L(s) = 1 | + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (−0.415 + 0.909i)5-s + (−0.959 − 0.281i)7-s + (−0.841 + 0.540i)8-s + (−0.959 + 0.281i)10-s + (0.654 − 0.755i)11-s + (−0.959 + 0.281i)13-s + (−0.415 − 0.909i)14-s + (−0.959 − 0.281i)16-s + (0.142 + 0.989i)17-s + (−0.142 + 0.989i)19-s + (−0.841 − 0.540i)20-s + 22-s + (−0.654 − 0.755i)25-s + (−0.841 − 0.540i)26-s + ⋯ |
L(s) = 1 | + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (−0.415 + 0.909i)5-s + (−0.959 − 0.281i)7-s + (−0.841 + 0.540i)8-s + (−0.959 + 0.281i)10-s + (0.654 − 0.755i)11-s + (−0.959 + 0.281i)13-s + (−0.415 − 0.909i)14-s + (−0.959 − 0.281i)16-s + (0.142 + 0.989i)17-s + (−0.142 + 0.989i)19-s + (−0.841 − 0.540i)20-s + 22-s + (−0.654 − 0.755i)25-s + (−0.841 − 0.540i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01594480590 + 1.272495347i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01594480590 + 1.272495347i\) |
\(L(1)\) |
\(\approx\) |
\(0.7837627152 + 0.7685873589i\) |
\(L(1)\) |
\(\approx\) |
\(0.7837627152 + 0.7685873589i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 5 | \( 1 + (-0.415 + 0.909i)T \) |
| 7 | \( 1 + (-0.959 - 0.281i)T \) |
| 11 | \( 1 + (0.654 - 0.755i)T \) |
| 13 | \( 1 + (-0.959 + 0.281i)T \) |
| 17 | \( 1 + (0.142 + 0.989i)T \) |
| 19 | \( 1 + (-0.142 + 0.989i)T \) |
| 29 | \( 1 + (0.142 + 0.989i)T \) |
| 31 | \( 1 + (0.841 - 0.540i)T \) |
| 37 | \( 1 + (0.415 + 0.909i)T \) |
| 41 | \( 1 + (-0.415 + 0.909i)T \) |
| 43 | \( 1 + (0.841 + 0.540i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (0.959 + 0.281i)T \) |
| 59 | \( 1 + (0.959 - 0.281i)T \) |
| 61 | \( 1 + (0.841 - 0.540i)T \) |
| 67 | \( 1 + (-0.654 - 0.755i)T \) |
| 71 | \( 1 + (0.654 + 0.755i)T \) |
| 73 | \( 1 + (-0.142 + 0.989i)T \) |
| 79 | \( 1 + (-0.959 + 0.281i)T \) |
| 83 | \( 1 + (-0.415 - 0.909i)T \) |
| 89 | \( 1 + (-0.841 - 0.540i)T \) |
| 97 | \( 1 + (0.415 - 0.909i)T \) |
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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
−31.14676286446958129704826692816, −29.95308631248031828432446791918, −28.85538862428073564103723275823, −28.108565978211339278654113702603, −27.04199218396655859066801616622, −25.20961794025842199812981198985, −24.33419472847554624393332950641, −23.01430154772086575279032811494, −22.28990000779017034752595968588, −20.9550254965658679386908291211, −19.827840674393859137580737095907, −19.308229163063601392944789402806, −17.569081949740993339935814076461, −16.045479103773865935839169112753, −15.0273612249198926735349714103, −13.5028138347521609123744817626, −12.44433488236117129511384663823, −11.75619872633928672757989594822, −9.92529773152285568214862800851, −9.08103237246441801027863302932, −6.97395205662791469023523599964, −5.306980499881969333617715184469, −4.154275150075052354092237916, −2.54014861416951217054938350095, −0.51307809705519744920004310773,
3.026984064622387764783315125939, 4.07462061953857083987613051575, 6.060143741768380347643686675204, 6.906072922648469822925374419008, 8.2291075202367752310088511715, 9.95843621077639631021960030837, 11.5794178488577670335041284259, 12.76391715899163987847729538844, 14.13253291384912185478999402853, 14.9374796744864671631532386940, 16.24239408969097248060333140893, 17.118828335972164244771336535112, 18.74075061197156891540275766270, 19.7439137645674778133185956490, 21.575388933814025159687439388786, 22.34995446552456728188311829149, 23.23900448479697628762224924040, 24.34674070358077011849598390880, 25.60513181663884056564485449024, 26.47445006493770391099542146705, 27.31724711616952529911063053136, 29.337733195682958425522369467391, 30.04339631030474722019608148535, 31.26469442806450195980079649119, 32.16344838284954587665783056038