L(s) = 1 | + (−0.587 − 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)6-s + (0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + i·12-s + (0.587 + 0.809i)13-s + (−0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (0.587 − 0.809i)17-s + (−0.951 − 0.309i)18-s + (−0.309 − 0.951i)19-s + 21-s + ⋯ |
L(s) = 1 | + (−0.587 − 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)6-s + (0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + i·12-s + (0.587 + 0.809i)13-s + (−0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (0.587 − 0.809i)17-s + (−0.951 − 0.309i)18-s + (−0.309 − 0.951i)19-s + 21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.492593171 - 0.8775273885i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.492593171 - 0.8775273885i\) |
\(L(1)\) |
\(\approx\) |
\(1.134410250 - 0.4760026104i\) |
\(L(1)\) |
\(\approx\) |
\(1.134410250 - 0.4760026104i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.587 - 0.809i)T \) |
| 3 | \( 1 + (0.951 - 0.309i)T \) |
| 7 | \( 1 + (0.951 + 0.309i)T \) |
| 13 | \( 1 + (0.587 + 0.809i)T \) |
| 17 | \( 1 + (0.587 - 0.809i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + (-0.309 + 0.951i)T \) |
| 31 | \( 1 + (-0.809 + 0.587i)T \) |
| 37 | \( 1 + (0.951 + 0.309i)T \) |
| 41 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.951 + 0.309i)T \) |
| 53 | \( 1 + (0.587 + 0.809i)T \) |
| 59 | \( 1 + (-0.309 + 0.951i)T \) |
| 61 | \( 1 + (-0.809 - 0.587i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.951 - 0.309i)T \) |
| 79 | \( 1 + (0.809 - 0.587i)T \) |
| 83 | \( 1 + (-0.587 + 0.809i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.587 - 0.809i)T \) |
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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
−33.102647145561186220004923671876, −32.17939025580797655974000544206, −31.01410630007374458687430639255, −29.79700067056472024929188400146, −27.882745045618870381991344122353, −27.33459567641497503809013521224, −26.14904011330336230870423152494, −25.266962258399316822122604148882, −24.26418193226319800363495286162, −23.08058983790223978170185926005, −21.2851923221573890229629011765, −20.20423138395625298963682816932, −19.02574812729203055218465338960, −17.85188192476284325873394535468, −16.563774671691519227382394855623, −15.21978777055287923460195806350, −14.48727094924965266602020644053, −13.263286813045280611520812577951, −10.88266487502327322643253360844, −9.74243815770659510254693360076, −8.28417889980620217857127900880, −7.66162290265441259641387734479, −5.65811874086761071458690937887, −3.99873869803303770716218160583, −1.59920888992399799804775150087,
1.419626679528417059099578114665, 2.79582447961465225648512508179, 4.447523573447786564205406489409, 7.16672996654210566574848130846, 8.44725429010390591402632294352, 9.29054917863484728887706647753, 10.94001723346847809653248257638, 12.18477963449148910679315618431, 13.51951634149810576751047022725, 14.66329475080473583535113728002, 16.3756901364290351407299021380, 18.04409639426043710236950385413, 18.67787599193744499373687612103, 19.984671452173336621493725002773, 20.88897698991125651230020479348, 21.77379946917152715632509496160, 23.671756902306521552329232717196, 24.991510557120913753123156737324, 26.02445554084137276827871358272, 27.04842864704104213174069278689, 28.08385535402607753066178440833, 29.40809437569849366886665774400, 30.576122163968513196531127377111, 31.070255736441503795803184987521, 32.29579714592882237453125762347