L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + 5-s − 8-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + 23-s + 25-s + (−0.5 − 0.866i)26-s + (0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s + (0.5 + 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + 5-s − 8-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + 23-s + 25-s + (−0.5 − 0.866i)26-s + (0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s + (0.5 + 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.494771297 - 2.957147692i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.494771297 - 2.957147692i\) |
\(L(1)\) |
\(\approx\) |
\(1.331515804 - 0.9805618352i\) |
\(L(1)\) |
\(\approx\) |
\(1.331515804 - 0.9805618352i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−22.832269780453827655351402281, −21.9131744043714955172714174045, −21.339765291259354092787927503867, −20.74895409517223487836122143799, −19.37425375739560020552811466343, −18.42032181232578662874761346343, −17.63868833902703803590362259013, −16.96262418618433125372397346672, −16.25042605912389790024572204596, −15.263396254626674914740777184218, −14.461538727327951569648738993256, −13.67984367763454705867852840987, −13.13297181915354785851188621995, −12.16263112825669888874211703635, −11.115931454645714003172219811649, −9.93133813464756331804782587341, −9.05400321769381458181441533617, −8.35604738917888358099407389606, −7.061655173127840237499053555804, −6.465025963425608000623396682030, −5.53651526724043814737768274650, −4.75826013709516611264111459831, −3.616665459647481502674658777553, −2.54483960091909108840216640769, −1.17941194204718264695414217238,
0.71146846000462944745295697003, 1.62691709063300759730490016314, 2.7780014157460956226630157903, 3.504917242771221685244941827362, 4.90511935052472290788149960672, 5.55113415131491800227195686446, 6.36735907507657771735512960429, 7.708239177164717339792502409431, 9.04519395650293896239714971901, 9.61196699213698033196952184988, 10.561381630704177321016623521813, 11.15226368364941416655440253326, 12.435005767024436743339194230008, 12.87605821556058857551280122649, 13.94414605501065982499566654225, 14.321490478976943950114383494035, 15.42099954458853499959656720470, 16.44586237683268794903576330097, 17.5727788006614431349538328943, 18.23510150669294907434840799790, 18.896580238615411130061328326627, 19.99568865231041473095412377569, 20.84706916669036287472776002696, 21.088054107972184374724397538470, 22.28187942726248753804446335327