11-s − i·13-s − i·17-s − 19-s − 21-s − i·23-s − i·27-s + 29-s − 31-s + i·33-s + i·37-s + 39-s + ⋯ |
L(s) = 1 | + i·3-s + i·7-s − 9-s + 11-s − i·13-s − i·17-s − 19-s − 21-s − i·23-s − i·27-s + 29-s − 31-s + i·33-s + i·37-s + 39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7015832744 + 0.3911582944i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7015832744 + 0.3911582944i\) |
\(L(1)\) |
\(\approx\) |
\(0.9119401687 + 0.3227714304i\) |
\(L(1)\) |
\(\approx\) |
\(0.9119401687 + 0.3227714304i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 + iT \) |
| 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 | \( 1 - T \) |
| 73 | \( 1 \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 \) |
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
−35.122125834453898743831884437606, −33.80457709813138321773176703112, −32.58753197454238590740533921319, −31.12003428298199346923354335974, −30.0971994351986827050315741741, −29.28666097719713521204783741303, −27.883428506007810131250784550945, −26.38337483426292802936698400847, −25.269445915875329627552442127233, −23.92639158745413475851254561378, −23.21696771785133477076154526384, −21.593589323327567001176671697400, −19.887703187581400930829557858825, −19.176543805830934711741344483751, −17.54685466546225082919988802852, −16.71672928645412687732821755996, −14.57334623724062779575792727986, −13.56783189492185872773033389516, −12.22307268743900450537621130756, −10.868897953192585114440795293532, −8.940371155949579377574244484917, −7.37110915929680674050723007875, −6.26373631835534623225079304783, −3.96812450725671685803413562738, −1.64202537801390229418067778151,
2.89056781780750596559976448631, 4.67014511488030982111174235052, 6.14211875190494735953280392413, 8.45883954006282991693725964976, 9.57509793875861435371820853105, 11.054318087978597375813628502568, 12.39862134062951689437667133845, 14.38530492672731610600971896733, 15.36983053429381221235339712706, 16.57602095753138741705364509985, 17.96880266645786107177546621635, 19.583334729135430289489453237354, 20.82583463722073568508810719242, 22.013127749235026651765169384278, 22.83112681279834309789155808254, 24.776076615038713622041184822249, 25.67646987378714662053549730758, 27.280467228065419895394290647645, 27.83902827305057233268188607501, 29.20694324423191248769592152510, 30.76022116148276051216075375023, 31.9983823454100829952983009412, 32.76546716492736566693430288692, 34.14985841778778744419684566672, 34.98937625757406440227690267113