L(s) = 1 | + 2-s + 4-s + (−0.5 + 0.866i)5-s + 8-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + 16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 − 0.866i)26-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + (−0.5 + 0.866i)5-s + 8-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + 16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 − 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.436415598 + 0.1716779885i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.436415598 + 0.1716779885i\) |
\(L(1)\) |
\(\approx\) |
\(1.544080465 + 0.1212792210i\) |
\(L(1)\) |
\(\approx\) |
\(1.544080465 + 0.1212792210i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + T \) |
| 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
−31.97528462018729448844350280799, −31.48669907677979793529516492932, −30.38838507045460863059745361144, −28.99937466633912832789599636201, −28.28428480725796869335824495436, −26.72131737528539753035523890827, −25.27189730300615054107016496546, −24.317844932477225142751654136640, −23.39647963853528082536616738209, −22.37708662019528224026103520488, −20.92130282078276344863347994774, −20.306110030425296125436977741168, −18.987152355584578371776042771983, −17.06571045062561068492299915515, −16.03532399263962021408568402645, −14.988363826995783892640581326566, −13.62693597437755972136620955645, −12.44344646685592426687320555254, −11.64719657936340570885271950233, −9.94039552823299905250001110737, −8.1231298607054143440724777139, −6.77027576449304127125513693512, −5.04673640052099617779085922190, −4.14022906279380746878277374178, −2.1529959313093544231969574101,
2.594916183043727279383412342332, 3.806710223879517089444640946548, 5.4925545335228443821026322492, 6.82895774292570260806191718216, 8.08882544457826254956186566496, 10.45170623480493104006616456509, 11.27680424580001237602032757318, 12.67201770052863198631015700803, 13.85946696344231432630432418067, 15.08716606832238207327774198588, 15.82121162187913886014945947311, 17.46716861508634850381543071392, 19.09723212100276036939750918257, 19.988366946696056638161342711651, 21.53636394778561573028411369018, 22.24780988518929291858495626411, 23.441559335291343922269138706, 24.25648926244939257041347423235, 25.68221593104835352201364562805, 26.667110846739718933720825999003, 28.13207389467879360277721741227, 29.5991720505558032551659137816, 30.198307482762071458900535754864, 31.38354802722667209021170228863, 32.19656956688511423171633757470