L(s) = 1 | + (−0.179 + 0.983i)2-s + (−0.716 − 0.697i)3-s + (−0.935 − 0.352i)4-s + (−0.0771 − 0.997i)5-s + (0.815 − 0.579i)6-s + (0.423 − 0.905i)7-s + (0.514 − 0.857i)8-s + (0.0257 + 0.999i)9-s + (0.994 + 0.102i)10-s + (0.978 − 0.204i)11-s + (0.423 + 0.905i)12-s + (0.0257 + 0.999i)13-s + (0.815 + 0.579i)14-s + (−0.640 + 0.767i)15-s + (0.751 + 0.660i)16-s + (−0.894 − 0.447i)17-s + ⋯ |
L(s) = 1 | + (−0.179 + 0.983i)2-s + (−0.716 − 0.697i)3-s + (−0.935 − 0.352i)4-s + (−0.0771 − 0.997i)5-s + (0.815 − 0.579i)6-s + (0.423 − 0.905i)7-s + (0.514 − 0.857i)8-s + (0.0257 + 0.999i)9-s + (0.994 + 0.102i)10-s + (0.978 − 0.204i)11-s + (0.423 + 0.905i)12-s + (0.0257 + 0.999i)13-s + (0.815 + 0.579i)14-s + (−0.640 + 0.767i)15-s + (0.751 + 0.660i)16-s + (−0.894 − 0.447i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6161322409 - 0.4804284005i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6161322409 - 0.4804284005i\) |
\(L(1)\) |
\(\approx\) |
\(0.7415376785 - 0.1108941878i\) |
\(L(1)\) |
\(\approx\) |
\(0.7415376785 - 0.1108941878i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 367 | \( 1 \) |
good | 2 | \( 1 + (-0.179 + 0.983i)T \) |
| 3 | \( 1 + (-0.716 - 0.697i)T \) |
| 5 | \( 1 + (-0.0771 - 0.997i)T \) |
| 7 | \( 1 + (0.423 - 0.905i)T \) |
| 11 | \( 1 + (0.978 - 0.204i)T \) |
| 13 | \( 1 + (0.0257 + 0.999i)T \) |
| 17 | \( 1 + (-0.894 - 0.447i)T \) |
| 19 | \( 1 + (0.952 - 0.304i)T \) |
| 23 | \( 1 + (0.751 - 0.660i)T \) |
| 29 | \( 1 + (0.128 - 0.991i)T \) |
| 31 | \( 1 + (-0.935 - 0.352i)T \) |
| 37 | \( 1 + (-0.784 - 0.620i)T \) |
| 41 | \( 1 + (0.978 + 0.204i)T \) |
| 43 | \( 1 + (0.952 + 0.304i)T \) |
| 47 | \( 1 + (-0.843 + 0.536i)T \) |
| 53 | \( 1 + (-0.0771 - 0.997i)T \) |
| 59 | \( 1 + (-0.558 + 0.829i)T \) |
| 61 | \( 1 + (0.815 + 0.579i)T \) |
| 67 | \( 1 + (-0.558 - 0.829i)T \) |
| 71 | \( 1 + (-0.935 + 0.352i)T \) |
| 73 | \( 1 + (-0.967 - 0.254i)T \) |
| 79 | \( 1 + (-0.640 - 0.767i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.843 + 0.536i)T \) |
| 97 | \( 1 + (0.751 - 0.660i)T \) |
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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
−24.972876371117755717430756592556, −23.58975511423173572786980494637, −22.55436022825425944322213036451, −22.2086790233226449740124539447, −21.58606615113731509246593479220, −20.5367831427910833576457521987, −19.64713206638333693682344112416, −18.58321820929251590583727290067, −17.7885856740825476918733732181, −17.372606417548883441903904473566, −15.88057319266064471924583145912, −14.96408089162319609503886166213, −14.25688920096084345149260331782, −12.77147427190675444382392161166, −11.88489504304583519840165551311, −11.187547348887151528492593707635, −10.53202456368221718011942743766, −9.50375111433685981412543373665, −8.72296048319562260532367225626, −7.267924638509347906575196758114, −5.90781970347993168077593403446, −5.00660642236503797185430665793, −3.751013603584518718113010755923, −2.95598009217368655783995526971, −1.496283098866069989624912264075,
0.63117052191450136359086021531, 1.5918473756276580130797095702, 4.19637617002615704371331251350, 4.75965205658624290430368408971, 5.917405931291321854502107667935, 6.91311771866988448542472099200, 7.56896889244749101660201692525, 8.7306840357834326640986983740, 9.50477041381310492404612852723, 11.00026112206290257018062041266, 11.83436971712467512041459112425, 13.04608740649707883035938848453, 13.69329112369216358163356308567, 14.503782830742892108562694329546, 16.11282873748779090078465755680, 16.48479145442451508599459927590, 17.36791788146161749223298942136, 17.841110328410237852032744977793, 19.132430835091242889832168687640, 19.778659729444042536441021165782, 21.025885316361285238261571956108, 22.306775873374817811169131168033, 22.95932367200079268288232231752, 23.95731587999966646199305271285, 24.41165276498891046881623188088