L(s) = 1 | + (−0.951 + 0.309i)3-s − i·7-s + (0.809 − 0.587i)9-s + (0.809 + 0.587i)11-s + (0.587 + 0.809i)13-s + (0.951 + 0.309i)17-s + (0.309 − 0.951i)19-s + (0.309 + 0.951i)21-s + (0.587 − 0.809i)23-s + (−0.587 + 0.809i)27-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (−0.951 − 0.309i)33-s + (−0.587 − 0.809i)37-s + (−0.809 − 0.587i)39-s + ⋯ |
L(s) = 1 | + (−0.951 + 0.309i)3-s − i·7-s + (0.809 − 0.587i)9-s + (0.809 + 0.587i)11-s + (0.587 + 0.809i)13-s + (0.951 + 0.309i)17-s + (0.309 − 0.951i)19-s + (0.309 + 0.951i)21-s + (0.587 − 0.809i)23-s + (−0.587 + 0.809i)27-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (−0.951 − 0.309i)33-s + (−0.587 − 0.809i)37-s + (−0.809 − 0.587i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8189813763 + 0.02573752661i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8189813763 + 0.02573752661i\) |
\(L(1)\) |
\(\approx\) |
\(0.8632529409 + 0.002418697518i\) |
\(L(1)\) |
\(\approx\) |
\(0.8632529409 + 0.002418697518i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.951 + 0.309i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (0.809 + 0.587i)T \) |
| 13 | \( 1 + (0.587 + 0.809i)T \) |
| 17 | \( 1 + (0.951 + 0.309i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + (0.587 - 0.809i)T \) |
| 29 | \( 1 + (-0.309 - 0.951i)T \) |
| 31 | \( 1 + (-0.309 + 0.951i)T \) |
| 37 | \( 1 + (-0.587 - 0.809i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.951 - 0.309i)T \) |
| 53 | \( 1 + (0.951 - 0.309i)T \) |
| 59 | \( 1 + (-0.809 + 0.587i)T \) |
| 61 | \( 1 + (-0.809 - 0.587i)T \) |
| 67 | \( 1 + (-0.951 - 0.309i)T \) |
| 71 | \( 1 + (-0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.587 + 0.809i)T \) |
| 79 | \( 1 + (0.309 + 0.951i)T \) |
| 83 | \( 1 + (0.951 + 0.309i)T \) |
| 89 | \( 1 + (0.809 + 0.587i)T \) |
| 97 | \( 1 + (-0.951 + 0.309i)T \) |
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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
−29.75134497896497924188733929759, −29.03807759897790761052726191491, −27.65833201598717615167126248123, −27.4846744619477263227896054420, −25.49140502196141903299958682400, −24.79873171819603764199974176147, −23.671669920623123485607582635679, −22.58099123726501916654939377274, −21.87194723635468622821665523687, −20.655985623178627664768748627004, −18.99399470145447725385614703782, −18.43068605032943237362972211063, −17.1867108405310791788383110636, −16.231391100036494771766250270246, −15.09471448734655827241744259926, −13.61490649015771129123187723241, −12.331081066506890717813585915630, −11.61840136870746610817257269291, −10.37075448725982881120174553089, −8.93513808453674627269607924228, −7.52648614870437713718767241863, −6.02199089557285058334749582405, −5.3600153803888763914398888020, −3.42761153632103008991521704159, −1.40591797495635540487682333433,
1.252207630791939854476865396509, 3.77373672129255718228815685134, 4.788258772374089186236626011, 6.36649099390229779396272494640, 7.271963046223547151164731628437, 9.17136930469354971096349127851, 10.32154081592810066780959865316, 11.31514049934427781729455966756, 12.38967638140588294963865818397, 13.73700353936956698148899087927, 14.99799374441442619338468919017, 16.38939919655287679779555654895, 17.01686215314292491795645451873, 18.06175006417411582800086336023, 19.3944831300154978376526725295, 20.646948695771474564392637578540, 21.62007648978252239587132095664, 22.85426147720275760530074366890, 23.41869644842605579225625124927, 24.55506554930713135169632655925, 26.03813029430556337126227272641, 26.92201253402264885845502277454, 27.996930739966873129018179952113, 28.75431744763901995096685175974, 29.96850085821985079698546614731