L(s) = 1 | + i·7-s − i·11-s + 13-s + i·17-s + i·19-s + i·23-s − i·29-s + 31-s + 37-s + 41-s − 43-s + i·47-s − 49-s + 53-s + i·59-s + ⋯ |
L(s) = 1 | + i·7-s − i·11-s + 13-s + i·17-s + i·19-s + i·23-s − i·29-s + 31-s + 37-s + 41-s − 43-s + i·47-s − 49-s + 53-s + i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.165301054 + 0.3761857562i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.165301054 + 0.3761857562i\) |
\(L(1)\) |
\(\approx\) |
\(1.086321679 + 0.1565124024i\) |
\(L(1)\) |
\(\approx\) |
\(1.086321679 + 0.1565124024i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
| 31 | \( 1 \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 + iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 \) |
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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
−26.116990129171269696674054284484, −25.284798755151197244201041203662, −24.1925833329323869664526877964, −23.18393632008777854287205736141, −22.701055806746891399357852491830, −21.36671649485440195641028379322, −20.36025248887901328061742285571, −19.902179355592794346962388199556, −18.46874773542171334604233715100, −17.75082144197663660264814580141, −16.68386166430663181383142419552, −15.820708112109810431818334890982, −14.69730784605379660708218876964, −13.68482146335122702141021886387, −12.899631153879954212038234924469, −11.62602451410469203866110896726, −10.665051338305563479219745251070, −9.71700436738572592892992104016, −8.55242984379699403586801252901, −7.29999129135756840778744168824, −6.56807993653776378044060312572, −4.96850547342363034422630019954, −4.06293875711406236777354909626, −2.65563838318113078372092398857, −1.03022770119630283725345199408,
1.47754921937299700719787944741, 2.9459470046296532838002711612, 4.07633315396502243284125313163, 5.7506082120748285042007523227, 6.15716002377605250846110977318, 7.96451688598810200717962301684, 8.62947243043469988804899110435, 9.78627046137616971993549843509, 11.02484286307279456674151060180, 11.84718801549491243705507678741, 12.97062855840641409999074205099, 13.90182808491216382292626928898, 15.066346111013513985098982145142, 15.865803803193787555818179400022, 16.83498273831770704114383783394, 18.01884641814183110831082291452, 18.84817041287311207474443933991, 19.57085941084321155950921032861, 21.079858867356359854388926732296, 21.44438911889570355351838575040, 22.58986240660801866234101722540, 23.54215828469549828087022308010, 24.526581113623232116180490936004, 25.313637484579677270347951111276, 26.220382064257015913646353528744