L(s) = 1 | + (0.587 − 0.809i)3-s − i·7-s + (−0.309 − 0.951i)9-s + (−0.309 + 0.951i)11-s + (0.951 − 0.309i)13-s + (−0.587 − 0.809i)17-s + (−0.809 + 0.587i)19-s + (−0.809 − 0.587i)21-s + (0.951 + 0.309i)23-s + (−0.951 − 0.309i)27-s + (0.809 + 0.587i)29-s + (0.809 − 0.587i)31-s + (0.587 + 0.809i)33-s + (−0.951 + 0.309i)37-s + (0.309 − 0.951i)39-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)3-s − i·7-s + (−0.309 − 0.951i)9-s + (−0.309 + 0.951i)11-s + (0.951 − 0.309i)13-s + (−0.587 − 0.809i)17-s + (−0.809 + 0.587i)19-s + (−0.809 − 0.587i)21-s + (0.951 + 0.309i)23-s + (−0.951 − 0.309i)27-s + (0.809 + 0.587i)29-s + (0.809 − 0.587i)31-s + (0.587 + 0.809i)33-s + (−0.951 + 0.309i)37-s + (0.309 − 0.951i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9945252018 - 0.6311448850i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9945252018 - 0.6311448850i\) |
\(L(1)\) |
\(\approx\) |
\(1.116965167 - 0.4113014413i\) |
\(L(1)\) |
\(\approx\) |
\(1.116965167 - 0.4113014413i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.587 - 0.809i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-0.309 + 0.951i)T \) |
| 13 | \( 1 + (0.951 - 0.309i)T \) |
| 17 | \( 1 + (-0.587 - 0.809i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
| 23 | \( 1 + (0.951 + 0.309i)T \) |
| 29 | \( 1 + (0.809 + 0.587i)T \) |
| 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 | \( 1 + (-0.587 + 0.809i)T \) |
| 53 | \( 1 + (-0.587 + 0.809i)T \) |
| 59 | \( 1 + (0.309 + 0.951i)T \) |
| 61 | \( 1 + (0.309 - 0.951i)T \) |
| 67 | \( 1 + (0.587 + 0.809i)T \) |
| 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 | \( 1 + (-0.309 + 0.951i)T \) |
| 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
−30.442396680572865243323744729439, −28.79254188393758299026699538777, −28.084793244518938355504643594897, −26.96033710137051902574459784523, −26.044038671150277299527457970939, −25.149626675453035944987505204665, −24.03613814297844306132975612615, −22.61653148050436344654674739782, −21.42454082407070938031486570647, −21.10067834332063788600381611349, −19.496644029062362117430060513509, −18.80134619550598543721281790129, −17.274736384800885513090390438552, −15.91000843901192083815359423309, −15.35092085432207074743579797057, −14.08005584533068657696870549940, −12.990296391428464606209369242651, −11.36117924760869609443738317533, −10.42592513389010368002039980541, −8.772269643842701744198712947479, −8.52288534515483407010158719891, −6.372505071926371222241753510, −5.05556461014858999001318651088, −3.61010660602096720920508715451, −2.323827694017852473815328687709,
1.34867547335343612410304478024, 2.98410970323721462349261115813, 4.475976160153197778372340822069, 6.425810703736440169816117976168, 7.39633804327965915488430324046, 8.489993877272501323009328338120, 9.888639003003479412881757581506, 11.21939014038266345950741975891, 12.70464140403861060176890012050, 13.46355711731038195812812127609, 14.52621042918832994064214819404, 15.74162662378227791084954850144, 17.26690547100191439525245243594, 18.10937713733076699907133975502, 19.27108363709598539116334817331, 20.32387887200159186321133784531, 20.945811001347378095684041394862, 22.97324105110013362210526954680, 23.353264299525707380585593195249, 24.69543251457298384007298432488, 25.59938670430543376575185156074, 26.43882222609579719114201471778, 27.65294207460979484784066573674, 29.02018384578455923276437368670, 29.832900266768813539034257711680