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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.327682415 - 0.5745394857i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.327682415 - 0.5745394857i\) |
\(L(1)\) |
\(\approx\) |
\(1.248155800 - 0.3242805703i\) |
\(L(1)\) |
\(\approx\) |
\(1.248155800 - 0.3242805703i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
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
−27.047948562069520105780792112192, −25.98935145268416485410506456059, −25.50599152674223927975625556485, −24.24206676758245369065108651699, −23.00597953611929736004173889543, −22.43001135013575351664336986364, −20.99870020097720795780757882345, −20.507560386555340266104624648855, −19.70060012943226893970477431605, −18.46320545862450584504111845155, −17.25644083241733637027345243336, −16.2674754650757809850045995568, −15.54176065990060839645011271404, −14.10191372905668073657746259611, −13.89811808352816342003662764153, −12.29891901633936347478656032970, −11.02228197847996483105985777604, −10.06250474482479072987748094494, −9.28776633679493697608595144277, −7.97993755478958190832438737693, −7.030874433402290154292534219338, −5.380111751809857909163306449695, −4.16433421536977530274035891398, −3.38210537357619214229291301878, −1.663824548384778397172246840463,
1.27602448912838674014834969047, 2.67114654466296244269418856288, 3.70851430659581806328897034640, 5.69830701228057917901575452330, 6.37057928080190450799778847587, 7.946394048040783555167129590964, 8.56280466287258145093048656492, 9.63099356663590350260381135114, 11.25853317522945616244219098300, 12.11483600912327968825894950085, 13.17630267441952030404750195040, 14.01050325542393040532471425197, 15.07166924571179995885796108190, 16.01105696324983190066011689040, 17.37971631987862278176805602138, 18.46256308766382323669849622499, 18.94496931652766911193762916932, 20.005278689639437348649542811240, 21.03408527947478219870398569794, 21.98388420966712284711947533046, 23.12851803886739554597121113980, 24.28485596696979118398664186336, 24.718876069243617777593256370191, 25.82784600403345562231676352776, 26.45738831113117726735993463129