L(s) = 1 | + (0.809 + 0.587i)3-s − i·7-s + (0.309 + 0.951i)9-s + (−0.951 − 0.309i)11-s + (0.309 + 0.951i)13-s + (0.587 + 0.809i)17-s + (0.587 + 0.809i)19-s + (0.587 − 0.809i)21-s + (0.951 + 0.309i)23-s + (−0.309 + 0.951i)27-s + (0.587 − 0.809i)29-s + (0.809 − 0.587i)31-s + (−0.587 − 0.809i)33-s + (0.309 + 0.951i)37-s + (−0.309 + 0.951i)39-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)3-s − i·7-s + (0.309 + 0.951i)9-s + (−0.951 − 0.309i)11-s + (0.309 + 0.951i)13-s + (0.587 + 0.809i)17-s + (0.587 + 0.809i)19-s + (0.587 − 0.809i)21-s + (0.951 + 0.309i)23-s + (−0.309 + 0.951i)27-s + (0.587 − 0.809i)29-s + (0.809 − 0.587i)31-s + (−0.587 − 0.809i)33-s + (0.309 + 0.951i)37-s + (−0.309 + 0.951i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.635194932 + 0.5742395879i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.635194932 + 0.5742395879i\) |
\(L(1)\) |
\(\approx\) |
\(1.355397784 + 0.2640152612i\) |
\(L(1)\) |
\(\approx\) |
\(1.355397784 + 0.2640152612i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.809 + 0.587i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-0.951 - 0.309i)T \) |
| 13 | \( 1 + (0.309 + 0.951i)T \) |
| 17 | \( 1 + (0.587 + 0.809i)T \) |
| 19 | \( 1 + (0.587 + 0.809i)T \) |
| 23 | \( 1 + (0.951 + 0.309i)T \) |
| 29 | \( 1 + (0.587 - 0.809i)T \) |
| 31 | \( 1 + (0.809 - 0.587i)T \) |
| 37 | \( 1 + (0.309 + 0.951i)T \) |
| 41 | \( 1 + (-0.309 - 0.951i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.587 - 0.809i)T \) |
| 53 | \( 1 + (0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.951 + 0.309i)T \) |
| 61 | \( 1 + (-0.951 - 0.309i)T \) |
| 67 | \( 1 + (-0.809 + 0.587i)T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.951 - 0.309i)T \) |
| 79 | \( 1 + (-0.809 - 0.587i)T \) |
| 83 | \( 1 + (0.809 - 0.587i)T \) |
| 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
−24.54637530206502110495869059391, −23.45597879370588163958002176146, −22.70830817917925611876710941916, −21.454757874418473187031437304086, −20.78317239120092963203074487492, −19.94072944120668724027463895057, −18.999192940388084219235927622059, −18.17590617013334112775768668610, −17.75899591871135979262531481384, −16.04450723347830803745966330637, −15.42416086310920424121721510957, −14.58940507638872771078068553172, −13.547329052873907760851423120567, −12.744258060956240179261796052592, −12.057483848590720503622906034129, −10.77879361917582143623316983845, −9.592080952462336079543578268576, −8.77471424674138586793016076076, −7.882400321125824277839269323633, −7.04225705512778242732026245535, −5.76875728237226410133268468261, −4.812651460234101181123047387594, −2.95554199635257844673464572283, −2.733809592859278877465987826218, −1.10089616190565854819133761504,
1.41139430107975892647173847220, 2.82674858175727573275162463949, 3.814214076984006879840374304755, 4.645814802486849200536089654789, 5.93616403326962223527416141006, 7.36205127852082212477246875607, 8.03586756143037178322842884782, 9.07127625096497931122391358603, 10.17098500473771160909346548523, 10.63820625967885888817132625247, 11.90075800261550112323969268354, 13.36214376395878045057729817161, 13.72685620175295976873348159769, 14.7250416718062135917689787097, 15.661098644036351798585968245486, 16.50711237098117579005805241780, 17.22043980824617392130445769951, 18.73703998023324666070635821870, 19.18972368164098586792634758801, 20.34000971381034561401146559727, 20.96606330446608823554473851932, 21.551525778273594242383683862181, 22.825516187516026789404073205925, 23.59217953425487196589914652836, 24.48178462181982390860273797104