L(s) = 1 | + (−0.809 − 0.587i)3-s + 7-s + (0.309 + 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.309 − 0.951i)13-s + (0.809 − 0.587i)17-s + (0.809 − 0.587i)19-s + (−0.809 − 0.587i)21-s + (0.309 − 0.951i)23-s + (0.309 − 0.951i)27-s + (−0.809 − 0.587i)29-s + (0.809 − 0.587i)31-s + (0.809 − 0.587i)33-s + (−0.309 − 0.951i)37-s + (−0.309 + 0.951i)39-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)3-s + 7-s + (0.309 + 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.309 − 0.951i)13-s + (0.809 − 0.587i)17-s + (0.809 − 0.587i)19-s + (−0.809 − 0.587i)21-s + (0.309 − 0.951i)23-s + (0.309 − 0.951i)27-s + (−0.809 − 0.587i)29-s + (0.809 − 0.587i)31-s + (0.809 − 0.587i)33-s + (−0.309 − 0.951i)37-s + (−0.309 + 0.951i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.152259700 - 0.7312462417i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.152259700 - 0.7312462417i\) |
\(L(1)\) |
\(\approx\) |
\(0.9358598968 - 0.2540250827i\) |
\(L(1)\) |
\(\approx\) |
\(0.9358598968 - 0.2540250827i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (-0.309 + 0.951i)T \) |
| 13 | \( 1 + (-0.309 - 0.951i)T \) |
| 17 | \( 1 + (0.809 - 0.587i)T \) |
| 19 | \( 1 + (0.809 - 0.587i)T \) |
| 23 | \( 1 + (0.309 - 0.951i)T \) |
| 29 | \( 1 + (-0.809 - 0.587i)T \) |
| 31 | \( 1 + (0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.809 - 0.587i)T \) |
| 53 | \( 1 + (0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.309 - 0.951i)T \) |
| 61 | \( 1 + (0.309 - 0.951i)T \) |
| 67 | \( 1 + (-0.809 + 0.587i)T \) |
| 71 | \( 1 + (0.809 + 0.587i)T \) |
| 73 | \( 1 + (-0.309 + 0.951i)T \) |
| 79 | \( 1 + (0.809 + 0.587i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 + (0.309 - 0.951i)T \) |
| 97 | \( 1 + (0.809 + 0.587i)T \) |
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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
−29.62348675175455948657143170016, −28.862864317648190569354303924637, −27.67254527094758985097710054763, −27.03945801957379322856626174956, −25.99040166984662601954589267086, −24.288622393715059497473679419531, −23.77168627725284455223865792646, −22.500746527933434973022986903622, −21.34887291688026555442951841509, −20.93087518459753714318474474348, −19.17987175836405276145918954940, −18.07547485728295299584238691502, −17.00851786843568440386897379672, −16.17547052947023803600766315794, −14.90968176023100092777196599135, −13.850566925811140406548961270076, −12.127258559032661927367061469768, −11.327677058938488326085917184956, −10.28633586044067894083763015481, −8.949389890869736027196310587588, −7.51536704981603626340804836678, −5.903977872380190143192313822978, −4.94659941145312019039306062752, −3.555418614831584361195134171829, −1.298207143180879839028718267543,
0.792213449768409302693341609757, 2.38876140162935409427990490726, 4.690024050922154656117754448364, 5.568335664713866196274496847317, 7.2124093265236748327092897608, 7.975732734127437543419132903096, 9.867910571362079489000546002057, 11.05339551512235533069173483399, 12.05884164505776065298772794622, 13.05380073458289411470742942131, 14.40144086701674480739211980527, 15.61894319877056381564962476881, 17.01505058877383710544906544375, 17.8323258625520663390390987578, 18.592914512003524128651541180257, 20.08890603279608935027598439530, 21.09606106817307240756140497399, 22.51269811540296234566486856716, 23.14420343092907230534774892912, 24.41251454024979104337849073111, 24.97156220988477854066874124064, 26.51888450800937232553276592063, 27.79284901168348543510186134930, 28.29389592309771258936881911290, 29.648336246280855743902313948205